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Journal of Convex Analysis 27 (2020), No. 3, 845--879
Copyright Heldermann Verlag 2020

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

Marco Pozzetta
Dip. di Matematica, UniversitÓ di Pisa, 56127 Pisa, Italy


\newcommand{\con}{\subset} \newcommand{\cH}{\mathcal{H}} \newcommand{\fp}{\cF_p} \newcommand{\cF}{\mathcal{F}} \newcommand{\pa}{\partial} Under suitable regularity assumptions, the $p$-elastic energy of a planar set $E\con\mathbb{R}^2$ is defined as \begin{equation*} \fp(E)=\int_{\pa E} 1 + |k_{\pa E}|^p \,\, d\cH^1, \end{equation*} 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; \emph{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.

Keywords: Curvature varifolds, p-elastic energy, relaxation.

MSC: 49Q15, 49Q20, 49Q10, 53A07.

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