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| style="text-align: center;" | <math>\begin{array}{lcl}\left\vert \displaystyle \int _{\Omega }wv\right\vert \leq  \Vert w\Vert _{L^2(\Omega )} \Vert v\Vert _{L^2(\Omega )} \leq C_P^2\Vert w\Vert _{*} \Vert v\Vert _{*} \leq  \alpha ^2C_P^2\Vert w\Vert _{H^1(\Omega )} \Vert v\Vert _{H^1(\Omega )}  \leq \beta ^2\alpha ^2C_P^2\Vert w\Vert _{V}\Vert v\Vert _{V}, </math>
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| style="text-align: center;" | <math>\begin{array}{lcl}\left\vert \displaystyle \int _{\Omega }wv\right\vert \leq  \Vert w\Vert _{L^2(\Omega )} \Vert v\Vert _{L^2(\Omega )} \leq C_P^2\Vert w\Vert _{*} \Vert v\Vert _{*} \leq  \alpha ^2C_P^2\Vert w\Vert _{H^1(\Omega )} \Vert v\Vert _{H^1(\Omega )}  \leq \beta ^2\alpha ^2C_P^2\Vert w\Vert _{V}\Vert v\Vert _{V}, \end{array} </math>
 
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Revision as of 01:34, 13 December 2020

Formulación variacional de un modelo continuo en EDP's de la angiogénesis inducida por tumor

Ana Kristhel Esteban LópezFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): ^{1} , Justino Alavez-RamírezFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): ^{1,*}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): ^{1} División Académica de Ciencias Básicas, UJAT

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): ^{*} Autor por correspondencia: justino.alavez@ujat.mx

Resumen

En este trabajo retomamos el modelo matemático de Anderson y Chaplain (1998) formado por tres ecuaciones diferenciales, que describe la respuesta migratoria inicial de las células endoteliales al factor angiogénico tumoral (TAF) y a la fibronectina. Aplicamos el método variacional a los problemas de contorno mixto homogéneo y no homogéneo, que surgen del modelo antes citado, para demostrar la existencia, unicidad y dependencia continua respecto de los datos de la solución débil de dichos problemas.

Palabras clave: Células endoteliales, TAF, fibronectina, problema mixto homogéneo, formulación variacional.

2 Introducción

El cáncer no es una enfermedad nueva. Papiros egipcios que datan de aproximadamente el año Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 1600} 

a. C. ya la describían [1]. Sin embargo, fue hasta la invención del microcospio en el siglo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle XIX}
que comenzó el estudio patológico moderno de cáncer [2]. El cáncer se ha vuelto un problema global de salud pública. Según estimaciones de la Organización Mundial de la Salud (OMS) en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 2015}

, el cáncer es la primera o la segunda causa de muerte antes de los Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 70} 

años en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 91}
de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 172}
países, y ocupa el tercer o cuarto lugar en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 22}
países adicionales [3]. Algunos factores de riesgo de cáncer se pueden vincular estrechamente con la herencia, los productos químicos, las radiaciones ionizantes, las infecciones o virus y los traumas. Los investigadores estudian cómo estos diferentes factores pueden interactuar de una manera multifactorial y secuencial para producir tumores malignos [2]. Los tumores pueden ser benignos o malignos. La células de los tumores malignos presentan dos características que la distinguen de las normales: se reproducen de manera descontrolada, y son capaces de invadir y colonizar tejidos y órganos distantes, en lugares donde normalmente no pueden crecer [4]. La combinación desafortunada de estas características es la que hace tan peligrosa y mortal a la mayoría de las formas del cáncer. Afortunadamente, existen muchos modelos matemáticos que permiten describir, bajo ciertas condiciones, la evolución de las células cancerígenas y el efecto que sobre ellas produce una terapia elegida con la intención de eliminar o, al menos, contener el crecimiento de un tumor [5].

La hipótesis de que el crecimiento tumoral depende de la angiogénesis fue introducida por primera vez en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 1971} 

[6]. El término angiogénesis significa literalmente formación de nuevos vasos sanguíneos a partir de una vasculatura existente. En donde las células endoteliales (CEs) migran y proliferan, organizándose hasta formar estructuras tubulares que eventualmente se unirán, para finalmente madurar en vasos sanguíneos estables [7]. Anderson y Chaplain (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 1998}

) [8], presentaron un modelo matemático continuo que describe la formación de la red de brotes capilares en respuesta a estímulos químicos (factores angiogénicos tumorales, TAF) suministrados por un tumor sólido. El modelo también tiene en cuenta las interacciones esenciales entre las células endoteliales y la matriz extracelular mediante la inclusión de la macromolécula de la matriz fibronectina.

Todos los tejidos y órganos contienen una mezcla de células y componentes no celulares, que forman redes bien organizadas llamadas matrices extracelulares (MEC). Las MEC proporcionan no sólo estructuras físicas en las que se incrustan las células, sino que también regulan muchos procesos celulares, como el crecimiento, la migración, la diferenciación, la supervivencia, la homeostasis y la morfogénesis [9]. La MEC está conformada por una gran variedad de moléculas, las cuales interaccionan entre sí, generando una estructura tridimensional a la cual las células se adhieren ya sea por receptores específicos o ligandos. Las macromoléculas que constituyen la MEC incluyen, entre otras, a la familia de las colágenas, que son las responsables de la resistencia mecánica de los tejidos conjuntivos, la elastina que le confiere cualidades de flexibilidad y elasticidad, proteínas de adhesión como fibronectinas y lamininas y los proteoglicanos que son esenciales para la adhesividad [4]. Las fibronectinas constituyen una familia de glucoproteínas multifuncionales, que se pueden encontrar tanto en forma insoluble, formando parte de la MEC, como en forma soluble circulando en el plasma [4]. Las células pueden ensamblar fibronectina soluble derivado de fibronectina en plasma en fibras. Alternativamente, las células pueden producir su propia fibronectina que es secretada y formada en fibras [9]. La fibronectina desempeña un papel fisiológico importante no sólo en la formación de la matriz extracelular, sino además en la adhesión y migración celular, en la coagulación de la sangre y en la cicatrización de las heridas, entre otras [10].

En cuanto al contenido de este artículo, en la sección 3, retomamos el modelo continuo dado por Anderson y Chaplain (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 1998} ) [8], el cual consiste en un sistema de ecuaciones diferenciales que describen la respuesta migratoria inicial de las células endoteliales al TAF y la fibronectina. En la sección 4, realizamos la discretización en el tiempo mediante el método de diferencias finitas, que nos permite transformar el problema continuo formado por tres ecuaciones diferenciales parciales en un problema discreto en el que en cada tiempo se debe resolver una sola ecuación diferencial parcial de segundo orden de tipo elíptico, a la cual le aplicamos el método variacional al problema de contorno mixto homogéneo, que surge del modelo, para demostrar la existencia, unicidad y dependencia continua respecto de los datos iniciales de su solución débil.

