Opuscula Math. 28, no. 4 (2008), 561-566

Opuscula Mathematica

# 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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• Wacław Pielichowski
• Cracow University of Technology, Institute of Mathematics, ul. Warszawska 24, 31-155 Cracow, Poland
• Revised: 2008-03-03.
• Accepted: 2008-03-03.