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

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).