Electronic Journal of Differential Equations,
Vol. 2000(2000), No. 38, pp. 1-17.
Title: The maximum principle for equations with composite coefficients
Author: Gary M. Lieberman (Iowa State University, Ames, Iowa, USA)
Abstract:
It is well-known that the maximum of the solution of a linear elliptic equation
can be estimated in terms of the boundary data provided the coefficient of the
gradient term is either integrable to an appropriate power or blows up like a
small negative power of distance to the boundary. Apushkinskaya and Nazarov
showed that a similar estimate holds if this term is a sum of such functions
provided the boundary of the domain is sufficiently smooth and a Dirichlet
condition is prescribed. We relax the smoothness of the boundary and also
consider non-Dirichlet boundary conditions using a variant of the method of
Apushkinskaya and Nazarov. In addition, we prove a Holder estimate for
solutions of oblique derivative problems for nonlinear equations satisfying
similar conditions.
Submitted April 24, 2000. Published May 22, 2000.
Math Subject Classifications: 35J25, 35B50, 35J65, 35B45, 35K20.
Key Words: elliptic differential equations; oblique boundary conditions;
maximum principles; Holder estimates; Harnack inequality;
parabolic differential equations