The average distance problem with an Euler elastica penalization

Abstract

We consider the minimization of an average distance functional defined on a two-dimensional domain $\Omega$ with an Euler elastica penalization associated with $\pd Ω$, the boundary of $\Omega$. The average distance is given by \begin{equation*} \int_Ω \dist^p(x,\pd Ω)\d x \end{equation*} where $p\ge 1$ is a given parameter, and $\dist(x,\pd Ω)$ is the Hausdorff distance between $\{x\}$ and $\pd Ω$. The penalty term is a multiple of the Euler elastica (i.e., the Helfrich bending energy or the Willmore energy) of the boundary curve ${\pd Ω}$, which is proportional to the integrated squared curvature defined on $\pd Ω$, as given by \begin{equation*} \la \int_{\pd Ω} κ_{\pd Ω}^2\d\H_{\llcorner \pd Ω}^1, \end{equation*} where $κ_{\pd Ω}$ denotes the (signed) curvature of $\pd Ω$ and $\la>0$ denotes a penalty constant. The domain $\Omega$ is allowed to vary among compact, convex sets of ${\mathbb{R}}^2$ with Hausdorff dimension equal to $2$\tcr{.} Under no a priori assumptions on the regularity of the boundary $\pd Ω$, we prove the existence of minimizers of $E_{p,\la}$. Moreover, we establish the $C^{1,1}$-regularity of its minimizers. An original construction of a suitable family of competitors plays a decisive role in proving the regularity.