The average distance problem with perimeter-to-area ratio penalization

TL;DR

This paper investigates the minimization of a functional combining the average distance to the boundary and the perimeter-to-area ratio for convex sets, establishing existence and regularity of solutions.

Contribution

It proves the existence and $C^{1,1}$ regularity of minimizers for a new shape optimization functional involving distance and perimeter-to-area ratio.

Findings

Existence of minimizers is established.

Minimizers are proven to be $C^{1,1}$ regular.

The functional balances distance to boundary with perimeter-to-area ratio.

Abstract

In this paper we consider the functional \begin{equation*} E_{p,\la}(\Omega):=\int_\Omega \dist^p(x,\pd \Omega )\d x+\la \frac{\H^1(\pd \Omega)}{\H^2(\Omega)}. \end{equation*} Here $p\geq 1$, $\la>0$ are given parameters, the unknown $\Omega$ varies among compact, convex, Hausdorff two-dimensional sets of $\R^2$, $\pd \Omega$ denotes the boundary of $\Omega$, and $\dist(x,\pd \Omega):=\inf_{y\in\pd \Omega}|x-y|$. The integral term $\int_\Omega \dist^p(x,\pd \Omega )\d x$ quantifies the "easiness" for points in $\Omega$ to reach the boundary, while $\frac{\H^1(\pd \Omega)}{\H^2(\Omega)}$ is the perimeter-to-area ratio. The main aim is to prove existence and $C^{1,1}$-regularity of minimizers of $\E$.