Bubbling of almost critical points of anisotropic isoperimetric problems with degenerating ellipticity

TL;DR

This paper investigates the behavior of almost-critical sets for anisotropic surface energies as the underlying norms degenerate, showing they tend to unions of tangent Wulff shapes.

Contribution

It establishes the rigidity and limit behavior of volume-constrained almost-critical points under degenerating anisotropic norms, extending understanding of anisotropic isoperimetric problems.

Findings

Limits are finite unions of tangent Wulff shapes.

Almost-critical sets converge in L^1 to these unions.

Results hold for arbitrary limiting norms.

Abstract

Given a sequence of uniformly convex norms $ \phi_h $ on $ \mathbf{R}^{n+1} $ converging to an arbitrary norm $ \phi $, we prove rigidity of $ L^1 $-accumulation points of sequences of sets $ E_h \subseteq \mathbf{R}^{n+1} $ of finite perimeter, that are volume-constrained almost-critical points of the anisotropic surface energy functionals associated with $ \phi_h $. Here, almost criticality is measured in terms of the $ L^n $-deviation from being constant of the distributional anisotropic mean $ \phi_h $-curvature of (the varifold associated to) of the reduced boundaries of $ E_h $. We prove that such limits are finite union of disjoint, but possibly mutually tangent, $ \phi $-Wulff shapes.