IJPAM: Volume 58, No. 1 (2010)

THE OPTIMIZATION OF EIGENVALUE PROBLEMS FOR
OPERATORS INVOLVING THE $\boldsymbol{p}$-LAPLACIAN

Wac\law Pielichowski
Department of Physics, Mathematics and Computer Science
Cracow University of Technology
24, Warszawska Street, Cracow, 31-155, POLAND
e-mail: wpielich@pk.edu.pl


Abstract.In this paper we survey results concerning the following optimization problem: given a bounded domain $\Omega\subset\mathbb{R}^n$, numbers $p>1$, $\alpha\geq 0$ and $A\in [0,\vert\Omega\vert]$, find a subset $D\subset\Omega$, of measure $A$, for which the first eigenvalue of the operator

\begin{displaymath}u\mapsto-\text{div}(\vert\nabla u\vert^{p-2}\nabla u)+\alpha\chi_D \vert u\vert^{p-2}u\end{displaymath}

with the Dirichlet boundary condition is as small as possible. We also address the question of symmetry of optimal solutions.

Received: December 2, 2009

AMS Subject Classification: 35P30, 35J65, 35J70

Key Words and Phrases: $p$-Laplacian, the first eigenvalue, Steiner symmetry

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2010
Volume: 58
Issue: 1