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

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

(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.