Linear stability implies nonlinear stability for Faber–Krahn-type inequalities

Abstract

For a domain $\Omega \subseteq \mathbb{R}^n$ and a small number $\mathfrak{T} > 0$, let $$ \mathcal{E}0(\Omega) = \lambda\_1(\Omega) + \mathfrak{T} \operatorname{tor}(\Omega) = \inf{u, w \in H^1\_0(\Omega)\setminus {0}} \frac{\int |\nabla u|^2}{\int u^2} + \mathfrak{T} !\int \frac{1}{2} |\nabla w|^2 - w $$ 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 $B\_R$ 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.