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). DOI: 10.57262/ade/1355926587


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.


Download 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


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

zbMATH: 1073.35120
MathSciNet: MR1989355
Digital Object Identifier: 10.57262/ade/1355926587

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

Rights: Copyright © 2003 Khayyam Publishing, Inc.


This article is only available to subscribers.
It is not available for individual sale.

Vol.8 • No. 8 • 2003
Back to Top