The average-distance problem with an Euler elastica penalization

We consider the minimization of an average distance functional defined on a two-dimensional domain Ω with an Euler elastica penalization associated with ∂Ω, the boundary of Ω. The average distance is given by ∫ Ω dist(x, ∂Ω) dx where p ≥ 1 is a given parameter, and dist(x, ∂Ω) is the Hausdorff distance between {x} and ∂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 ∂Ω, which is proportional to the integrated squared curvature defined on ∂Ω, as given by λ ∫ ∂Ω κ∂Ω dH x∂Ω, where κ∂Ω denotes the (signed) curvature of ∂Ω and λ > 0 denotes a penalty constant. The domain Ω is allowed to vary among compact, convex sets of R2 with Hausdorff dimension equal to 2. Under no a priori assumptions on the regularity of the boundary ∂Ω, we prove the existence of minimizers of Ep,λ. Moreover, we establish the C1,1-regularity of its minimizers. An original construction of a suitable family of competitors plays a decisive role in proving the regularity.