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Remarks on the Dirichlet and state-constraint problems for quasilinear parabolic equations. (English) Zbl 1073.35120
The authors prove two different types of comparison results between semicontinuous viscosity sub- and supersolutions of the generalized Dirichlet problem 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 results concerns a mixed Dirichlet-state-constrainsts problems for quasilinear parabolic equations in annular domains without rotational symmetry; in this case, the authors do not obtain 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 they obtain the existence and the uniqueness of either a continuous or a discontinuous solution.

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35K55 Nonlinear parabolic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35B50 Maximum principles in context of PDEs
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games