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

Bellettini, Giovanni; Carano, Simone; Scala, Riccardo

ESAIM: Control, Optimisation and Calculus of Variations, Tome 28 (2022), article no. 56

Given a bounded open set Ω ⊂ ℝ², we study the relaxation of the nonparametric area functional in the strict topology in BV(Ω; ℝ²). and compute it for vortex-type maps, and more generally for maps in W^1,1(Ω;S¹) having a finite number of topological singularities. We also extend the analysis to some specific piecewise constant maps in BV(Ω;S¹), including the symmetric triple junction map.

Classification : 49J45, 49Q05, 49Q15, 28A75
Keywords: Area functional, relaxation, Cartesian currents, total variation of the Jacobian, S¹-valued singular maps

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