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Linear Stability Implies Nonlinear Stability for Faber-Krahn Type Inequalities

For a domain $Ω\subset \mathbb{R}^n$ and a small number $\frak{T} > 0$, let \[ \mathcal{E}_0(Ω) = λ_1(Ω) + {\frak{T}} {\text{tor}}(Ω) = \inf_{u, w \in H^1_0(Ω)\setminus \{0\}} \frac{\int |\nabla u|^2}{\int u^2} + {\frak{T}} \int \frac{1}{2} |\nabla w|^2 - w \] be a modification of the first Dirichlet eigenvalue of $Ω$. It is well-known that over all $Ω$ 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 $Ω$ 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, \[ \int |u_Ω - u_{B_R}|^2 + |Ω\triangle B_R|^2 \leq C [\mathcal{E}_0(Ω) - \mathcal{E}_0(B_R)] \] (i.e. the Faber-Krahn inequality is linearly stable), then the same is true for any $Ω$ with the same volume and barycenter as $B_R$ without any smoothness assumptions (i.e. it is nonlinearly stable). Here $u_Ω$ stands for an $L^2$-normalized first Dirichlet eigenfunction of $Ω$. 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.