The average-distance problem with an Euler elastica penalization

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