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

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

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