Gradient Young measures, varifolds, and a generalized Willmore functional

TL;DR

This paper introduces a new framework combining varifolds and Young measures to analyze the relaxation of a generalized Willmore functional in BV spaces, extending the understanding of geometric variational problems.

Contribution

It develops a novel approach integrating varifolds and Young measures for the relaxation of the generalized Willmore functional in BV, advancing geometric measure theory methods.

Findings

Established a relaxation framework for the generalized Willmore functional.

Connected varifold and Young measure techniques for BV function analysis.

Provided insights into the lower semicontinuity and relaxation of geometric energies.

Abstract

Being Omega an open and bounded Lipschitz domain of R^n, we consider the generalized Willmore functional on Omega defined, for smooth functions, as the p-Willmore energy of each isolevel set integrated over all levels. We propose a new framework, that combines varifolds and Young measures, to study the relaxation of this functional in BV(Omega) with respect to the strong topology of L^1.