Remarks on the Dirichlet and State-constraint Problems for Quasilinear Parabolic Equations

We prove two different types of comparison results between semicontinuous viscosity suband supersolutions of the generalized Dirichlet problem (in the sense of viscosity solutions theory) for quasilinear parabolic equations: the first one is an extension of the Strong Comparison Result obtained previously by the second author for annular domains, to domains with a more complicated geometry. The key point in the proof is a localization argument based on a “strong maximum principle” type property. The second type of comparison result concerns a mixed Dirichlet-State-constraints problems for quasilinear parabolic equations in annular domains without rotational symmetry; in this case, we do not obtain a Strong Comparison Result but a weaker one on the envelopes of the discontinuous solutions. As a consequence of these results and the Perron’s method we obtain the existence and the uniqueness of either a continuous or a discontinuous solution.