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Parabolic interior Schauder estimates by the maximum principle

  • Barry F. Knerr
Article

Abstract

In [1] Brandt proved all of the assertions of the parabolic interior Schauder estimates regarding Hölder continuity in x (exponent α) by a very simple maximum principle argument. In this paper we give a simple maximum principle proof of Hölder continuity in t (exponent α/2). In fact we show that each derivative D x 2 u is Hölder continuous in t (exponent α/2) even if the coefficients and nonhomogeneous term are not.

Keywords

Neural Network Complex System Nonlinear Dynamics Maximum Principle Electromagnetism 
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References

  1. 1.
    Brandt, A., Interior Schauder estimates for parabolic differential-(or difference-) equations. Israel J. Math. 7, 254–262 (1969).Google Scholar
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    Friedman, A., Partial differential equations of the parabolic type. Englewood Cliffs, W.J.: Prentice-Hall 1964.Google Scholar
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    Krushkov, S. N., Results concerning the nature of the continuity of solutions of parabolic equations and some of their applications. Math. Notes 6, 517–523 (1969).Google Scholar
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    Gilding, B. H., Hölder continuity of solutions of parabolic equations. J. London Math. Soc. (2) 13, 103–106 (1976).Google Scholar
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    Ladyženskaja, O. A., & V. A. Solonnikov, & N. N. Uralceva, Linear and quasilinear equations of the parabolic type. Transl. Math. Monographs, Vol. 23, Amer. Math. Soc., 1968.Google Scholar

Copyright information

© Springer-Verlag 1980

Authors and Affiliations

  • Barry F. Knerr
    • 1
  1. 1.University of KentuckyLexington

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