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

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Abstract

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.

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Notes

  1. We refer to [20, Sec. 3.1] for the definition of Brouwer degree of a Sobolev map, see also [4, Sec. 2.3].

  2. The strata of \(G_{u_h}\) must converge to the corresponding ones of \(T_u\) in measure and in total variation.

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Acknowledgements

We thank Andrea Marchese for useful discussions. The authors are members of the GNAMPA of INDAM.

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  1. SISSA-Scuola Internazionale Superiore di Studi Avanzati, Via Bonomea 265, 34136, Trieste, Italy

    Simone Carano

  2. DSMFI, Università di Parma, Parco Area delle Scienze 53/A, 43124, Parma, Italy

    Domenico Mucci

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  1. Simone Carano
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  2. Domenico Mucci
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Correspondence to Domenico Mucci.

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Carano, S., Mucci, D. Strict BV relaxed area of Sobolev maps into the circle: the high dimension case. Nonlinear Differ. Equ. Appl. 31, 54 (2024). https://doi.org/10.1007/s00030-024-00941-8

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