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

where $p\ge 1$ is a given parameter and $\mathrm{dist}(x,\partial \Omega )$ 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

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.