3 Modelo matemático

Partimos del modelo matemático de Anderson y Chaplain (1998) [8]. Este modelo describe cómo las células endoteliales que emergen de un vaso padre, responden y migran a través de la motilidad aleatoria, de la quimiotaxis a través de gradientes del factor angiogénico tumoral (TAF) liberado por el tumor, y de la haptotaxis a través de gradientes de fibronectina en la matriz extracelular. En este modelo, la densidad de las células endoteliales (en o cerca de una punta de brote capilar) por unidad de área se denota por Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n} , que está influenciada por la motilidad aleatoria, la quimiotaxis y la haptotaxis. La concentración de TAF y la concentración de fibronectina están representadas por Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle c} 

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f}

, respectivamente. La quimiotaxis está en respuesta a los gradientes de TAF y la haptotaxis está en respuesta a los gradientes de fibronectina. El sistema de ecuaciones diferenciales parciales en el modelo está dado por

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{array}{lcl}\dfrac{\partial n}{\partial t} &=& D_{n}\nabla ^{2}n - \nabla \cdot \left(\dfrac{\chi _{0}k_1}{k_1+c}n \nabla c\right)- \nabla \cdot \left(\rho _{0}n \nabla f\right) \\ \dfrac{\partial f}{\partial t} &=& wn - \mu nf \\ \dfrac{\partial c}{\partial t} &=& -\lambda nc \end{array}
(1)

donde, como se dijo antes, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n} 

es la densidad de las células endoteliales, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle c}
es la concentración de TAF y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f}
es la concentración de fibronetina. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle D_{n}}
es el coeficiente de motilidad aleatoria de la célula, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \chi _{0}}
es el coeficiente quimiotáctico, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \rho _{0}}
es el coeficiente haptotáctico, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k_1}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \lambda } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle w} 

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mu }
son constantes positivas.

Considerando una geometría bidimensional en la que las ecuaciones del modelo están definidas en el dominio espacial cuadrado de lado Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle L} 

(un cuadrado de tejido corneal), Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle [0,L]\times [0,L]}

, se reescala la distancia del vaso parental al tumor con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{x}=x/L} 

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{y}=y/L}

, con lo cual el vaso parental permanece en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle x=0} 

y el tumor en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle x=1}

. Adimensionalizando el tiempo con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tau =L^2/D_c} , donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle D_c} 

es el coeficiente de difusión del TAF; la densidad inicial de células endoteliales con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n^{0}}
y las concentraciones iniciales del TAF y fibronectina con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle c^{0}}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f^{0}}

, respectivamente. Haciendo el cambio de variables:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \tilde{c}=\frac{c}{c^{0}}, \quad \tilde{f}=\frac{f}{f^{0}},\quad \tilde{n}=\frac{n}{n^{0}},\quad \tilde{t}=\frac{t}{\tau },
(2)

se obtiene el siguiente sistema adimensionalizado:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{array}{lcl}\dfrac{\partial \tilde{n}}{\partial \tilde{t}} &=& D \nabla ^{2}\tilde{n}-\nabla \cdot \left(\frac{\chi }{1+\alpha \tilde{c}}\tilde{n} \nabla \tilde{c}\right)-\nabla \cdot \left(\rho \tilde{n}\nabla \tilde{f}\right) \\ \dfrac{\partial \tilde{f}}{\partial \tilde{t}} &=& \beta \tilde{n}-\gamma \tilde{n}\tilde{f} \\ \dfrac{\partial \tilde{c}}{\partial \tilde{t}} &=& -\eta \tilde{n}\tilde{c}, \end{array}
(3)

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle D=D_{n}/D_{c}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \chi =\chi _{0}c^{0}/D_{c}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \alpha =c^{0}/k_{1}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \rho =\rho _{0}f^{0}/D_{c}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \beta =\omega L^{2}n^{0}/f^{0}D_{c}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \gamma =\mu L^{2}n^{0}/D_{c}} 

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \eta =\lambda L^{2}n^{0}/D_{c}}

. El modelo adimensionalizado (3) con condiciones iniciales, parámetros y condiciones de contorno apropiadas, es el que usan Anderson y Chaplain [8] para simular la migración (evolución) de las células endoteliales hacia la fuente de la señal tumoral. Es muy importante notar que solamente la primera ecuación del sistema (3), es la que contiene derivadas parciales espaciales, mientras que la segunda y tercera ecuación solo contienen la primera derivada en el tiempo. Así que al discretizar estas dos últimas ecuaciones con respecto al tiempo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t} , darán lugar a ecuaciones algebraicas. Por otro lado, el dominio espacial del modelo adimensionalizado (3) es el conjunto Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Omega = \left\{(x,y)\in \mathbb{R}^{2}:0 < x < 1,\; 0 < y < 1\right\}} 

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Gamma = \partial \Omega = \cup _{m=1}^{4}\Gamma _{m}}

, donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Gamma _{1}=\{ (x,y)\in \Gamma : 0\leq x<1,\; y=0\} } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Gamma _{2}=\{ (x,y)\in \Gamma : x=1,\; 0\leq y<1\} } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Gamma _{3}=\{ (x,y)\in \Gamma : 0<x\leq 1,\; y=1\} } 

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle  \Gamma _{4}=\{ (x,y)\in \Gamma : x=0,\; 0<y\leq 1\} }

.

4 Discretización en el tiempo y formulación variacional del modelo adimensionalizado

Discretizamos las derivadas parciales que aparecen en el lado izquierdo del sistema (3), usando un esquema progresivo para aproximar las derivadas parciales con respecto al tiempo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t} . Para simplificar notaciones, escribiremos Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f} 

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle c}
en lugar de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{n}}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{f}} 

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \tilde{c}}

, respectivamente, teniendo siempre en cuenta las relaciones dadas en (2). Así tenemos por ejemplo, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \frac{n(x,y,t+\Delta t)-n(x,y,t)}{\Delta t} \cong \frac{\partial n(x,y,t)}{\partial t}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta t\neq 0} . Consideramos un intervalo de tiempo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle [0, T]} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle N} 

entero positivo, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Delta t=T/N}
y puntos Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle t_k=k\Delta t}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k=0,1,\ldots , N} , [11,12]; y denotemos por Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n^k=n(x,y,t_k)} , la primera ecuación del sistema (3) se discretiza en el tiempo como

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): n^k-\Delta tD \nabla ^{2}n^k = n^{k-1} - \Delta t\nabla \cdot \left(\frac{\chi }{1+\alpha c^{k-1}}n^{k-1}\nabla c^{k-1}\right)- \Delta t\nabla \cdot \left(\rho n^{k-1}\nabla f^{k-1}\right)
(4)

para Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k=1,2,\ldots ,N} 

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n^0}
dado. Similarmente, para la segunda y tercera ecuación de (3), de donde resulta el esquema discreto:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{array}{rcl}-\Delta tD\nabla ^2n^k + n^k & = & F \\ f^k &=& \Delta t \beta n^{k-1} + \left(1-\Delta t\gamma n^{k-1}\right)f^{k-1} \\ c^k &=& \left(1-\Delta t\eta n^{k-1}\right)c^{k-1}, \end{array}

o bien si utilizamos un esquema de discretizacion en el tiempo hacia adelante, resulta el esquema:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{array}{rcl}\Delta tD\nabla ^2n^k + n^k & = & F \\ f^k &=& \dfrac{f^{k-1}+\Delta t \beta n^{k}}{1 + \Delta t\gamma n^{k}} \\ c^k &=& \dfrac{c^{k-1}}{1+\Delta t\eta n^{k}}, \end{array}

para Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k=1,2,\ldots ,N} , y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n^0} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f^0} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle c^0} 

dados, donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle F=n^{k-1} - \Delta t\nabla \cdot \left(\frac{\chi }{1+\alpha c^{k-1}}n^{k-1}\nabla c^{k-1}\right)- \Delta t\nabla \cdot \left(\rho n^{k-1}\nabla f^{k-1}\right)}

.

