Lieberman, Gary M.2019-12-202019-12-202000-05-22Lieberman, G. M. (2000). The maximum principle for equations with composite coefficients. <i>Electronic Journal of Differential Equations, 2000</i>(38), pp. 1-17.1072-6691https://hdl.handle.net/10877/9128It 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.Text17 pages1 file (.pdf)enAttribution 4.0 InternationalElliptic differential equationsOblique boundary conditionsMaximum principlesHolder estimatesHarnack inequalityParabolic differential equationsArticle