Publication: Homogenization of elliptic problems in thin domains with oscillatory boundaries
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2016-08-16
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Universidad Complutense de Madrid
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Abstract
Los dominios finos, es decir, dominios sustancialmente más pequeños en alguna o varias de sus direcciones que en el resto, aparecen en muchos campos de la ciencia. Por ejemplo, dinámica de fluídos (lubricación, conducción de fluídos en tubos delgados, dinámica de oceanos...), mecánica de sólidos (barras delgadas, placas o cáscaras) o incluso en fisiología (circulación de la sangre). Así, el amplio número de posibles aplicaciones a situaciones reales ha hecho que la investigación de modelos de ecuaciones en derivadas parciales en dominios finos se convierta en un tema muy estudiado en los últimos años. Desde un punto de vista matemático, el estudio de las soluciones de una EDP en un dominio fino es un caso particular de la cuestión general relativa a cómo la variación de los dominios afecta al comportamiento de las soluciones de la EDP. En este marco, obtener la ecuación límite del modelo considerado, comparar la solución de la ecuación límite y las soluciones del problema en el dominio fino, analizar los coeficientes de la ecuación límite y comprender cómo la geometría del dominio afecta a la ecuación límite son algunos de los objetivos que deberían ser alcanzados. De hecho, es importante señalar que este tipo de cuestiones no sólo proporcionan importantes resultados teóricos sino que son muy relevantes desde el punto de vista de las aplicaciones. Por ejemplo, ser capaz de reducir el problema original a un problema mucho más sencillo, problema límite, que refleje las principales características del problema de partida puede ser muy útil para ingenieros y físicos...
Thin domains, that is, domains where one or several of their characteristic directions are substantially smaller than the others, appear in many fields of science, like fluid dynamics (lubrication, conduction of fluids in thin tubes, ocean dynamics...), solid mechanics (thin rods, plates or shells) or even physiology (blood circulation). Thus, the wide possibilities of applying the mathematical results to real situations has made that partial differential equations on thin domains becomes a very studied topic over the last years. From a mathematical point of view, the study of the solutions of a PDE on thin domains is a particular case of the general question concerning the effects of the variation in domains on the behavior of the solutions of the PDE. In this framework, obtaining the limit equation of the model considered on the thin domain, comparing the limit solution and the solutions of the equation defined on the thin domain, analyzing the coefficients of the limit equation and understanding how the geometry of the thin domains affects the limit equation are some of the main goals that should be reached. In fact, answering this kind of questions not only provide important theoretical results, it is also very interesting for the applications. For instance, being able to reduce the original problem to an easier to handle limit problem, which reflects most of important features of the original one is very useful for engineers and applied scientists...
Thin domains, that is, domains where one or several of their characteristic directions are substantially smaller than the others, appear in many fields of science, like fluid dynamics (lubrication, conduction of fluids in thin tubes, ocean dynamics...), solid mechanics (thin rods, plates or shells) or even physiology (blood circulation). Thus, the wide possibilities of applying the mathematical results to real situations has made that partial differential equations on thin domains becomes a very studied topic over the last years. From a mathematical point of view, the study of the solutions of a PDE on thin domains is a particular case of the general question concerning the effects of the variation in domains on the behavior of the solutions of the PDE. In this framework, obtaining the limit equation of the model considered on the thin domain, comparing the limit solution and the solutions of the equation defined on the thin domain, analyzing the coefficients of the limit equation and understanding how the geometry of the thin domains affects the limit equation are some of the main goals that should be reached. In fact, answering this kind of questions not only provide important theoretical results, it is also very interesting for the applications. For instance, being able to reduce the original problem to an easier to handle limit problem, which reflects most of important features of the original one is very useful for engineers and applied scientists...
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Tesis inédita de la Universidad Complutense de Madrid, Facultad de Ciencias Matemáticas, Departamento de Matemática Aplicada, leída el 01-02-2016