Some remarks on the optimization of eigenvalue problems involving the p-Laplacian

Wacław Pielichowski

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.

Mathematics Subject Classification: 35P30, 35J65, 35J70.

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