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.

Abstract

In this thesis, we will work around the eigenvalue problem 8< : 􀀀 pu = jujp􀀀2u 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(jrujp􀀀2ru): 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.