Calculus of Variations and Geometric Measure Theory

S. Carano - D. Mucci

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

created by carano on 30 Nov 2023
modified on 30 Mar 2024


Accepted Paper

Inserted: 30 nov 2023
Last Updated: 30 mar 2024

Journal: NoDEA
Year: 2023


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

Keywords: relaxation, distributional jacobian, area functional, Cartesian currents, strict convergence, $\mathbb{S}^1$ valued singular maps