Electron. J. Diff. Eqns., Vol. 2000(2000), No. 38, pp. 1-17.

The maximum principle for equations with composite coefficients

Gary M. Lieberman

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.

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Gary M. Lieberman
Department of Mathematics
Iowa State University
Ames, Iowa 50011, USA
e-mail: lieb@iastate.edu

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