Condiciones iniciales. El primer evento de angiogénesis inducida por tumor es la secreción de TAF por las células tumorales, que se difunde en la matriz extracelular y se establece un gradiente de concentración entre el tumor y el vaso progenitor. Anderson y Chaplain (1998) [8] consideran un campo de concentración inicial de TAF de la forma Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle c(x,y,0)=\exp (-(1-x)^{2}/\epsilon _{1})} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (x,y) \in \Omega } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \epsilon _{1}=0.45} . Una vez que el TAF ha alcanzado el vaso sanguíneo principal, las células endoteliales dentro del vaso se forman en grupos que finalmente se convierten en brotes. Los autores antes citados suponen que inicialmente se forman tres grupos a lo largo del eje Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle y} 

(Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 0\leq y\leq 1}

) en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle x\approx 0} , con el tumor localizado en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (1,1/2)} 

y el vaso progenitor de las células endoteliales también en el eje Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle y}
(Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 0\leq y\leq 1}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle x=0} ). El grupo inicial de células endoteliales se genera con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n(x,y,0)=\exp (-x^{2}/\epsilon _{3})\sen ^{2}(7\pi y)} 

si Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle y\in [0.15,0.28]\cup [0.43,0.58]\cup [0.72,0.85]}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n(x,y,0)=0}
en otro caso; Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \epsilon _3=0.001}

. Una vez que las células endoteliales han sido activadas por el TAF, degradan la lámina basal del vaso principal. Este daño inicial resulta en una mayor capacidad de permeabilidad del vaso que permite que la fibronectina plasmática de la sangre se escape del vaso parental y se difunda en el tejido corneal. Posteriormente, esta fibronectina plasmática se une a la matriz extracelular del tejido corneal, creando una alta concentración inicial de fibronectina en y alrededor del buque matriz. Los autores antes citados toman el perfil de concentración inicial de fibronectina como Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f(x,y,0)=k\exp (-x^{2}/\epsilon _{2})} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (x,y)\in \Omega } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle k=0.75} 

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \epsilon _{2}=0.45}

.

Condiciones de contorno. Veamos ahora las condiciones de contorno que se deben satisfacer en la frontera Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Gamma } 

de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Omega }

, para que se pueda plantear el problema de existencia, unicidad y dependencia continua de los datos del problema de la solución del sistema (3). Anderson y Chaplain (1998) [8] proponen una condición de no flujo en la frontera Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Gamma } 

de la forma Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \zeta \cdot \left(-D_n\nabla n + n\left[\chi (c)\nabla c+\rho _0\nabla f\right]\right)=0}

, donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \zeta } 

es el vector normal unitario exterior a la frontera Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Gamma }

. Orme y Chaplain (1997) [13], consideran condiciones de frontera mixta. Por ejemplo, una vez que el proceso de angiogénesis ha comenzado, es decir, la membrana basal del vaso parental se rompe para formar brotes, y como se concentra la atención exclusivamente en las células endoteliales que están cerca del brote, ya que éstas son las que emigran hacia el tumor, entonces se puede suponer que la densidad inicial de las células endoteliales es Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n(0,y,t)=1} 

en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Gamma _4}
y condiciones de flujo cero en las demás fronteras, es decir, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \frac{\partial n}{\partial x}(1,y,t)=0}
en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Gamma _2}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \frac{\partial n}{\partial y}(x,0,t)=0} 

en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Gamma _1}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \frac{\partial n}{\partial y}(x,1,t)=0}
en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Gamma _3}

, ya que como lo indican Paweletz y Knierim (1989) [14], las células endoteliales no cruzan la membrana basal en ninguna otra parte. Por otro lado, ya que el TAF está siendo secretado por el tumor, su concentración permanece constante en ese punto y decae en cero en el vaso parental, por lo tanto, se consideran las condiciones de frontera: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle c(1,y,t)=1} 

en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Gamma _2}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle c(0,y,t)=0} 

en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Gamma _4}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \frac{\partial c}{\partial y}(x,0,t)=0} 

en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Gamma _1}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \frac{\partial c}{\partial y}(x,1,t)=0}
en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Gamma _3}

. Análogamente, para la concentración de fibronectina, ya que las células endoteliales liberan fibronectina, por lo que las condiciones de frontera para este caso serán: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f(0,y,t)=1} 

en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Gamma _4}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \frac{\partial f}{\partial x}(1,y,t)=0} 

en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Gamma _2}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \frac{\partial f}{\partial y}(x,0,t)=0} 

en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Gamma _1}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \frac{\partial f}{\partial y}(x,1,t)=0}
en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Gamma _3}

.

Si hacemos Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r=\Delta tD>0} 

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle u=n^k}

, la ecuación (4) toma la forma Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle -r\nabla ^{2}u + u = F} , que es una ecuación diferencial parcial de segundo orden de tipo elíptico.

4.1 Formulación variacional del problema mixto

Como observamos antes, Orme y Chaplain (1997) [13] sugieren estudiar un problema mixto. Para ello, dividimos la frontera Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Gamma } 

de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Omega }
en dos partes disjuntas, es decir, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Gamma = \Gamma _D \cup \Gamma _N}
con Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Gamma _D \cap \Gamma _N=\emptyset }

. El problema mixto lo formulamos como sigue:

Problema mixto no homogéneo. Sean Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle F:\overline{\Omega }\rightarrow \mathbb{R}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle g:\overline{\Gamma }_D\rightarrow \mathbb{R}} 

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle h:\overline{\Gamma }_N\rightarrow \mathbb{R}}
continuas y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r>0}

, determinar Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle u\in C^2(\Omega )} 

y continua en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \overline{\Omega }}
tal que

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{array}{rclcl}-r\nabla ^2u + u &=& F & \hbox{ en} & \Omega \\ u &=& g & \hbox{ sobre} & \Gamma _D \\ \dfrac{\partial u}{\partial \mathbf{n}} &=& h & \hbox{ sobre} & \Gamma _N, \\ \end{array}
(5)

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{n}} 

es el vector normal exterior unitario sobre la frontera Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Gamma _N}

.

