COMPARISON RESULTS FOR QUASILINEAR EQUATIONS IN ANNULAR DOMAINS AND APPLICATIONS1*

Abstract

ABSTRACT We study the “generalized” Dirichlet problem (in the sense of viscosity solutions) for quasilinear elliptic and parabolic equations in the case when losses of boundary conditions can actually occur. We prove for such problems comparison results between semicontinuous viscosity sub- and supersolutions (Strong Comparison Principle) in annular domains. As a consequence of the Strong Comparison Principle and the Perron's method we obtain the existence and the uniqueness of a continuous solution. Our approach allow us to handle also the case of “singular” equations, in particular the geometric equations arising in the level-sets approach for defining the motions of hypersurfaces with different types of normal velocities. We are able to provide a level-sets approach for equations set in bounded domains with generalized Dirichlet boundary conditions. *This work was partially supported by the TMR Programme “Viscosity Solutions and their Applications.”