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

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