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Given a bounded open set $\Omega \subset \mathbb{R}^2$, we study therelaxation of the nonparametric area functional in the strict topology in$BV(\Omega;\mathbb{R}^2)$, and compute it for vortex-type maps, and moregenerally for maps in $W^{1,1}(\Omega;\mathcal{S}^1)$ having a finite number oftopological singularities. We also extend the analysis to some specificpiecewise constant maps in $BV(\Omega;\mathcal{S}^1)$, including the symmetrictriple junction map.

The relaxed area of $\mathcal{S}^1$-valued singular maps in the strict $BV$-convergence