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We deal with the relaxed area functional in the strict $BV$-convergence ofnon-smooth maps defined in domains of generic dimension and taking values intothe unit circle. In case of Sobolev maps, a complete explicit formula isobtained. Our proof is based on tools from Geometric Measure Theory andCartesian currents. We then discuss the possible extension to the wider classof maps with bounded variation. Finally, we show a counterexample to thelocality property in case of both dimension and codimension larger than two.

Strict BV relaxed area of Sobolev maps into the circle: the high dimension case