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 $\\partial\\Omega$, the boundary of $\\Omega$. The average distance is given by $$ \\int\_{\\Omega}\\operatorname{dist}^p(x,\\partial\\Omega)\\operatorname{d}x, $$ where $p\\geq 1$ is a given parameter and $\\operatorname{dist}(x,\\partial\\Omega)$ is the Hausdorff distance between ${x}$ and $\\partial\\Omega$. The penalty term is a multiple of the Euler elastica (i.e., the Helfrich bending energy or the Willmore energy) of the boundary curve ${\\partial\\Omega}$, which is proportional to the integrated squared curvature defined on $\\partial\\Omega$, as given by $$ \\lambda\\int{\\partial\\Omega} \\kappa{\\partial\\Omega}^2 \\operatorname{d}\\mathcal{H}\_{\\llcorner\\partial\\Omega}^1, $$ where $\\kappa{\\partial\\Omega}$ denotes the (signed) curvature of $\\partial\\Omega$ and $\\lambda>0$ denotes a penalty constant. The domain $\\Omega$ is allowed to vary among compact, convex sets of $\\mathbb{R}^2$ with Hausdorff dimension equal to two. Under no a priori assumptions on the regularity of the boundary $\\partial\\Omega$, we prove the existence of minimizers of $E{p,\\lambda}$. 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.