Optimal domains for the Cheeger inequality
Dorin Bucur, Giuseppe Buttazzo, Alexis de Villeroché
TL;DR
This work studies the scale-invariant shape functional ${\mathcal F}_{p,q}(\Omega)$ linking the first Dirichlet eigenvalues of the $p$- and $q$-Laplacians via $p>q$, establishing existence of optimal domains in $\mathbb{R}^d$ for both the infimum and supremum and clarifying when the supremum is finite (namely, $M(p,q)<\infty$ iff $q>d$). It shows that optimizers may be unbounded and, for $p>q>d$, can have complements that are discrete sets; in the special case $p=\infty$, it provides $\mathcal F_{\infty,q}(\Omega)=\frac{1}{\rho(\Omega)\lambda_q^{1/q}(\Omega)}$ and analyzes the constrained problem under a fixed box via $p$-capacitary measures and $\gamma_p$-convergence. Under a bounded constraint $D$, the paper proves the existence of optimal $p$-quasi-open domains by passing to $\gamma_p$- and $\gamma_q$-limits, highlighting the loss of scale invariance and the need for capacitary tools. It also investigates a nonlinear gap $J_{p,q}=\lambda_p-\lambda_q$ and shows that, up to homothety, minimizers and maximizers agree with the $\mathcal F_{p,q}$-optimal shapes, and it outlines several open questions, suggesting a rich interplay between spectral optimization, capacitary measures, and geometric configurations.
Abstract
In this paper we consider the scale invariant shape functional $${\mathcal{F}}_{p,q}(Ω)=\frac{λ_p^{1/p}(Ω)}{λ_q^{1/q}(Ω)},$$ where $1\le q<p\le+\infty$ and $λ_p(Ω)$ (respectively $λ_q(Ω)$) is the first eigenvalue of the $p$-Laplacian $-Δ_p$ (respectively $-Δ_q$) with Dirichlet boundary condition on $\partialΩ$. We study both the maximization and minimization problems for ${\mathcal{F}}_{p,q}$, and show the existence of optimal domains in ${\mathbb{R}}^d$, along with some of their qualitative properties. Surprisingly, the case of a bounded box $D$ constraint $$\max\Big\{λ_q(Ω)\ :\ Ω\subset D,\ λ_p(Ω)=1\Big\},$$ leads to a problem of different nature, for which the existence of a solution is shown by analyzing optimal capacitary measures. In the last section we list some interesting questions that, in our opinion, deserve to be investigated.