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A Relaxation Result in the Vectorial Setting and Power Law Approximation for Supremal Functionals

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Abstract

We provide relaxation for not lower semicontinuous supremal functionals defined on vectorial Lipschitz functions, where the Borel level convex density depends only on the gradient. The connection with indicator functionals is also enlightened, thus extending previous lower semicontinuity results in that framework. Finally, we discuss the power law approximation of supremal functionals, with nonnegative, coercive densities having explicit dependence also on the spatial variable, and satisfying minimal measurability assumptions.

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Acknowledgements

EZ is indebted with Dipartimento di Matematica of University of Ferrara for its kind support and hospitality. The authors thank the editors and the anonymous referee for their suggestions and comments. Both the authors are members of GNAMPA-INdAM, whose support is gratefully acknowledged.

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Authors and Affiliations

  1. Dip. di Matematica e Informatica, Università di Ferrara, Via Machiavelli 35, 44121, Ferrara, Italy

    Francesca Prinari

  2. Dip. di Ingegneria Industriale, Università degli Studi di Salerno, Via Giovanni Paolo II, 132, 84084, Fisciano, SA, Italy

    Elvira Zappale

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  1. Francesca Prinari
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  2. Elvira Zappale
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Correspondence to Francesca Prinari or Elvira Zappale.

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Communicated by Giuseppe Buttazzo.

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Prinari, F., Zappale, E. A Relaxation Result in the Vectorial Setting and Power Law Approximation for Supremal Functionals. J Optim Theory Appl 186, 412–452 (2020). https://doi.org/10.1007/s10957-020-01712-y

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