On the existence and regularity of an optimal shape for the non-linear first eigenvalue problem with Dirichlet condition

TL;DR

This paper proves the existence, openness, and boundary regularity of optimal shapes minimizing the first eigenvalue for a nonlinear spectral problem involving p-Laplacian and Laplacian operators under volume constraints.

Contribution

It establishes the existence, local continuity, openness, and boundary regularity of optimal shapes for a nonlinear eigenvalue shape optimization problem.

Findings

Existence of a minimizer shape under volume constraint.

Eigenfunction continuity implies the optimal shape is open.

Reduced boundary of the optimal shape is regular.

Abstract

We study a shape optimization problem associated with the first eigenvalue of a nonlinear spectral problem involving a mixed operator ($p-$Laplacian and Laplacian) with a constraint on the volume. First, we prove the existence of a quasi-open $\Omega^*\subset D$ minimizer of the first eigenvalue under a volume constraint. Next, the local continuity of the eigenfunction associated with the eigenvalue on $\Omega^*$ is proved. This allows us to conclude that $\Omega^*$ is open when $D$ is connected. This is an important first step for regularizing the optimal shape themselves. Finally, there is a proof that the reduced boundary of the optimal shape is regular.