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Sharp quantitative stability of the Dirichlet spectrum near the ball.

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Titel: Sharp quantitative stability of the Dirichlet spectrum near the ball.
Autoren: Bucur, Dorin1 (AUTHOR) dorin.bucur@univ-savoie.fr, Lamboley, Jimmy2,3 (AUTHOR), Nahon, Mickaël4 (AUTHOR), Prunier, Raphaël5 (AUTHOR)
Quelle: Communications on Pure & Applied Mathematics. Nov2025, p1. 70p.
Schlagwörter: *EIGENVALUES, *SPECTRAL theory, *LAPLACIAN operator, *BOUNDARY value problems, *MATHEMATICAL inequalities, *PERTURBATION theory
Abstract: Let Ω⊂Rn$\Omega \subset \mathbb {R}^n$ be an open set with the same volume as the unit ball B$B$ and let λk(Ω)$\lambda _k(\Omega)$ be the k$k$‐th eigenvalue of the Laplace operator of Ω$\Omega$ with Dirichlet boundary conditions on ∂Ω$\partial \Omega$. In this work, we answer the following question: If λ1(Ω)−λ1(B)$\lambda _1(\Omega)-\lambda _1(B)$ is small, how large can |λk(Ω)−λk(B)|$|\lambda _k(\Omega)-\lambda _k(B)|$ be? We establish quantitative bounds of the form |λk(Ω)−λk(B)|≤C(λ1(Ω)−λ1(B))α$|\lambda _k(\Omega)-\lambda _k(B)|\le C (\lambda _1(\Omega)-\lambda _1(B))^\alpha$ with sharp exponents α$\alpha$ depending on the multiplicity of λk(B)$\lambda _k(B)$. We first show that such an inequality is valid with α=1/2$\alpha =1/2$ for any k$k$, 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$\alpha =1$ if λk(B)$\lambda _{k}(B)$ is simple. We also obtain a similar result for the whole cluster of eigenvalues when λk(B)$\lambda _{k}(B)$ 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.If λ1(Ω)−λ1(B)$\lambda _1(\Omega)-\lambda _1(B)$ is small, how large can |λk(Ω)−λk(B)|$|\lambda _k(\Omega)-\lambda _k(B)|$ be? [ABSTRACT FROM AUTHOR]
Datenbank: Academic Search Index
Beschreibung
Abstract:Let Ω⊂Rn$\Omega \subset \mathbb {R}^n$ be an open set with the same volume as the unit ball B$B$ and let λk(Ω)$\lambda _k(\Omega)$ be the k$k$‐th eigenvalue of the Laplace operator of Ω$\Omega$ with Dirichlet boundary conditions on ∂Ω$\partial \Omega$. In this work, we answer the following question: <italic>If</italic> λ1(Ω)−λ1(B)$\lambda _1(\Omega)-\lambda _1(B)$ <italic>is small, how large can</italic> |λk(Ω)−λk(B)|$|\lambda _k(\Omega)-\lambda _k(B)|$ <italic>be?</italic> We establish quantitative bounds of the form |λk(Ω)−λk(B)|≤C(λ1(Ω)−λ1(B))α$|\lambda _k(\Omega)-\lambda _k(B)|\le C (\lambda _1(\Omega)-\lambda _1(B))^\alpha$ with sharp exponents α$\alpha$ depending on the multiplicity of λk(B)$\lambda _k(B)$. We first show that such an inequality is valid with α=1/2$\alpha =1/2$ for any k$k$, 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$\alpha =1$ if λk(B)$\lambda _{k}(B)$ is simple. We also obtain a similar result for the whole cluster of eigenvalues when λk(B)$\lambda _{k}(B)$ 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.<italic>If</italic> λ1(Ω)−λ1(B)$\lambda _1(\Omega)-\lambda _1(B)$ <italic>is small, how large can</italic> |λk(Ω)−λk(B)|$|\lambda _k(\Omega)-\lambda _k(B)|$ <italic>be?</italic> [ABSTRACT FROM AUTHOR]
ISSN:00103640
DOI:10.1002/cpa.70021