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Regularity of weak solutions to the model Venttsel problem for linear parabolic systems with nonsmooth in time principal matrix: A(t)-caloric approximation method

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Abstract

We consider the model Venttsel type problem for linear parabolic systems of equations. The Venttsel type boundary condition is fixed on the flat part of the lateral surface of a given cylinder. It is defined by a parabolic operator (with respect to the tangential derivatives) and the conormal derivative. The Hölder continuity of every weak solution of the problem is proved under optimal assumptions on the data. In particular, only boundedness in the time variable of the principal matrices of the system and the boundary operator is assumed. All results are obtained by so-called A(t)-caloric approximation method (Arkhipova et al. in Nonlinear Anal 95:421–435, 2014).

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Acknowledgments

The author’s research has been financially supported by the Russian Fond of the Basic Research (RFBR), Grant 15-01-07650. The author wish to thank the referee for a careful reading of the paper and for numerous helpful comments.

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  1. Saint Petersburg State University, Universitetskaya nab. 7/9, St. Petersburg, Russia, 199034

    Arina A. Arkhipova

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  1. Arina A. Arkhipova
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Correspondence to Arina A. Arkhipova.

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Arkhipova, A.A. Regularity of weak solutions to the model Venttsel problem for linear parabolic systems with nonsmooth in time principal matrix: A(t)-caloric approximation method. manuscripta math. 151, 519–548 (2016). https://doi.org/10.1007/s00229-016-0838-y

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  • DOI: https://doi.org/10.1007/s00229-016-0838-y

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