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On the properties of the set where a generalized function of bounded variation takes infinite value

Alessandro Cucinotta

Abstract

We study the properties of the set where a generalized function of bounded variation has infinite approximate limit, highlighting in this way the main geometric difference with functions of bounded variation. To this aim we prove a new result on strict approximation of sets of finite perimeter from the outside with open sets.

On the properties of the set where a generalized function of bounded variation takes infinite value

Abstract

We study the properties of the set where a generalized function of bounded variation has infinite approximate limit, highlighting in this way the main geometric difference with functions of bounded variation. To this aim we prove a new result on strict approximation of sets of finite perimeter from the outside with open sets.
Paper Structure (3 sections, 28 theorems, 44 equations)

This paper contains 3 sections, 28 theorems, 44 equations.

Table of Contents

  1. Notation and preliminaries
  2. Polar sets
  3. Outer approximation of sets of finite perimeter with open sets

Key Result

Theorem 1.1

GMT If $u \in L^0(\Omega)$ then for $\lambda^N$-almost every $x \in \Omega$ the approximate limit of $u$ at $x$ exists and is finite.

Theorems & Definitions (51)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 1.6
  • Proposition 1.7
  • Proposition 1.8
  • Definition 1.9
  • Proposition 1.10
  • ...and 41 more