Linear Stability Implies Nonlinear Stability for Faber-Krahn Type Inequalities

TL;DR

This paper proves that linear stability of Faber-Krahn inequalities implies nonlinear stability for domains with the same volume, extending known results from smooth perturbations to more general shapes, with applications to Riemannian manifolds.

Contribution

It establishes that linear stability of Faber-Krahn inequalities guarantees nonlinear stability without smoothness assumptions, using analysis of free boundary problems.

Findings

Linear stability implies nonlinear stability for Faber-Krahn inequalities.

Results extend to Riemannian manifolds.

Analysis based on Bernoulli-type free boundary problems.

Abstract

For a domain $\Omega \subset \mathbb{R}^n$ and a small number $\frak{T} > 0$, let \[ \mathcal{E}_0(\Omega) = \lambda_1(\Omega) + {\frak{T}} {\text{tor}}(\Omega) = \inf_{u, w \in H^1_0(\Omega)\setminus \{0\}} \frac{\int |\nabla u|^2}{\int u^2} + {\frak{T}} \int \frac{1}{2} |\nabla w|^2 - w \] be a modification of the first Dirichlet eigenvalue of $\Omega$. It is well-known that over all $\Omega$ with a given volume, the only sets attaining the infimum of $\mathcal{E}_0$ are balls $B_R$; this is the Faber-Krahn inequality. The main result of this paper is that, if for all $\Omega$ with the same volume and barycenter as $B_R$ and whose boundaries are parametrized as small $C^2$ normal graphs over $\partial B_R$ with bounded $C^2$ norm, \[ \int |u_{\Omega} - u_{B_R}|^2 + |\Omega \triangle B_R|^2 \leq C [\mathcal{E}_0(\Omega) - \mathcal{E}_0(B_R)] \] (i.e. the Faber-Krahn inequality is linearly stable), then the same is true for any $\Omega$ with the same volume and barycenter as $B_R$ without any smoothness assumptions (i.e. it is nonlinearly stable). Here $u_{\Omega}$ stands for an $L^2$-normalized first Dirichlet eigenfunction of $\Omega$. Related results are shown for Riemannian manifolds. The proof is based on a detailed analysis of some critical perturbations of Bernoulli-type free boundary problems. The topic of when linear stability is valid, as well as some applications, are considered in a companion paper.