Resumen:
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.