Upper bounds for the relaxed area of $\mathbb S^1$-valued Sobolev maps and its countably subadditive interior envelope

TL;DR

This paper establishes an upper bound for the relaxed area functional of circle-valued Sobolev maps in Lipschitz domains and proves a modified De Giorgi conjecture related to the countably subadditive envelope of this functional.

Contribution

It provides a new upper bound for the relaxed area functional of b1-valued Sobolev maps and introduces a modified De Giorgi conjecture for the associated set function.

Findings

The relaxed Cartesian area functional is finite for b1-valued Sobolev maps.

An upper bound for the relaxed area functional is derived.

A modified De Giorgi conjecture is proved for the countably subadditive envelope.

Abstract

Given a bounded open connected Lipschitz set $\Omega \subset \mathbb R^2$, we show that the relaxed Cartesian area functional $\overline{\mathcal A}(u,\Omega)$ of a map $u\in W^{1,1}(\Omega;\mathbb S^1)$ 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 $W^{1,1}(\Omega;\mathbb S^1)$, on the largest countably subadditive set function $\overline {\overline{\mathcal A}}(u, \cdot)$ smaller than or equal to $\overline{\mathcal A}(u,\cdot)$.