On the existence and regularity of an optimal shape for the non-linear first eigenvalue problem with Dirichlet condition
Rocard Michel Gouton, Aboubacar Marcos, Diaraf Seck · arXiv (Cornell University) · 2023
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 $Ω^*\subset D$ minimizer of the first eigenvalue under a volume constraint. Next, the local continuity of the eigenfunction associated with the eigenvalue on $Ω^*$ is proved. This allows us to conclude that $Ω^*$ 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.