@Article{Pielichowski2008,
author = "Pielichowski, Wacław",
title = "Some remarks on the optimization of eigenvalue problems involving the p-Laplacian",
journal = "Opuscula Math.",
year = "2008",
volume = "28",
number = "4",
pages = "561--566",
doi = "http://dx.doi.org/10.7494/OpMath.2008.28.4.561",
url = "http://dx.doi.org/10.7494/OpMath.2008.28.4.561",
abstract = "Given a bounded domain \(\Omega \subset \mathbb{R}^n\), numbers \(p \gt 1\), \(\alpha \geq 0\) and \(A \in [0,|\Omega |]\), consider the optimization problem: find a subset \(D \subset \Omega \), of measure \(A\), for which the first eigenvalue of the operator \(u\mapsto -\text{div} (|\nabla u|^{p-2}\nabla u)+ \alpha \chi_D |u|^{p-2}u \) with the Dirichlet boundary condition is as small as possible. We show that the optimal configuration \(D\) is connected with the corresponding positive eigenfunction \(u\) in such a way that there exists a number \(t\geq 1\) for which \(D=\{u \leq t\}\). We also give a new proof of symmetry of optimal solutions in the case when \(\Omega \) is Steiner symmetric and \(p = 2\).",
keywords = "\(p\)-Laplacian, the first eigenvalue, Steiner symmetry",
issn = "1232-9274"
}