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

Abstract

Given a connected bounded open 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 we provide a useful upper bound for its value. Using this estimate, we prove a modified version of a De Giorgi conjecture 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 $\overline{\mathcal{A}}(u,\cdot )$.