Linear Stability Implies Nonlinear Stability for Faber-Krahn Type Inequalities

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.