Lipschitz Regularity of the Eigenfunctions on Optimal Domains

AbstractWe study the optimal sets $${\Omega^\ast\subseteq\mathbb{R}^d}$$Ω*⊆Rd for spectral functionals of the form $${F\big(\lambda_1(\Omega),\ldots,\lambda_p(\Omega)\big)}$$F(λ1(Ω),…,λp(Ω)), which are bi-Lipschitz with respect to each of the eigenvalues $${\lambda_1(\Omega), \lambda_2(\Omega)}, \ldots, {\lambda_p(\Omega)}$$λ1(Ω),λ2(Ω),…,λp(Ω) of the Dirichlet Laplacian on $${\Omega}$$Ω, a prototype being the problem $$\min{\big\{\lambda_1(\Omega)+\cdots+\lambda_p(\Omega)\;:\;\Omega\subseteq\mathbb{R}^d,\ |\Omega|=1\big\}}.$$min{λ1(Ω)+⋯+λp(Ω):Ω⊆Rd,|Ω|=1}.We prove the Lipschitz regularity of the eigenfunctions $${u_1,\ldots,u_p}$$u1,…,up of the Dirichlet Laplacian on the optimal set $${\Omega^\ast}$$Ω* and, as a corollary, we deduce that $${\Omega^\ast}$$Ω* is open. For functionals depending only on a generic subset of the spectrum, as for example $${\lambda_k(\Omega)}$$λk(Ω), our result proves only the existence of a Lipschitz continuous eigenfunction in correspondence to each of the eigenvalues involved.