Una forma de estudiar el problema mixto no homogéneo (5) es transformándolo en un problema equivalente que sea de tipo Dirichlet homogéneo sobre Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Gamma _D} , mediante un cambio de variable apropiado [15,12]. Para ello, notemos que si Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle u\in H^1(\Omega )} , el teorema de la traza B garantiza que existe un mapeo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \gamma _0:H^1(\Omega )\rightarrow L^2(\Omega )} 

tal que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \gamma _0(u)=u\vert _{\Gamma _D}}

. Más explícitamente, el rango de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \gamma _0} 

es el espacio Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H^{1/2}(\Gamma _D)}

. Así, podemos suponer que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle g\in H^{1/2}(\Gamma _D)} . Así que de nuevo por el teorema de la traza B, existe Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \widehat{g}\in H^1(\Omega )} 

tal que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \widehat{g}\vert _{\Gamma _D} = \gamma _0\left(\widehat{g}\right)=g}

. Proponemos el cambio de variable

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): w = u-\widehat{g}.
(6)

Transformemos ahora la primera y tercera ecuación de (5) en términos de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle w}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): F = -r\nabla ^2u + u = -r\nabla ^2w-r\nabla ^2\widehat{g}+w+\widehat{g},

de donde

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): -r\nabla ^2w + w = \widehat{F},\quad \widehat{F}=F+r\nabla ^2\widehat{g}-\widehat{g}.
(7)

Y

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): h = \frac{\partial u}{\partial \mathbf{n}} = (\nabla u)\cdot \mathbf{n} = (\nabla w)\cdot \mathbf{n} + (\nabla \widehat{g})\cdot \mathbf{n} = \frac{\partial w}{\partial \mathbf{n}} + \frac{\partial \widehat{g}}{\partial \mathbf{n}},

de donde

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \frac{\partial w}{\partial \mathbf{n}} = \widehat{h},\quad \widehat{h} = h-\frac{\partial \widehat{g}}{\partial \mathbf{n}}.
(8)

Así, bajo el cambio de variable (6) y teniendo en cuenta (7) y (8), el problema mixto no homogéneo (5) es equivalente al problema:

Problema mixto homogéneo. Sean Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \widehat{F}:\overline{\Omega }\rightarrow \mathbb{R}} 

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \widehat{h}:\overline{\Gamma }_N\rightarrow \mathbb{R}}
continuas y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r>0}

, determinar Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle w\in C^2(\Omega )} 

y continua en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \overline{\Omega }}
tal que

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{array}{rclcl}-r\nabla ^2w + w &=& \widehat{F} & \hbox{ en} & \Omega \\ w &=& 0 & \hbox{ sobre} & \Gamma _D \\ \dfrac{\partial w}{\partial \mathbf{n}} &=& \widehat{h} & \hbox{ sobre} & \Gamma _N. \end{array}
(9)

Para realizar la formulación variacional del problema mixto homogéneo (9), las condiciones de frontera sugieren de manera natural buscar soluciones débiles en el espacio de Sobolev Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle V = H_{0,\Gamma _D}^1(\Omega ) = \left\{v \in H^{1}(\Omega ) : v=0 \;\;\hbox{sobre}\;\;\Gamma _{D}\right\}} . Puesto que se satisface la desigualdad de Poincaré (teorema A) en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle V} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle V} 

es un espacio de Hilbert con la norma
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \Vert v\Vert _V = \left(\int _{\Omega }\vert \nabla v\vert ^2\right)^{1/2} = \left(\sum _{i=1}^2\int _{\Omega }\left(\frac{\partial v}{\partial x_i}\right)^2\right)^{1/2},\quad v\in V.

Multiplicando ambos lados de la primera ecuación de (9) por una función de prueba Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle v \in V} 

e integrando sobre Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Omega }
tenemos:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): -r\int _{\Omega }\left(\nabla ^2 w\right)v + \int _{\Omega } wv = \int _{\Omega }\widehat{F}v.
(10)

Puesto que

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left(\nabla ^2 w\right)v = v\frac{\partial ^2 w}{\partial x^2} + v\frac{\partial ^2 w}{\partial y^2}.
(11)

Entonces, bajo el supuesto que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Omega } 

satisface las hipótesis del teorema de Green A, se sigue que
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{array}{lcl}\displaystyle \int _{\Omega }v\dfrac{\partial ^2w}{\partial x^2} &=& \displaystyle \int _{\Gamma }v\vert _{\Gamma }\left.\dfrac{\partial w}{\partial x}\right\vert _{\Gamma }n_1 - \displaystyle \int _{\Omega }\dfrac{\partial v}{\partial x}\dfrac{\partial w}{\partial x} \\ &=& \displaystyle \int _{\Gamma _D}v\vert _{\Gamma _D}\left.\dfrac{\partial w}{\partial x}\right\vert _{\Gamma _D}n_1 + \displaystyle \int _{\Gamma _N}v\vert _{\Gamma _N}\left.\dfrac{\partial w}{\partial x}\right\vert _{\Gamma _N}n_1 - \displaystyle \int _{\Omega }\dfrac{\partial v}{\partial x}\dfrac{\partial w}{\partial x}, \end{array}

de donde,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \int _{\Omega }v\frac{\partial ^2w}{\partial x^2} = \int _{\Gamma _N}v\frac{\partial w}{\partial (n_1,0)} - \int _{\Omega }\frac{\partial v}{\partial x}\dfrac{\partial w}{\partial x}.
(12)

Similarmente,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \int _{\Omega }v\frac{\partial ^2w}{\partial y^2} = \int _{\Gamma _N}v\frac{\partial w}{\partial (0,n_2)} - \int _{\Omega }\frac{\partial v}{\partial y}\dfrac{\partial w}{\partial y}.
(13)

Tomando Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{n}=(n_1,n_2)} 

que es el vector normal exterior unitario a la frontera Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Gamma _N}
e integrando (11) sobre Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Omega }

, y teniendo en cuenta (12) y (13), obtenemos

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{array}{lcl}\displaystyle \int _{\Omega }\left(\nabla ^2 w\right)v &=& \displaystyle \int _{\Gamma _N}v\left[\dfrac{\partial w}{\partial (n_1,0)} + \dfrac{\partial w}{\partial (0,n_2)}\right]- \displaystyle \int _{\Omega }\left[\dfrac{\partial v}{\partial x}\dfrac{\partial w}{\partial x} + \dfrac{\partial v}{\partial y}\dfrac{\partial w}{\partial y}\right] \\ &=& \displaystyle \int _{\Gamma _N}v\dfrac{\partial w}{\partial \mathbf{n}} -\displaystyle \int _{\Omega }\nabla w\cdot \nabla v, \end{array}

de donde,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \displaystyle \int _{\Omega }\left(\nabla ^2 w\right)v = \displaystyle \int _{\Gamma _N}v\widehat{h} -\displaystyle \int _{\Omega }\nabla w\cdot \nabla v.
(14)


Sustituyendo (14) en (10) y agrupando términos, resulta

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): r\int _{\Omega }\nabla w\cdot \nabla v + \int _{\Omega }wv = r\int _{\Gamma _N}\widehat{h}v + \int _{\Omega }\widehat{F}v.

Por lo tanto, la formulación variacional o débil del problema mixto homogéneo (9) se formula como sigue:

Problema variacional mixto homogéneo. Sean Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \widehat{F}\in L^2(\Omega )} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \widehat{h}\in L^2(\Gamma _N)} 

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r>0}

, determinar Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle w\in V} 

tal que

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): r\int _{\Omega }\nabla w\cdot \nabla v + \int _{\Omega }wv = r\int _{\Gamma _N}\widehat{h}v + \int _{\Omega }\widehat{F}v, \quad \forall \, v\in V.
(15)

La existencia, unicidad y dependencia continua de la solución con respecto a los datos y parámetros del problema variacional mixto homogéneo (15), lo damos en el siguiente teorema vía el teorema de Lax-Milgram A.

