Advances in Differential Equations

Remarks on the Dirichlet and state-constraint problems for quasilinear parabolic equations

Guy Barles and Francesca Da Lio

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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.

Article information

Adv. Differential Equations, Volume 8, Number 8 (2003), 897-922.

First available in Project Euclid: 19 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35K55: Nonlinear parabolic equations
Secondary: 35B05: Oscillation, zeros of solutions, mean value theorems, etc. 35B50: Maximum principles 49L25: Viscosity solutions


Barles, Guy; Da Lio, Francesca. Remarks on the Dirichlet and state-constraint problems for quasilinear parabolic equations. Adv. Differential Equations 8 (2003), no. 8, 897--922.

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