A varifold perspective on the $p$-elastic energy of planar sets

TL;DR

This paper introduces a varifold-based framework to analyze the $p$-elastic energy of planar sets, providing new tools, relaxation characterizations, and applications to inpainting, advancing geometric measure theory and shape analysis.

Contribution

It develops a varifold approach to define and study the $p$-elastic energy, including existence, bounds, and relaxation, with applications to inpainting and shape regularity.

Findings

Established existence and bounds for curvature varifolds with finite elastic energy.

Characterized the $L^1$-relaxation of the $p$-elastic energy using varifold methods.

Applied the framework to inpainting problems and analyzed properties of finite energy sets.

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.