Regularity and singularities of Optimal Convex shapes in the plane

TL;DR

This paper investigates the regularity and singularities of optimal convex shapes in the plane for shape optimization problems, establishing conditions under which solutions are smooth or polygonal using first and second order optimality conditions.

Contribution

It provides new regularity results for optimal convex shapes, showing they are either smooth or polygonal depending on the convexity or concavity of the shape functional.

Findings

Optimal shapes are $W^{2,p}$-sets under convexity assumptions.

Optimal shapes are polygons under concavity assumptions.

Results apply to functionals involving area, energy, eigenvalues, and perimeter.

Abstract

We focus here on the analysis of the regularity or singularity of solutions $\Om_{0}$ to shape optimization problems among convex planar sets, namely: $$ J(\Om_{0})=\min\{J(\Om),\ \Om\ \textrm{convex},\ \Omega\in\mathcal S_{ad}\}, $$ where $\mathcal S_{ad}$ is a set of 2-dimensional admissible shapes and $J:\mathcal{S}_{ad}\rightarrow\R$ is a shape functional. Our main goal is to obtain qualitative properties of these optimal shapes by using first and second order optimality conditions, including the infinite dimensional Lagrange multiplier due to the convexity constraint. We prove two types of results: i) under a suitable convexity property of the functional $J$, we prove that $\Omega_0$ is a $W^{2,p}$-set, $p\in[1,\infty]$. This result applies, for instance, with $p=\infty$ when the shape functional can be written as $ J(\Omega)=R(\Omega)+P(\Omega), $ where $R(\Om)=F(|\Om|,E_{f}(\Om),\la_{1}(\Om))$ involves the area $|\Omega|$, the Dirichlet energy $E_f(\Omega)$ or the first eigenvalue of the Laplace-Dirichlet operator $\lambda_1(\Omega)$, and $P(\Omega)$ is the perimeter of $\Om$, ii) under a suitable concavity assumption on the functional $J$, we prove that $\Omega_{0}$ is a polygon. This result applies, for instance, when the functional is now written as $ J(\Om)=R(\Omega)-P(\Omega), $ with the same notations as above.