About H\"older-regularity of the convex shape minimizing {\lambda}2

TL;DR

This paper investigates the regularity of shapes minimizing the second Laplace eigenvalue under area and convexity constraints, proving optimal shapes are Hölder continuous with exponent 1/2 and not smoother.

Contribution

It establishes the precise Hölder regularity of optimal convex shapes for a classical eigenvalue minimization problem, extending results to broader boundary value problems.

Findings

Optimal shapes are ^{1,1/2}

Optimal shapes are not ^{1,eta} for any eta > 1/2

Regularity results apply to general boundary value problems

Abstract

In this paper, we consider the well-known following shape optimization problem: $$\lambda_2(\Omega^*)=\min_{\stackrel{|\Omega|=V_0} {\Omega\textrm{ convex}}} \lambda_2(\Omega),$$ where $\lambda_2(\Om)$ denotes the second eigenvalue of the Laplace operator with homogeneous Dirichlet boundary conditions in $\Om\subset\R^2$, and $|\Om|$ is the area of $\Om$. We prove, under some technical assumptions, that any optimal shape $\Omega^*$ is $\mathcal{C}^{1,\frac{1}{2}}$ and is not $\C^{1,\alpha}$ for any $\alpha>\frac{1}{2}$. We also derive from our strategy some more general regularity results, in the framework of partially overdetermined boundary value problems, and we apply these results to some other shape optimization problems.