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Existence of Anisotropic Minkowski Content

Filip Fryš

TL;DR

This work establishes a sharp criterion for the existence and value of the anisotropic Minkowski content of the topological boundary $\partial E$ of a finite-perimeter set $E$ in an open domain $\Omega$. The author proves that $\mathcal{M}(\partial E; \Omega)$ exists and equals $\mathcal{H}^{n-1}(\Omega\cap\partial^*E)$ precisely when the $C$-anisotropic Minkowski content $\mathcal{M}_C(\partial E; \Omega)$ exists and equals the arithmetic mean $\tfrac12\int_{\Omega\cap\partial^*E}(h_C(\nu_E)+h_C(-\nu_E))\,d\mathcal{H}^{n-1}$, linking boundary regularity to anisotropic perimeters. The main result shows a bi-conditional relationship: the boundary Minkowski content existence is equivalent to the corresponding anisotropic boundary content taking the half-sum of the anisotropic perimeters of $E$ and its complement, a conclusion proved via Besicovitch covering and a key geometric lemma. As a consequence, if anisotropic outer Minkowski content exists for $E$ and for its complement, then the ordinary outer Minkowski content exists for $E$ and its complement. The framework generalizes known results (e.g., Chambolle) by providing converse-type implications and clarifying how the reduced boundary and Lebesgue density structure control these anisotropic notions, with potential implications for variational problems in anisotropic media and materials science.

Abstract

This paper is devoted to the existence of anisotropic Minkowski content and anisotropic outer Minkowski content. Our result is that the Minkowski content of the topological boundary of a given set of finite perimeter $E$ coincides with the perimeter of $E$ if and only if the anisotropic Minkowski content of the topological boundary of $E$ coincides with half of the sum of the anisotropic perimeter of $E$ and the anisotropic perimeter of the complement of $E.$ As a consequence, we find that the existence of anisotropic outer Minkowski content of a given set of finite perimeter and its complement ensures the existence of outer Minkowski content of the set and its complement.

Existence of Anisotropic Minkowski Content

TL;DR

This work establishes a sharp criterion for the existence and value of the anisotropic Minkowski content of the topological boundary of a finite-perimeter set in an open domain . The author proves that exists and equals precisely when the -anisotropic Minkowski content exists and equals the arithmetic mean , linking boundary regularity to anisotropic perimeters. The main result shows a bi-conditional relationship: the boundary Minkowski content existence is equivalent to the corresponding anisotropic boundary content taking the half-sum of the anisotropic perimeters of and its complement, a conclusion proved via Besicovitch covering and a key geometric lemma. As a consequence, if anisotropic outer Minkowski content exists for and for its complement, then the ordinary outer Minkowski content exists for and its complement. The framework generalizes known results (e.g., Chambolle) by providing converse-type implications and clarifying how the reduced boundary and Lebesgue density structure control these anisotropic notions, with potential implications for variational problems in anisotropic media and materials science.

Abstract

This paper is devoted to the existence of anisotropic Minkowski content and anisotropic outer Minkowski content. Our result is that the Minkowski content of the topological boundary of a given set of finite perimeter coincides with the perimeter of if and only if the anisotropic Minkowski content of the topological boundary of coincides with half of the sum of the anisotropic perimeter of and the anisotropic perimeter of the complement of As a consequence, we find that the existence of anisotropic outer Minkowski content of a given set of finite perimeter and its complement ensures the existence of outer Minkowski content of the set and its complement.

Paper Structure

This paper contains 7 sections, 21 theorems, 150 equations.

Table of Contents

  1. Preliminaries
  2. Notation
  3. Geometric Measure Theory
  4. Anisotropic (Outer) Minkowski Content
  5. Basic Properties of Anisotropic (Outer) Minkowski Content
  6. Main Result
  7. Main Result

Key Result

Theorem 1

Let $\Omega\subseteq\mathbb{R}^n$ be open, let $E\subseteq\Omega$ be a set of finite perimeter in $\Omega$, and let $C$ be a convex body a convex body whose interior contains the origin. If then

Theorems & Definitions (50)

  • Theorem : chambolle
  • Definition 1.1
  • Remark 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 1.7
  • Definition 1.8
  • Remark 1.9
  • ...and 40 more