Teorema 1: Sea Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Omega } 

un conjunto abierto y acotado de clase Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle C^2}
en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbb{R}^2}
con frontera Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Gamma =\Gamma _D\cup \Gamma _N}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Gamma _D\cap \Gamma _N=\emptyset } . Si Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \widehat{F}\in L^2(\Omega )} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \widehat{h}\in L^2(\Gamma _N)} 

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r>0}

, entonces el problema variacional mixto homogéneo (15) tiene una única solución Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle w\in V} . Además, existe una constante Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle C>0} 

tal que

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \Vert w\Vert _V \leq \frac{C}{r}\left(\Vert \widehat{F}\Vert _{L^2(\Omega )} + \Vert \widehat{h}\Vert _{L^2(\Omega _N)}\right).
(16)

Más aún, la funcional

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): J(v) = \frac{r}{2}\int _{\Omega }\vert \nabla v\vert ^2+\frac{1}{2}\int _{\Omega }v^2 - r\int _{\Gamma _N}\widehat{h}v - \int _{\Omega }\widehat{F}v, \quad \forall \, v\in V
(17)

alcanza su mínimo en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle w} .

Con el propósito de aplicar el teorema de Lax-Milgram A, definimos: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle B(w,v) = r\int _{\Omega }\nabla w\cdot \nabla v + \int _{\Omega }wv} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (w,v)\in V\times V} 

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \langle \mathcal{F},v\rangle = r\int _{\Gamma _N}\widehat{h}v + \int _{\Omega }\widehat{F}v}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle v\in V} . El objetivo es determinar Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle w\in V} 

tal que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle B(w,v) = \langle \mathcal{F},v\rangle }

, para todo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle v\in V} . En efecto. a) Es claro que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{F}} 

es una funcional lineal en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle V}

.

b) Veamos que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{F}} 

es continua en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle V}

. Aplicando primero la desigualdad de Schwarz para integrales (véase página 63 de [16]), luego el teorema de la traza B y finalmente la desigualdad de Poincaré A, obtenemos:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{array}{rcl} \vert\langle \mathcal{F},v\rangle \vert & \leq & r\left\vert \displaystyle \int _{\Gamma _N}\widehat{h} v\right\vert + \left\vert \displaystyle \int _{\Omega }\widehat{F}v\right\vert \\ & \leq & r\left(\displaystyle \int _{\Gamma _N}\widehat{h}^2\right)^{1/2}\left(\displaystyle \int _{\Gamma _N}v^2\right)^{1/2} + \left(\displaystyle \int _{\Omega }\widehat{F}^2\right)^{1/2}\left(\displaystyle \int _{\Omega }v^2\right)^{1/2} \\ & = & \Vert \widehat{h}\Vert _{L^2(\Gamma _N)}\Vert v\Vert _{L^2(\Gamma _N)} + \Vert \widehat{F}\Vert _{L^2(\Omega )}\Vert v\Vert _{L^2(\Omega )} \end{array}

de donde,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \vert \langle \mathcal{F},v\rangle \vert \leq \Vert \widehat{h}\Vert _{L^2(\Gamma _N)}C_T\Vert v\Vert _{H^1(\Omega )} + \Vert \widehat{F}\Vert _{L^2(\Omega )}C_P\Vert v\Vert _{*}.
(18)

La desigualdad de Poincaré A, garantiza que la norma Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Vert v\Vert _{*}=\sum _{i=1}^2\Vert \frac{\partial v}{\partial x_i}\Vert _{L^2(\Omega )}} 

es equivalente a la norma Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Vert \cdot \Vert _{H^1(\Omega )}}
en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle V}

. Así, existe Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \alpha{>0}} 

tal que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Vert v\Vert _{*} \leq \alpha \Vert v\Vert _{H^1(\Omega )}}
para todo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle v\in V}

. Por otro lado, la misma desigualdad de Poincaré A, nos dice que las normas Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Vert \cdot \Vert _{V}} 

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Vert \cdot \Vert _{H^1(\Omega )}}
son equivalentes, por lo que también existe Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \beta{>0}}
tal que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Vert v\Vert _{H^1(\Omega )} \leq \beta \Vert v\Vert _V}
para todo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle v\in V}

. Concluimos de (18) que

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \vert \langle \mathcal{F},v\rangle \vert \leq \beta C_T\Vert \widehat{h}\Vert _{L^2(\Gamma _N)}\Vert v\Vert _V + \beta \alpha C_P\Vert \widehat{F}\Vert _{L^2(\Omega )}\Vert v\Vert _V = M\Vert v\Vert _V \quad \forall \,v\in V,
(19)

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle M = \beta C_T\Vert \widehat{h}\Vert _{L^2(\Gamma _N)} + \beta \alpha C_P \Vert \widehat{F}\Vert _{L^2(\Omega )}>0} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle C_T>0} 

es la constante de la traza y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle C_P>0}
es la constante de Poincaré. Por consiguiente, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{F}}
es continua en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle V}

.

c) También es directo verificar que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle B} 

es una forma bilineal y simétrica sobre Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle V\times V}

.

d) Veamos que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle B} 

es continua. Notemos primero que

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \vert B(w,v)\vert \leq r \left\vert \displaystyle \int _{\Omega }\nabla w\cdot \nabla v\right\vert + \left\vert \displaystyle \int _{\Omega }wv\right\vert{.}
(20)

Trabajemos con la primera integral. Aplicamos la desigualdad de Schwarz para integrales (véase página 63 de [16]), luego la desigualdad de Schwarz para Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle n} 

pares de números reales o complejos (véase pág. 16 de [17]), tenemos:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{array}{lcl}\left\vert \displaystyle \int _{\Omega }\nabla w\cdot \nabla v\right\vert &\leq & \left\vert \displaystyle \int _{\Omega }\dfrac{\partial w}{\partial x}\dfrac{\partial v}{\partial x}\right\vert + \left\vert \displaystyle \int _{\Omega }\dfrac{\partial w}{\partial y}\dfrac{\partial v}{\partial y}\right\vert \leq \displaystyle \int _{\Omega }\left\vert \dfrac{\partial w}{\partial x}\right\vert \,\left\vert \dfrac{\partial v}{\partial x}\right\vert + \displaystyle \int _{\Omega }\left\vert \dfrac{\partial w}{\partial y}\right\vert \,\left\vert \dfrac{\partial v}{\partial y}\right\vert \\ &\leq & \left[\displaystyle \int _{\Omega }\left(\dfrac{\partial w}{\partial x}\right)^2\right]^{1/2}\left[\displaystyle \int _{\Omega }\left(\dfrac{\partial v}{\partial x}\right)^2\right]^{1/2} + \left[\displaystyle \int _{\Omega }\left(\dfrac{\partial w}{\partial y}\right)^2\right]^{1/2}\left[\displaystyle \int _{\Omega }\left(\dfrac{\partial v}{\partial y}\right)^2\right]^{1/2} \\ &\leq & \left(\displaystyle \int _{\Omega }\left(\dfrac{\partial w}{\partial x}\right)^2 + \displaystyle \int _{\Omega }\left(\dfrac{\partial w}{\partial y}\right)^2\right)^{1/2} \left(\displaystyle \int _{\Omega }\left(\dfrac{\partial v}{\partial x}\right)^2 + \displaystyle \int _{\Omega }\left(\dfrac{\partial v}{\partial y}\right)^2\right)^{1/2}, \end{array}

de donde

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left\vert \displaystyle \int _{\Omega }\nabla w\cdot \nabla v\right\vert \leq \Vert w\Vert _V \Vert v\Vert _V, \quad \forall \,(w,v)\in V\times V.
(21)

