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