an:02002553
Zbl 1073.35120
Barles, Guy; Da Lio, Francesca
Remarks on the Dirichlet and state-constraint problems for quasilinear parabolic equations
EN
Adv. Differ. Equ. 8, No. 8, 897-922 (2003).
00101131
2003
j
35K60 35K55 35B05 35B50 49L25
generalized Dirichlet problem; maximum principle; viscosity solutions; quasilinear elliptic equations; comparison results; Perron's method
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.
Jorge Ferreira (S??o Jo??o del-Rei)