Domain variations of the first eigenvalue via a strict Faber-Krahn type inequality

Abstract

For $d\ge 2$ and $\frac{2d+2}{d+2}<p<\infty$, we prove a strict Faber-Krahn type inequality for the first eigenvalue ${\lambda }_1(\Omega )$ of the $p$-Laplace operator on a bounded Lipschitz domain $\Omega \subset {\mathbb{R}}^d$ (with mixed boundary conditions) under the polarizations. We apply this inequality to the obstacle problems on the domains of the form $\Omega \setminus \mathcal{O}$, where $\mathcal{O}\subset \subset \Omega$ is an obstacle. Under some geometric assumptions on $\Omega$ and $\mathcal{O}$, we prove the strict monotonicity of ${\lambda }_1(\Omega \setminus \mathcal{O})$ with respect to certain translations and rotations of $\mathcal{O}$ in $\Omega$.

21 pages, 6 figures

References in corpus (2)

• A shape variation result via the geometry of eigenfunctions
• On the optimization of the first weighted eigenvalue

Cited by in corpus (3)

• A stability result for the first Robin-Neumann eigenvalue: A double perturbation approach
• Strict monotonicity of the first $q$-eigenvalue of the fractional $p$-Laplace operator over annuli
• Strict Faber-Krahn type inequality for the mixed local-nonlocal operator under polarization