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On the Regularity of Minima of Non-autonomous Functionals

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Abstract

We consider regularity issues for minima of non-autonomous functionals in the Calculus of Variations exhibiting non-uniform ellipticity features. We provide a few sharp regularity results for local minimizers that also cover the case of functionals with nearly linear growth. The analysis is carried out provided certain necessary approximation-in-energy conditions are satisfied. These are related to the occurrence of the so-called Lavrentiev phenomenon that non-autonomous functionals might exhibit, and which is a natural obstruction to regularity. In the case of vector valued problems, we concentrate on higher gradient integrability of minima. Instead, in the scalar case, we prove local Lipschitz estimates. We also present an approach via a variant of Moser’s iteration technique that allows to reduce the analysis of several non-uniformly elliptic problems to that for uniformly elliptic ones.

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Acknowledgements

This work is supported by the Engineering and Physical Sciences Research Council (EPSRC): CDT Grant Ref. EP/L015811/1.

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Authors and Affiliations

  1. Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX26GG, UK

    Cristiana De Filippis

  2. Dipartimento SMFI, Università di Parma, Viale delle Scienze 53/a, Campus, 43124, Parma, Italy

    Giuseppe Mingione

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  1. Cristiana De Filippis
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  2. Giuseppe Mingione
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Correspondence to Giuseppe Mingione.

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De Filippis, C., Mingione, G. On the Regularity of Minima of Non-autonomous Functionals. J Geom Anal 30, 1584–1626 (2020). https://doi.org/10.1007/s12220-019-00225-z

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