Bounded variation and relaxed curvature of surfaces

TL;DR

This paper introduces a relaxed energy concept for non-parametric surfaces that incorporates area, mean, and Gauss curvature, analyzing BV and measure properties, and revisiting classical curvature counterexamples.

Contribution

It proposes a new relaxed energy framework for surfaces considering curvature and area, and studies the BV and measure properties of functions with finite relaxed energy.

Findings

Characterization of BV and measure properties of functions with finite relaxed energy

Analysis of Schwarz-Peano counterexample in the context of total curvature

Development of approximation methods using inscribed polyhedral surfaces

Abstract

We consider a relaxed notion of energy of non-parametric codimension one surfaces that takes account of area, mean curvature, and Gauss curvature. It is given by the best value obtained by approximation with inscribed polyhedral surfaces. The BV and measure properties of functions with finite relaxed energy are studied. Concerning the total mean and Gauss curvature, the classical counterexample by Schwarz-Peano to the definition of area is also analyzed.