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

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 ${\mathrm{\Omega }}^{\ast }\subset D$ minimizer of the first eigenvalue under a volume constraint. Next, the local continuity of the eigenfunction associated with the eigenvalue on ${\mathrm{\Omega }}^{\ast }$ is proved. This allows us to conclude that ${\mathrm{\Omega }}^{\ast }$ 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.