Minimization of anisotropic energies in classes of rectifiable varifolds

TL;DR

This paper studies the minimization of anisotropic energies within classes of rectifiable varifolds, proving convergence of minimizing sequences and conditions for the limit to be integral, advancing geometric measure theory.

Contribution

It establishes convergence results for anisotropic energy minimization in rectifiable varifolds and identifies conditions for the limit varifold to be integral.

Findings

Minimizing sequences with bounded density converge to a rectifiable varifold.

The limit varifold is integral if the sequence is integral with bounded anisotropic variation.

Provides new insights into anisotropic energy minimization in geometric measure theory.

Abstract

We consider the minimization problem of an anisotropic energy in classes of $d$-rectifiable varifolds in $\mathbb R^n$, closed under Lipschitz deformations and encoding a suitable notion of boundary. We prove that any minimizing sequence with density uniformly bounded from below converges (up to subsequences) to a $d$-rectifiable varifold. Moreover, the limiting varifold is integral, provided the minimizing sequence is made of integral varifolds with uniformly locally bounded anisotropic first variation.