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

We prove two different types of comparison results between semicontinuous viscosity sub- and 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.