Upper bounds for the relaxed area of \mathbb S¹-valued Sobolev maps and its countably subadditive interior envelope

2023/07/13 by Giovanni Bellettini, Bellettini, Giovanni, Riccardo Scala +3
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2307.06885

openalex publication_date 2023/07/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

Given a bounded open connected Lipschitz set Ω⊂ \mathbb R², we show that the relaxed Cartesian area functional \mathcal A(u,Ω) of a map u∈ W^1,1(Ω;\mathbb S¹) is finite, and provide a useful upper bound for its value. Using this estimate, we prove a modified version of a De Giorgi conjecture [17] adapted to W^1,1(Ω;\mathbb S¹), on the largest countably subadditive set function \mathcal A(u, ⋅) smaller than or equal to \mathcal A(u,⋅).

Related

• Strict BV relaxed area of Sobolev maps into the circle: the high dimension case
• The refined area formula for Sobolev mappings W^k,p
• Optimal limiting embeddings for Δ-reduced Sobolev spaces in L¹
• Relaxed area of graphs of piecewise Lipschitz maps in the strict BV-convergence
• Mountain Pass Critical Points of the Volume Constrained Area Functional