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).
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Literature Cited
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Ivochkina, N.M., Oskolkov, A.P. (1970). Nonlocal Estimates of First Derivatives of the Solutions of the Initial Boundary Problem for Nonuniformly Elliptic and Nonuniformly Parabolic Nondivergent Equations. In: Ladyzhenskaya, O.A. (eds) Boundary Value Problems of Mathematical Physics and Related Aspects of Function Theory. Seminars in Mathematics, vol 11. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-4666-2_1
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DOI: https://doi.org/10.1007/978-1-4757-4666-2_1
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