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2003 Remarks on the Dirichlet and state-constraint problems for quasilinear parabolic equations
Guy Barles, Francesca Da Lio
Adv. Differential Equations 8(8): 897-922 (2003).

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.

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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.

Information

Published: 2003
First available in Project Euclid: 19 December 2012

zbMATH: 1073.35120
MathSciNet: MR1989355

Subjects:
Primary: 35K55
Secondary: 35B05, 35B50, 49L25

Rights: Copyright © 2003 Khayyam Publishing, Inc.

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Vol.8 • No. 8 • 2003
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