Lipschitz regularity of the eigenfunctions on optimal domains

Abstract

We study the optimal sets ${\Omega }^{\ast }\subset {\mathbb{R}}^d$ for spectral functionals $F({\lambda }_1(\Omega ),\dots ,{\lambda }_p(\Omega ))$, which are bi-Lipschitz with respect to each of the eigenvalues ${\lambda }_1(\Omega ),\dots ,{\lambda }_p(\Omega )$ of the Dirichlet Laplacian on $\Omega$, a prototype being the problem $$\min \{{\lambda }_1(\Omega )+\cdots +{\lambda }_p(\Omega ):\Omega \subset {\mathbb{R}}^d,|\Omega |=1\}.$$ We prove the Lipschitz regularity of the eigenfunctions $u_1,\dots ,u_p$ of the Dirichlet Laplacian on the optimal set ${\Omega }^{\ast }$ and, as a corollary, we deduce that ${\Omega }^{\ast }$ is open. For functionals depending only on a generic subset of the spectrum, as for example ${\lambda }_k(\Omega )$ or ${\lambda }_{k_1}(\Omega )+\cdots +{\lambda }_{k_p}(\Omega )$, our result proves only the existence of a Lipschitz continuous eigenfunction in correspondence to each of the eigenvalues involved.

Cited by in corpus (5)

• Extremality conditions and regularity of solutions to optimal partition problems involving Laplacian eigenvalues
• Regularity of the free boundary for the vectorial Bernoulli problem
• Regularity of shape optimizers for some spectral fractional problems
• Liouville theorems and optimal regularity in elliptic equations
• Minimizing Eigenvalues of the Fractional Laplacian