Viscosity subsolutions and supersolutions for non-uniformly and degenerate elliptic equations

Aris S. Tersenov

University of Cyprus, Department of Mathematics and Statistics P.O. Box 20537, 1678 Nicosia, Cyprus
Sobolev Institute of Mathematics Siberian Branch of the Russian Academy of Sciences 4 Acad. Koptyug avenue, 630090 Novosibirsk, Russia


Abstract: In the present paper we study the Dirichlet boundary value problem for quasilinear elliptic equations including non-uniformly and degenerate ones. In particular, we consider mean curvature equation and pseudo p-Laplace equation as well. It is well-known that the proof of the existence of continuous viscosity solutions is based on Ishii’s implementation of Perron’s method. In order to use this method one has to produce suitable subsolution and supersolution. Here we introduce new methods to construct subsolutions and supersolutions for the above mentioned problems. Using these subsolutions and supersolutions one may prove the existence of unique continuous viscosity solution for a wide class of degenerate and non-uniformly elliptic equations.

AMSclassification: primary 35J60; secondary 35D05.

Keywords: viscosity subsolution, viscosity supersolution, mean curvature equation, pseudo p-Laplace equation.