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Nonlocal Estimates of First Derivatives of the Solutions of the Initial Boundary Problem for Nonuniformly Elliptic and Nonuniformly Parabolic Nondivergent Equations

  • N. M. Ivochkina
  • A. P. Oskolkov
Part of the Seminars in Mathematics book series (SM, volume 11)

Abstract

This paper is devoted to a study of nonlocal a priori estimates of maxima of moduli of the first derivatives of solutions of Dirichlet’s problem, and, correspondingly, the first initial-boundary problem for nonuniformly elliptic and nonuniformly parabolic nondivergent quasi-linear equations. It is closely related to known investigations of O. A. Ladyzhenskaya and N. N. Ural’tseva on quasi-linear elliptic and parabolic equations and systems [1, 2]. A characteristic peculiarity of the paper is the fact that the method, developed by O. A. Ladyzhenskaya and N. N. Ural’tseva, for obtaining a priori estimates of maxima of moduli of the first derivatives for solutions of uniformly elliptic and uniformly parabolic quasi-linear equations with divergent principal part, is used here for studying analogous estimates for solutions of nondivergent equations; moreover, the method enables one to investigate specific classes of nonuniformly elliptic and nonuniformly parabolic quasi-linear equations, including those not belonging to S. N. Bernshtein’s class (L).

Keywords

Volume Integral Surface Integral Cauchy Inequality Parabolic Case Obvious Inequality 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1970

Authors and Affiliations

  • N. M. Ivochkina
  • A. P. Oskolkov

There are no affiliations available

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