Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Mario Santilli

Abstract

Given a sequence of uniformly convex norms $ φ_h $ on $ \mathbf{R}^{n+1} $ converging to an arbitrary norm $ φ$, 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 $ φ_h $. Here, almost criticality is measured in terms of the $ L^n $-deviation from being constant of the distributional anisotropic mean $ φ_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, $ φ$-Wulff shapes.

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

Abstract

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

Paper Structure

This paper contains 4 sections, 8 theorems, 175 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Anisotropic Curvatures of closed sets
  4. Main results

Key Result

Corollary 1.1

Suppose $\phi$ is an arbitrary norm of $\mathbf{R}^{n+1}$ and $\{\phi_h\}_{h \in \mathscr{P}}$ is a sequence of uniformly convex $\mathscr{C}^{3}$-norms of $\mathbf{R}^{n+1}$ pointwise converging to $\phi$. Suppose $\lambda > 0$, $\overline{r} = \tfrac{n}{\lambda}$ and $\{\Omega_h\}_{h \in \mathscr{ where $H^{\phi_h}_{\Omega_h}$ is the mean $\phi_h$-curvature of $\partial\Omega_h$. Then there exis

Theorems & Definitions (19)

  • Corollary 1.1
  • Theorem 2.10
  • proof
  • Remark 2.11
  • Lemma 3.1: cf. HugSantilli
  • Remark 3.2
  • Lemma 3.3: cf. HugSantilli
  • Lemma 3.4
  • proof
  • Theorem 3.5
  • ...and 9 more