Electron. J. Diff. Equ., Vol. 2010(2010), No. 02, pp. 1-10.

Optimization in problems involving the p-Laplacian

Monica Marras

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 5, 2010.
Math Subject Classifications: 35J25, 49K20, 47A75.
Key Words: p-Laplacian; energy integral; eigenvalues; rearrangements; shape optimization

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Monica Marras
Dipartimento di Matematica e Informatica
Universitá di Cagliari
Via Ospedale 72, 09124 Cagliari, Italy
email: mmarras@unica.it

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