On smooth interior approximation of Sets of Finite Perimeter

2022/10/21 by Changfeng Gui, Gui, Changfeng, Yeyao Hu +3 · 1 citation
Computer Science · Mathematics · #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #Analysis of PDEs (math.AP) #FOS: Mathematics #Optimization and Variational Analysis

paper · pdf · doi:10.48550/arxiv.2210.11734

openalex publication_date 2022/10/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we prove that for any bounded set of finite perimeter Ω⊂ ℝⁿ, we can choose smooth sets E_k \Subset Ω such that E_k → Ω in L¹ and \limsup_{i → ∞} P(E_i) ≤ P(Ω)+C₁(n) \mathscrH^n-1(∂ Ω∩ Ω¹).In the above Ω¹ is the measure-theoretic interior of Ω, P(⋅) denotes the perimeter functional on sets, and C₁(n) is a dimensional constant. Conversely, we prove that for any sets E_k \Subset Ω satisfying E_k → Ω in L¹, there exists a dimensional constant C₂(n) such that the following inequality holds: \liminf_{k → ∞} P(E_k) ≥ P(Ω)+ C₂(n) \mathscrH^n-1(∂ Ω∩ Ω¹). In particular, these results imply that for a bounded set Ω of finite perimeter, \mathscrH^n-1(∂ Ω∩ Ω¹)=0 holds if and only if there exists a sequence of smooth sets E_k such that E_k \Subset Ω, E_k → Ω in L¹ and P(E_k) → P(Ω).

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• A Rough Divergence Theorem

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