Optimization in problems involving the p-Laplacian

Abstract

We minimize the energy integral $int_Omega | abla u|^p,dx$, where $g$ is a bounded positive function that varies in a class of rearrangements, $p>1$, and $u$ is a solution of $$displaylines{ -Delta_p u=g quadhbox{in } Omegacr u=0quad hbox{on } partialOmega,. }$$ Also we maximize the first eigenvalue $lambda=lambda_g$, where $$ -Delta_p u=lambda g u^{p-1}quadhbox{in }Omega,. $$ For both problems, we prove existence, uniqueness, and representation of the optimizers.

Keywords

• p-Laplacian
• energy integral
• eigenvalues
• rearrangements
• shape optimization