Sharp Quantitative Stability of the Dirichlet spectrum near the ball

TL;DR

This work analyzes how small deficits in the first Dirichlet eigenvalue $λ_1(Ω)$ around the ball control higher Dirichlet eigenvalues. By combining quantitative Saint-Venant and Faber–Krahn-type arguments with a vectorial free boundary framework, the authors obtain sharp exponents: a universal square-root bound $|λ_k(Ω)-λ_k(B)|≤C(·)igl(λ_1(Ω)-λ_1(B)igr)^{1/2}$ for all $k$, and a linear bound on clusters when $λ_k(B)$ is simple or when the cluster arises from a simple eigenvalue; a vectorial extension handles multiple eigenvalues. The ball is shown to persist as the minimizer for a broad class of spectral functionals and a full reverse Kohler–Jobin inequality is established, solving open questions in the field. The results provide quantitative, scale-invariant stability of the ball under spectral perturbations, with implications for spectral optimization and shape optimization problems. The techniques blend spectral growth estimates, torsion-based inequalities, and sophisticated regularity theory for (vectorial) free boundary problems.

Abstract

Let $Ω\subset\mathbb{R}^n$ be an open set with the same volume as the unit ball $B$ and let $λ_k(Ω)$ be the $k$-th eigenvalue of the Laplace operator of $Ω$ with Dirichlet boundary conditions on $\partialΩ$. In this work, we answer the following question: if $λ_1(Ω)-λ_1(B)$ is small, how large can $|λ_k(Ω)-λ_k(B)|$ be ? We establish quantitative bounds of the form $|λ_k(Ω)-λ_k(B)|\le C (λ_1(Ω)-λ_1(B))^α$ with sharp exponents $α$ depending on the multiplicity of $λ_k(B)$. We first show that such an inequality is valid with $α=1/2$ for any $k$, improving previous known results and providing the sharpest possible exponent. Then, through the study of a vectorial free boundary problem, we show that one can achieve the better exponent $α=1$ if $λ_{k}(B)$ is simple. We also obtain a similar result for the whole cluster of eigenvalues when $λ_{k}(B)$ is multiple, thus providing a complete answer to the question above. As a consequence of these results, we obtain the persistence of the ball as the minimizer for a large class of spectral functionals which are small perturbations of the fundamental eigenvalue on the one hand, and a full reverse Kohler-Jobin inequality on the other hand, solving an open problem formulated by M. Van Den Berg, G. Buttazzo and A. Pratelli.

Theorems & Definitions (93)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Theorem 1.5
• Corollary 1.6
• Proposition 2.1
• Lemma 2.2
• proof
• Lemma 2.3