Evolutionary, symmetric $p$-Laplacian. Interior regularity of time derivatives and its consequences

Jan  Burczak; P. Kaplický

doi:10.3934/cpaa.2016042

Article Contents

2016, Volume 15, Issue 6: 2401-2445. Doi: 10.3934/cpaa.2016042

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Evolutionary, symmetric $p$-Laplacian. Interior regularity of time derivatives and its consequences

Jan Burczak^1, and
P. Kaplický^2,

1.
Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-656 Warsaw
2.
Charles University, Faculty of Mathematics and Physics, Department of Mathematical Analysis, Sokolovská 83, 186 75 Prague 8

Received: February 29, 2016

Revised: June 30, 2016

Published: October 2016

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Abstract

We consider an evolutionary, non-degenerate, symmetric $p$-Laplacian. By symmetric we mean that the full gradient of $p$-Laplacian is replaced by its symmetric part, which causes a breakdown of the Uhlenbeck structure. We derive interior regularity of time derivatives of its local weak solution. To circumvent the space-time growth mismatch, we devise a new local regularity technique of iterations in Nikolskii-Bochner spaces. It is interesting by itself, as it may be modified to provide new regularity results for the full-gradient $p$-Laplacian case with lower-order dependencies. Finally, having our regularity result for time derivatives, we obtain respective regularity of the main part. The Appendix on Nikolskii-Bochner spaces, that includes theorems on their embeddings and interpolations, may be of independent interest.

Keywords:

Evolutionary systems of PDEs,
symmetric $p$-Laplacian,
local (interior) regularity of time derivatives,
iteration in Nikolskii-Bochner spaces.

Mathematics Subject Classification: 35K55, 35K59, 35K92, 35Q35, 35B65.

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