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.

Locations

• arXiv (Cornell University) - View - PDF
• Interfaces and Free Boundaries Mathematical Analysis Computation and Applications - View - PDF