Mixed boundary value problems for nonlinear elliptic equations of second order with measurable coefficients in high dimensional domains

Abstract

Cordres (Cordres, H.O., 1956, Über die Randwertaufgabe bei quasilinearen Differentialglei- chungen zweiter Ordnung in mehr als zwei Variablen. Mathematische Annalen, 131, 278–312) gave some a priori estimates of classical solutions of the Dirichlet problem for quasilinear elliptic equations of second order

Alkhutov and Mamedov (Alkhutov, Yu. A. and Mamedov, I.T., 1988, The first boundary value problem for nondivergence second order parabolic equations with discontinuous coefficients. Math. USSR Sbornik, 59, 471–495) 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 mixed boundary value problems for nonlinear uniformly elliptic equations of second order with measurable coefficients in higher dimensional domains, and then prove the solvability of the problem.

Keywords:

• Mixed derivative problems
• Nonlinear elliptic equations
• Measurable coefficients
• Higher dimensional domains

Keywords:

• AMS Subject Classifications: 35J65
• 35J55
• 35J45