*Preprint*

**Inserted:** 6 jul 2023

**Last Updated:** 10 jul 2023

**Year:** 2023

**Abstract:**

Given a bounded open connected Lipschitz set $\Omega\subset\mathbb R^2$, we show that the relaxed Cartesian area functional $\mathcal A(u,\Omega)$ 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 smaller than or equal to $\mathcal A(u,\cdot)$.

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