Calculus of Variations and Geometric Measure Theory
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M. Pozzetta

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

created by pozzetta1 on 27 Feb 2019
modified on 25 Jan 2021

[BibTeX]

Published Paper

Inserted: 27 feb 2019
Last Updated: 25 jan 2021

Journal: Journal of Convex Analysis
Volume: 27
Number: 3
Pages: 845-879
Year: 2020
Links: arXiv, PDF

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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