A Varifold Perspective on the p-Elastic Energy of Planar Sets

Under suitable regularity assumptions, the $p$-elastic energy of a planar set $E\con\mathbb{R}^2$ is defined as $$\fp(E)=\int_{\pa E} 1 + |k_{\pa E}|^p \,\, d\cH^1,$$ where $k_{\pa E}$ is the curvature of the boundary $\pa E$. In this work we use a varifold approach to investigate this energy, that can be well defined on varifolds with curvature. First we show new tools for the study of $1$-dimensional curvature varifolds, such as existence and uniform bounds on the density of varifolds with finite elastic energy. Then we characterize a new notion of $L^1$-relaxation of this energy by extending the definition of regular sets by an intrinsic varifold perspective, also comparing this relaxation with the classical one of G.\,Bellettini and L.\,Mugnai [{\it Characterization and representation of the lower semicontinuous envelope of the elastica functional}, Annales de l'Institut Henri Poincar\'{e} (C), Non Linear Analysis 21(6) (2004) 839--880; A varifolds representation of the relaxed elastica functional, J. Convex Analysis 14(3) (2007) 543--564]. Finally we discuss an application to the inpainting problem, examples and qualitative properties of sets with finite relaxed energy.