Aplicando de nuevo la desigualdad de Schwarz para integrales y luego la desigualdad de Poincaré A, a la segunda integral de (20), resulta:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{array}{lcl}\left\vert \displaystyle \int _{\Omega }wv\right\vert \leq \Vert w\Vert _{L^2(\Omega )} \Vert v\Vert _{L^2(\Omega )} \leq C_P^2\Vert w\Vert _{*} \Vert v\Vert _{*} \leq \alpha ^2C_P^2\Vert w\Vert _{H^1(\Omega )} \Vert v\Vert _{H^1(\Omega )} \leq \beta ^2\alpha ^2C_P^2\Vert w\Vert _{V}\Vert v\Vert _{V}, \end{array}
(22)

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Vert \cdot \Vert _*} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle C_P>0} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \alpha{>0}} 

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \beta{>0}}

, son como en (19). Sustituyendo (22) y (21) en (20), se sigue que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \vert B(w,v)\vert \leq \left(r+ \beta ^2\alpha ^2C_P^2\right)\Vert w\Vert _{V}\Vert v\Vert _{V}} , para todo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (w,v)\in V\times V} , lo que prueba que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle B} 

es continua.

e) La forma bilineal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle B} 

es coerciva ya que
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): B(v,v) = r\int _{\Omega }\vert \nabla v\vert ^2 + \int _{\Omega }v^2 \geq r\int _{\Omega }\vert \nabla v\vert ^2 = r\Vert v\Vert _V^2, \quad \forall \,v\in V.

Como se satisfacen todas la hipótesis del teorema de Lax-Milgram A, concluimos que el problema variacional mixto homogéneo (15), tiene una única solución Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle w\in V} .

f) Notemos que la constante de coercividad es Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r>0} , por lo que se sigue también del teorema de Lax-Milgram A y (19) que

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \begin{array}{lcl} \Vert w\Vert _V &\leq & \dfrac{1}{r}\,\hbox{ sup}\,\left\{\vert \langle \mathcal{F},v\rangle \vert : v\in V,\;\Vert v\Vert _V\leq 1\right\} \\ &\leq & \dfrac{1}{r}\,\hbox{ sup}\,\left\{\beta C_T\Vert \widehat{h}\Vert _{L^2(\Gamma _N)}\Vert v\Vert _V + \beta \alpha C_P\Vert \widehat{F}\Vert _{L^2(\Omega )}\Vert v\Vert _V : v\in V,\;\Vert v\Vert _V\leq 1\right\} \\ &\leq & \dfrac{1}{r}\left(\beta C_T\Vert \widehat{h}\Vert _{L^2(\Gamma _N)} + \beta \alpha C_P\Vert \widehat{F}\Vert _{L^2(\Omega )}\right)\leq \dfrac{C}{r}\left(\Vert \widehat{h}\Vert _{L^2(\Gamma _N)} + \Vert \widehat{F}\Vert _{L^2(\Omega )}\right), \end{array}

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle C=\beta \,\hbox{max}\,\left\{C_T, \alpha C_P\right\}>0} . Esto muestra la dependencia continua de la solución Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle w} 

con respecto a los datos Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \widehat{F}}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \widehat{h}} 

y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle r}
del problema. También con esto se prueba la desigualdad (16).

g) Finalmente, como Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle B} 

es simétrica en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle V\times V}

, se sigue una vez más del teorema de Lax-Milgram A que la funcional Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle J:V\rightarrow \mathbb{R}} 

por (17) alcanza su mínimo en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle w}

.

5 Conclusiones

La aportación más relevante del trabajo consistió en aplicar el método variacional para demostrar, en el teorema 1, la existencia, unicidad y dependencia continua de la solución débil del problema mixto homogéneo (9) con respecto a los datos y parámetros del problema. Como consecuencia, también se garantizó la existencia de la solución débil del problema no homogéneo (5).

Agradecimientos. A los árbitros por todas las sugerencias y recomendaciones para mejorar el manuscrito. A CONACYT por la beca de manutención que otorgó al primer autor durante el desarrollo del proyecto.

Apéndice

A Espacios de Sobolev y desigualdad de Poincaré

Definición A: Sea Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle L^{p}(\Omega )} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 1\leq p<\infty } , el espacio de las clases de todas las funciones Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \varphi :\Omega \rightarrow \mathbb{R}} 

medibles y tales que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \vert \varphi \vert ^{p}}
es integrable sobre Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Omega }

. Si Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \varphi \in L ^{p}(\Omega )} , se define la norma de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \varphi } 

como Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Vert \varphi \Vert _{L^p(\Omega )} = \left(\int _{\Omega } \vert \varphi \vert ^{p} d\boldsymbol{x}\right)^{1/p}, 1 \leq p < \infty }

. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle L^{\infty }(\Omega )} 

es el espacio de todas las clases de funciones Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \varphi :\Omega \rightarrow \mathbb{R}}
medibles y esencialmente acotadas sobre Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Omega }

, véase [18]. Si Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \varphi \in L ^{\infty }(\Omega )} 

se define la norma de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \varphi }
como Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Vert \varphi \Vert _{L^{\infty }(\Omega )} =\hbox{sup}^0\{ \vert \varphi (\boldsymbol{x})\vert :\boldsymbol{x}\in \Omega \} }

, donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \hbox{sup}^0} 

denota el supremo esencial de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \varphi }
sobre Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Omega }

.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle L^{2}(\Omega )} 

es un espacio de Hilbert con el producto interno Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \langle \varphi , \psi \rangle _{L^{2}(\Omega )} = \int _{\Omega }\varphi \psi \; d\boldsymbol{x}}
para todo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \varphi , \psi \in L^{2}(\Omega )}

.

Definición A: Se llama espacio de Sobolev Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H^{1}(\Omega )} 

al espacio de las funciones Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle u\in L^{2}(\Omega )}
cuyas derivadas parciales (en el sentido de las distribuciones) pertenecen a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle L^{2} (\Omega )}

, esto es, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H^{1}(\Omega )=\left\{u \in L^{2}(\Omega ):\frac{\partial u}{\partial x_{i}} \in L^{2}(\Omega ),\; 1\leq i\leq n \right\}} .

El espacio Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H^1(\Omega )} 

dotado del producto interno
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \langle u,v\rangle _{H^{1}(\Omega )}=\langle u,v\rangle _{L^{2}(\Omega )}+\sum _{i=1}^{n} \left\langle \frac{\partial u}{\partial x_{i}},\frac{\partial v}{\partial x_{i}}\right\rangle _{L^{2}(\Omega )} \; \forall u, v \in H^{1}(\Omega )

el cual induce la norma

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \Vert u \Vert _{H^{1}(\Omega )}=\left(\Vert u \Vert _{L^2(\Omega )}^{2}+ \sum _{i=1}^{n}\left\Vert \frac{\partial u}{\partial x_{i}}\right\Vert _{L^{2}(\Omega )}^{2}\right)^{1/2} \; \forall u \in H^{1}(\Omega )

es un espacio de Hilbert.

