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

Giovanni Bellettini, Riccardo Scala, Giuseppe Scianna

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}(Ω;\mathb…