Electronic Journal of Differential Equations,
Vol. 2010(2010), No. 02, pp. 1-10.
Title: Optimization in problems involving the p-Laplacian
Author: Monica Marras (Univ. di Cagliari, Italy)
Abstract:
We minimize the energy integral
$\int_\Omega |\nabla u|^p\,dx$, where $g$ is a bounded
positive function that varies in a class of rearrangements,
$p>1$, and $u$ is a solution of
$$\displaylines{
-\Delta_p u=g \quad\hbox{in } \Omega\cr
u=0\quad \hbox{on } \partial\Omega\,.
}$$
Also we maximize the first eigenvalue
$\lambda=\lambda_g$, where
$$
-\Delta_p u=\lambda g u^{p-1}\quad\hbox{in }\Omega\,.
$$
For both problems, we prove existence,
uniqueness, and representation of the optimizers.
Submitted November 2, 2009. Published January 05, 2010.
Math Subject Classifications: 35J25, 49K20, 47A75.
Key Words: p-Laplacian; energy integral; eigenvalues; rearrangements;
shape optimization