Si denotamos por Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{D}(\Omega )} 

al conjunto de todas las funciones de prueba sobre Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Omega }

, es decir, funciones de clase Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle C^{\infty }(\Omega )} 

y de soporte compacto contenido en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Omega }

, entonces Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{D}(\Omega ) \subset H^{1}(\Omega )} . Es especialmente útil el espacio Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H_{0}^{1}(\Omega )=\overline{\mathcal{D}(\Omega )}} , es decir, la adherencia respecto de la norma de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H^{1}(\Omega )} , del espacio de las funciones de prueba Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{D}(\Omega )} . Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H_{0}^{1}(\Omega )} 

con la norma que hereda de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H^{1}(\Omega )}
es también un espacio de Hilbert. Dicho de un modo un tanto impreciso, el espacio Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H_{0}^{1}(\Omega )}
es el formado por las funciones de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H^{1}(\Omega )}
que se anulan sobre la frontera de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Omega }

. Decimos de un modo un tanto impreciso dado que la frontera de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Omega } 

tiene medida nula, y dos funciones de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle L^{2}(\Omega )}
que son iguales salvo en un conjunto de medida cero son, como funciones de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle L^{2}(\Omega )}

, iguales. Para eliminar esta ambigüedad se introduce el concepto de traza de una función de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H^{1}(\Omega )} , véase por ejemplo [15]. Extendemos la definición del espacio de Sobolev para funciones en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle L^p(\Omega )} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p\geq 1} , como sigue:

Definición A: Sea Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle m\geq 1} 

entero y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 1\leq p\leq \infty }

. El espacio de Sobolev Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle W^{m,p}(\Omega )} 

se define como Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle W^{m,p}(\Omega )=\left\{u\in L^p(\Omega ):\partial ^{\alpha }u\in L^p(\Omega )\;\;\forall \;\vert \alpha \vert \leq m,\;\alpha \in \mathbb{N}^n\right\}}
dotada de la norma
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \Vert u\Vert _{m,p,\Omega } = \sum _{\vert \alpha \vert \leq m}\Vert \partial ^{\alpha }u\Vert _{L^p(\Omega )} \quad \forall \;u\in W^{m,p}(\Omega )

o equivalentemente, para Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 1<p<\infty } , de la norma

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \Vert u\Vert _{m,p,\Omega } = \left(\sum _{\vert \alpha \vert \leq m}\int _{\Omega }\vert \partial ^{\alpha }u\vert ^p\right)^{1/p} = \left(\sum _{\vert \alpha \vert \leq m}\Vert \partial ^{\alpha }u\Vert _{L^p(\Omega )}^p\right)^{1/p} \quad \forall \;u\in W^{m,p}(\Omega{).}

Un caso de especial importancia ocurre cuando Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p=2} , donde se obtiene el espacio Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H^m(\Omega )=W^{m,2}(\Omega )=\left\{u\in L^2(\Omega ):\partial ^{\alpha }u\in L^2(\Omega )\;\;\forall \;\vert \alpha \vert \leq m,\;\alpha \in \mathbb{N}^n\right\}} 

dotada de la norma

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \Vert u\Vert _{m,\Omega }=\Vert u\Vert _{m,2,\Omega } = \left(\sum _{\vert \alpha \vert \leq m}\int _{\Omega }\vert \partial ^{\alpha }u\vert ^2\right)^{1/2}.
(W)

El espacio Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H^m(\Omega )} 

posee un producto interno natural definido por
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \langle u,v\rangle _{m,\Omega } = \sum _{\vert \alpha \vert \leq m}\int _{\Omega }\partial ^{\alpha }u\partial ^{\alpha }v,\quad \forall \;u,v\in H^m(\Omega ),

que induce la norma (W), por lo que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H^m(\Omega )} 

es un  espacio de Hilbert.  Como antes, introducimos un importante subespacio de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle W^{m,p}(\Omega )}

. Si Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 1\leq p<\infty } , entonces Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{D}(\Omega )} 

es denso en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle L^p(\Omega )}

. También, si Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \varphi \in \mathcal{D}(\Omega )} , entonces Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \partial ^{\alpha }\varphi \in \mathcal{D}(\Omega )} 

para todo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \alpha \in \mathbb{N}^n}

, por lo que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{D}(\Omega )\subset W^{m,p}(\Omega )} 

para cualquier Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle m}
y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p}

. Si Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 1\leq p<\infty } , se define el espacio Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle W_0^{m,p}(\Omega )} 

como la adherencia de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{D}(\Omega )}
en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle W^{m,p}(\Omega )}

. Así Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle W_0^{m,p}(\Omega )} 

es un subespacio cerrado de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle W^{m,p}(\Omega )}
y sus elementos pueden ser aproximados en la norma de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle W^{m,p}(\Omega )}
por funciones de clase Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle C^{\infty }}
con soporte compacto contenido en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Omega }

. Cuando Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle p=2} , los espacios Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle W_0^{m,p}(\Omega )} 

se denotan como Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H_0^{m}(\Omega )}

. En general, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle W_0^{m,p}(\Omega )} 

es un subespacio estricto de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle W^{m,p}(\Omega )}

, salvo cuando Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Omega =\mathbb{R}^n} 

(véase [15]).

Teorema B: Sea Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Omega \subset \mathbb{R}^n} 

un conjunto abierto y acotado de clase Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle C^{m+1}}
con frontera Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Gamma }

. Entonces existe un mapeo traza Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \gamma =(\gamma _0,\gamma _1,\ldots ,\gamma _{m-1})} 

de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H^m(\Omega )}
en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (L^2(\Omega ))^m}
tal que
  1. Si Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle v\in C^{\infty }(\overline{\Omega })} , entonces Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \gamma _0(v)=v\vert _{\Gamma }, \gamma _1(v)=\left.\frac{\partial v}{\partial \mathbf{n}}\right\vert _{\Gamma }, \ldots , \gamma _{m-1}(v)=\left.\frac{\partial ^{m-1} v}{\partial \mathbf{n}^{m-1}}\right\vert _{\Gamma }} , donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{n}} es el vector normal exterior unitario sobre la frontera Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Gamma } .
  2. El rango de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \gamma } es el espacio Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \prod _{j=0}^{m-1}H^{m-j-1/2}(\Gamma )} .
  3. El núcleo de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \gamma } es el espacio Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H_0^{m}(\Omega )} .

Teorema A: Sea Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Omega \subset \mathbb{R}^n} 

un conjunto abierto y acotado de clase Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle C^{1}}
que yace sobre el mismo lado de su frontera Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Gamma }

. Sean Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle u,v\in H^1(\Omega )} . Entonces para Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle 1\leq i\leq n} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \int _{\Omega }u\,\frac{\partial v}{\partial x_i} = \int _{\Gamma }u\vert _{\Gamma }\, v\vert _{\Gamma }\, n_i - \int _{\Omega }\frac{\partial u}{\partial x_i}\,v} , donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbf{n}(\boldsymbol{x})=(n_1(\boldsymbol{x}),n_2(\boldsymbol{x}),\ldots ,n_n(\boldsymbol{x}))} 

es el vector normal exterior unitario sobre la frontera Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Gamma }

.

