Minimización del primer valor propio del p-Laplaciano de Dirichlet en ciertos tipos de dominios = Minimization of the first eigenvalue of the Dirichlet p-Laplacian in certain classes of domains.
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Date
2012
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Publisher
Universidad de Concepción
Abstract
In this thesis, we will work around the eigenvalue problem 8< : pu = jujp2u in u = 0 on @ where p is the p- Laplacian operator, with 1 < p < 1, which is a generalization of the Laplacian operator (p = 2) and it is defined for a function u in the Sobolev space W1;p 0 ( ) as pu = div(jrujp2ru): More specifically, we will study thoroughly the first eigenvalue 1( ) of p- Laplacian with Dirichlet condition, which is defined as the minimum of Rayleigh quotient for nonzero functions belonging to W1;p 0 ( ). i.e., 1( ) = min '2W1;p 0 ( );'6=0 R jr'jp R j'jp : We note that , 1 depends on the domain . We will show the principal properties of 1( ) and of its eigenfunctions, and later obtain results on the problem of minimization of 1( ) in certain classes of domains with the same volume or perimeter, similar to a classical problem. In the first chapter, which corresponds to the preliminaries, we will introduce some basic notions and definitions. We introduce the notion of a distribution, which allows us to define the concept of weak derivative of a function defined in a domain , among other notions. Moreover, in the first chapter we will define the Sobolev space W1;p( ), which is the set of all functions which belong to Lp( ), such that all its weak derivatives.
Description
Tesis presentada para optar al grado de Magíster en Matemática.
Keywords
Espacios de Sobolev, Medidas de Hausdorff, Teoría de la Medida, Problema de Dirichlet