Some boundary value problems for nonlinear elliptic equations of second order in high-dimensional domains§

Abstract

In [Codres, H.O., 1956, Über die Randwertaufgabe bei quasilinearen Differentialgleichungen zweiter Ordnung in mehr als zwei Variablen. Mathematische Annalen, 131, 278–312.], Cordes mainly gave some a priori estimates of classical solutions of the Dirichlet problem for quasilinear elliptic equations of second order §Dedicated to Professor Wei Lin on the occasion of his 70th birthday. In [Alkhutov, Yu.A. and Mamedov, I.T., 1988, The first boundary value problem for nondivergence second order parabolic equations with discontinuous coefficients. Mathematical USSR Sbornik, 59, 471–495.] Alkhutov and Mamedov discussed the solvability of the Dirichlet problem for the linear uniformly parabolic equation with measurable coefficients: where the coefficients satisfy the conditions: In the present article, we try to give estimates of solutions of some boundary value problems, mainly oblique derivative problems for nonlinear uniformly elliptic equations of second order with measurable coefficients in high-dimensional domains, and then prove the solvability of the problems.