The maximum principle for equations with composite coefficients. (English) Zbl 0952.35025

Lieberman, Gary M.

Electron. J. Differ. Equ. 2000, Paper No. 38, 17 p. (2000).

The paper deals with the maximum principle for elliptic and parabolic equations with composite coefficients of the gradient term. Precisely, as it is well known, the maximum of the solution of a linear elliptic equation can be estimated in terms of the boundary (Dirichlet) data, provided the coefficient of the gradient term of the equation is either integrable of an appropriate power, or blows up like a small negative power of the distance to the boundary. Recently, D. E. Apushkinskaya and A. I. Nazarov [Probl. Mat. Anal. 14, 3-27 (1995; Zbl 0892.35028)] showed that the same estimate remains valid if that term is a sum of such functions provided the boundary is sufficient regular.
The paper under review extends that result to the case of oblique derivative problems. As a result, Hölder estimates for the solutions are derived also for nonlinear equations.

Reviewer: Dian K.Palagachev (Bari)

MSC:

35J25	Boundary value problems for second-order elliptic equations
35B50	Maximum principles in context of PDEs
35J65	Nonlinear boundary value problems for linear elliptic equations
35B45	A priori estimates in context of PDEs
35K20	Initial-boundary value problems for second-order parabolic equations

Keywords:

elliptic equation; maximum principle; a priori estimates

Citations:

Zbl 0892.35028