The Maximum Principle for Equations with Composite Coefficients




Lieberman, Gary M.

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Southwest Texas State University, Department of Mathematics


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.



Elliptic differential equations, Oblique boundary conditions, Maximum principles, Holder estimates, Harnack inequality, Parabolic differential equations


Lieberman, G. M. (2000). The maximum principle for equations with composite coefficients. <i>Electronic Journal of Differential Equations, 2000</i>(38), pp. 1-17.


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