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