Calculus of Variations and Geometric Measure Theory

G. Bellettini - S. Carano - R. Scala

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

created by scala on 10 Feb 2022
modified by carano on 14 Mar 2023


Published Paper

Inserted: 10 feb 2022
Last Updated: 14 mar 2023

Journal: ESAIM: Control, Optimization and Calculus of Variations
Volume: 28
Number: 56
Pages: 38
Year: 2022


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

Keywords: relaxation, area functional, Cartesian currents, total variation of the Jacobian, $\mathcal{S}^1$-valued singular maps