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Relaxed area of 0-homogeneous maps in the strict BV-convergence

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Abstract

We compute the relaxed Cartesian area for a general 0-homogeneous map of bounded variation, with respect to the strict BV-convergence. In particular, we show that the relaxed area is finite for this class of maps and we provide an integral representation formula.

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Notes

  1. Clearly, (1.1) is finite if \(v\in W^{1,1}(\Omega ;\mathbb {R}^2)\) and \(Jv\in L^1(\Omega )\).

  2. Notice that \(u\in C^1(\Omega ;\mathbb {R}^2)\cap W^{1,1}(\Omega ;\mathbb {R}^2)\subset BV(\Omega ;\mathbb {R}^2)\). Neverthless \(Ju\notin L^1(\Omega ;\mathbb {R}^2)\), giving \(\overline{\mathcal {A}}_{L^1}(u;\Omega )=\mathcal {A}(u;\Omega )=+\infty \).

  3. If \({\bar{\tau }}=a\) or \({\bar{\tau }}=b\), E is a semi-open interval.

  4. If the number of jumps is finite, then \(\{t_i\}\) is definitively constant.

  5. \({{\mathbb {S}}}^1\) is identified with \([0,2\pi ]\).

  6. We identify \(\partial B_\varepsilon \) with \([0,2\pi \varepsilon ]\).

  7. See Theorem 2’ in [20]: notice that \(f^*_\rho =|\cdot |\) for every \(\rho \in (0,\ell )\), where \(f^*_\rho \) is the recession function associated to \(f_\rho \).

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Acknowledgements

The author is indebted to Giovanni Bellettini and Riccardo Scala for having suggested to write this paper and he thanks Paul Creutz for useful discussions. The author is member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the INdAM of Italy.

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  1. Area of Mathematical Analysis, Modelling, and Applications, Scuola Internazionale Superiore di Studi Avanzati “SISSA”, Via Bonomea, 265-34136, Trieste, Italy

    Simone Carano

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Carano, S. Relaxed area of 0-homogeneous maps in the strict BV-convergence. Annali di Matematica (2024). https://doi.org/10.1007/s10231-024-01435-1

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