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Given a bounded open connected Lipschitz set $\Omega \subset \mathbb R^2$, weshow that the relaxed Cartesian area functional $\overline{\mathcalA}(u,\Omega)$ of a map $u\in W^{1,1}(\Omega;\mathbb S^1)$ is finite, andprovide a useful upper bound for its value. Using this estimate, we prove amodified version of a De Giorgi conjecture [17] adapted to$W^{1,1}(\Omega;\mathbb S^1)$, on the largest countably subadditive setfunction $\overline {\overline{\mathcal A}}(u, \cdot)$ smaller than or equal to$\overline{\mathcal A}(u,\cdot)$.

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