Abstract
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.
Citation
Guy Barles. Francesca Da Lio. "Remarks on the Dirichlet and state-constraint problems for quasilinear parabolic equations." Adv. Differential Equations 8 (8) 897 - 922, 2003. https://doi.org/10.57262/ade/1355926587
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