Non-autonomous functionals, borderline cases and related function classes

Baroni, P.; Colombo, M.; Mingione, G.

doi:10.1090/spmj/1392

Non-autonomous functionals, borderline cases and related function classes
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by P. Baroni, M. Colombo and G. Mingione

St. Petersburg Math. J. 27 (2016), 347-379

DOI: https://doi.org/10.1090/spmj/1392

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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