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

TL;DR

This paper establishes a strict Faber-Krahn type inequality for the first eigenvalue of the p-Laplace operator on Lipschitz domains, demonstrating how domain variations affect eigenvalues, with applications to obstacle problems.

Contribution

It introduces a new strict inequality for the first eigenvalue under domain polarization and applies it to obstacle problems, showing eigenvalue monotonicity under geometric transformations.

Findings

Proves a strict Faber-Krahn inequality for p-Laplace eigenvalues.

Shows eigenvalue monotonicity with obstacle translations and rotations.

Applies results to obstacle problems with geometric assumptions.

Abstract

For $d\geq 2$ 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 $\Omega \setminus \mathscr{O}$, where $\mathscr{O}\subset \subset \Omega $ is an obstacle. Under some geometric assumptions on $\Omega $ and $\mathscr{O}$, we prove the strict monotonicity of $\lambda _1 (\Omega \setminus \mathscr{O})$ with respect to certain translations and rotations of $\mathscr{O}$ in $\Omega $.