*Accepted Paper*

**Inserted:** 27 feb 2019

**Last Updated:** 26 nov 2019

**Journal:** Journal of Convex Analysis

**Year:** 2020

**Abstract:**

Under suitable regularity assumptions the $p$-elastic energy of a planar set $E\subset\mathbb{R}^2$ is defined to be
$\int_{\partial E} 1 +

k_{\partial E}

^p \,\, d\mathcal{H}^1,$
where $k_{\partial E}$ is the curvature of the boundary $\partial E$. 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 ones. Finally we discuss an application to the inpainting problem, examples and qualitative properties of sets with ﬁnite relaxed energy.

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