Teorema A: Sea Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Omega } 

un conjunto abierto y acotado en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathbb{R}^n}

. Entonces existe una constante positiva Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle C=C(\Omega ,p)} 

tal que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Vert u\Vert _{L^p(\Omega )} \leq C\sum _{i=1}^n\left\Vert \frac{\partial u}{\partial x_i}\right\Vert _{L^p(\Omega )}}

, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle u\in W_0^{1,p}(\Omega )} . En particular, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle u\rightarrow \sum _{i=1}^n\left\Vert \frac{\partial u}{\partial x_i}\right\Vert _{L^p(\Omega )}} 

define una norma sobre Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle W_0^{1,p}(\Omega )}

, la cual es equivalente a la norma Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Vert \cdot \Vert _{1,p,\Omega }} . Sobre Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H_0^1(\Omega )} , la forma bilineal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \langle u,v\rangle = \int _{\Omega }\sum _{i=1}^n\frac{\partial u}{\partial x_i}\frac{\partial v}{\partial x_i}} 

define un producto interno que induce una norma equivalente a la norma Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Vert \cdot \Vert _{1,\Omega }}

.

B Formulación variacional abstracta y el teorema de Lax-Milgram

Una función Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f\in L^{2}(\Omega )} 

vista como una distribución sobre Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \mathcal{D}(\Omega )}

, se extiende a una forma lineal y continua sobre Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H_{0}^{1}(\Omega )} , por medio de la aplicación Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f:H_{0}^{1}(\Omega )\rightarrow \mathbb{R}} 

por Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle  \langle f,u \rangle = \int _{\Omega } f(\boldsymbol{x})u(\boldsymbol{x})d\boldsymbol{x}}

, para todo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle u \in H_{0}^{1}(\Omega )} .

Definición A: Sea Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (H,\Vert \cdot \Vert )} 

un espacio de Hilbert, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f:H\rightarrow \mathbb{R}}
una forma lineal y continua y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle B:H\times H\rightarrow \mathbb{R}}
una forma bilineal. Por problema variacional entendemos el problema de determinar Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle u\in H}
tal que
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): B(u,v)=\langle f,v\rangle \quad \hbox{para todo}\; v \in H.
(X)

La existencia, unicidad y dependencia continua respecto de los datos iniciales de la solución de (A), se obtiene a través del teorema de Lax-Milgram.

Definición A: Sea Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle B} 

una forma bilineal sobre un espacio normado Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (H,\Vert \cdot \Vert )}

. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle B} 

es continua si existe Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle M>0}
tal que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle |B(u,v)|\leq M\Vert u\Vert \Vert v \Vert }
para todo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle u, v \in H}

, y es coerciva o Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H} -elíptica si existe Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle m>0} 

tal que Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle |B(u,u)|\geq m\Vert u\Vert ^2}
para todo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle u\in H}

. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle B} 

es simétrica si Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle B(u,v)=B(v,u)}
para todo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle u,v\in H}

.

Teorema A: Sea Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle (H,\Vert \cdot \Vert )} 

un espacio de Hilbert, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle f:H\rightarrow \mathbb{R}}
una forma lineal y continua, y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle B:H\times H\rightarrow \mathbb{R}}
una forma bilineal continua y coerciva, entonces el problema variacional (A) tiene una única solución Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle u}
en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle H}

. Además, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Vert u\Vert \leq \frac{1}{m}\Vert f\Vert _{*}} , donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle \Vert f\Vert _{*}=\hbox{sup}\left\lbrace |\langle f,v\rangle |:\; v \in H \;\hbox{y}\;\Vert v\Vert \leq 1 \right\rbrace } . Si Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle B} 

es también simétrica, entonces la funcional Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle J:H\rightarrow \mathbb{R}}
por Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle  J(v)=\frac{1}{2}B(v,v)-\langle f,v\rangle }
para todo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle v\in H}
alcanza su mínimo en Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle u}

.

BIBLIOGRAFÍA

[1] Graña, A. (2015) "Breve evolución histórica del cáncer", Volume 5. Carcinos 1 26–31

[2] Aibar, S., Celano, C., Chambi, M.C., Estrada, S., Gandur, N., Gange, P., González, C., González, O., Grance, G., Junin, M., Kohen, N., Molina, J., Núñez, M.G., Sáenz, M., Troncoso, M., Vallejos, A. (2008) "Manual de Enfermería Oncológica". Instituto Nacional de Cáncer de Argentina. Argentina: Instituto Nacional del Cáncer. Publicación desconocida

[3] Bray, F., Ferlay, J., Soerjomataram, I., Siegel, R.L., Torre, L.A. and Jemal, A. (2018) "Global cancer statistics 2018: GLOBOCAN Estimates of incidence and mortality worldwide for 36 cancers in 185 countries", Volume 68. CA: A Cancer Journal for Clinicians 6 394–424

[4] Jiménez García, L.F., Merchant Larios, H. (2003) "Biología celular y molecular". México: Pearson Educación. Publicación desconocida 664

[5] Zapata Peña, J., Ortiz, A.C. (2010) "Uso de modelos matemáticos para la descripción del crecimiento de tumores cancerosos", Volume 8. NOVA-Publicación Científica en Ciencias Biomédicas 14 140–147

[6] Folkman, J. (1971) "Tumor angiogenesis: therapeutic implications", Volume 20. The New England Journal of Medicine 1182–1186

[7] Martínez-Ezquerro, J.D., Herrera, L.A. (2006) "Angiogénesis: VEGF/VEGFRs como blancos terapéuticos en el tratamiento contra el cáncer", Volume 1. Cancerología 83–96

[8] Anderson, A.R.A. and Chaplain, M.A.J. (1998) "Continuous and discrete mathematical models of tumor-induced angiogenesis", Volume 60. Addison-wesley. Bulletin of Mathematical Biology 857–900

[9] Theocharis, A.D., Skandalis, S.S., Gialeli, Ch. and Karamanos, N.K. (2015) "Extracellular Matrix Structure". Advanced Drug Delivery Reviews 1–76

[10] Kornblihtt, A.R., Pesce, C.G., Alonso, C.R., Cramer, P., Srebrow, A., Werbajh, S. and Muro, A.F. (1996) "The fibronectin gene as a model for splicing and transcription studies", Volume 10. The FASEB Journal 2 248–257

[11] Glowinski, R. (2003) "Numerical Methods for Fluids (Part 3): Finite Element Methods for Incompressible Viscous Flow". Elsevier Science. Publicación desconocida

[12] Reddy, J.N. (1986) "Applied functional analysis and variational methods in engineering". USA: McGraw-Hill. Publicación desconocida

[13] Orme, M.E. and Chaplain, M.A.J. (1997) "Two-dimesnional models of tumor angiogenesis and anti-angiogenesis strategies", Volume 14. IMA Journal of Mathematics Applied in Medicine and Biology 189–205

[14] Paweletz, N. and Knierim, M. (1989) "Tumor-related angiogenesis", Volume 9. Critical Reviews in Oncology Hematology 3 197–242

[15] Kesavan, S. (1989) "Topics in functional analysis and applications". India: Wiley. Publicación desconocida

[16] Rudin, W. (1987) "Real and complex analysis". USA: McGraw-Hill. Publicación desconocida, 3rd. Edition

[17] Rudin, W. (1980) "Principios de Análisis Matemático". México: McGraw-Hill. Publicación desconocida, 3a. Edition

[18] Taylor, A.E. and Lay, D.C. (1980) "Introduction to functional analysis". USA: John Wiley. Publicación desconocida, 2nd. Edition

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