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Non-autonomous functionals, borderline cases and related function classes

Authors: P. Baroni, M. Colombo and G. Mingione
Original publication: Algebra i Analiz, tom 27 (2015), nomer 3.
Journal: St. Petersburg Math. J. 27 (2016), 347-379
MSC (2010): Primary 49N60, 49J10, 35J20
Published electronically: March 30, 2016
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Abstract: The class of non-autonomous functionals under study is characterized by the fact that the energy density changes its ellipticity and growth properties according to the point; some regularity results are proved for related minimizers. These results are the borderline counterpart of analogous ones previously derived for non-autonomous functionals with $ (p,q)$-growth. Also, similar functionals related to Musielak-Orlicz spaces are discussed, in which basic properties like the density of smooth functions, the boundedness of maximal and integral operators, and the validity of Sobolev type inequalities are naturally related to the assumptions needed to prove the regularity of minima.

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Additional Information

P. Baroni
Affiliation: Dipartamento di Matematica e Application “R. Caccioppoli”, Università a degli studi di Napoli “Federico II”, I-80125 Napoli, Italy

M. Colombo
Affiliation: Institute for Theoretical Studies, ETH Zürich, Clausiusstrasse 47, CH-8092 Zürich, Switzerland; Institute for Mathematik, Universitaet Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland

G. Mingione
Affiliation: Dipartimento di Matematica e Informatica, Università di Parma, Parco Area delle Scienze 53/a, Campus, 43100 Parma, Italy

Keywords: Functionals with nonstandard growth, H\"older regularity of minimizers
Received by editor(s): October 10, 2014
Published electronically: March 30, 2016
Additional Notes: This work is partially supported by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). Part of this work was done while the first and last named authors were visiting Centro De Giorgi and Scuola Normale Superiore at Pisa. Last but not least, the authors would like to thank the referee for his/her valuable comments and his/her interest in the paper
Dedicated: To Nina Nikolaevna Ural’tseva, with gratitude and admiration for all her beautiful mathematics
Article copyright: © Copyright 2016 American Mathematical Society