Harnack estimates for quasi-linear degenerate parabolic differential equations
By E. DiBenedetto, Ugo Gianazza, V. Vespri. 2008
165 citations recorded by the source corpus.
We establish the intrinsic Harnack inequality for non-negative solutions of a class of degenerate, quasilinear, parabolic equations, including equations of the p-Laplacian and porous medium type. It is shown that the classical Harnack estimate, while failing for degenerate parabolic equations, it continues to hold in a space-time geometry intrinsic to the degeneracy. The proof uses only measure-theoretical arguments, it reproduces the classical Moser theory, for non-degenerate equations, and it is novel even in that context. Hölder estimates are derived as a consequence of the Harnack inequality. The results solve a long standing problem in the theory of degenerate parabolic equations.
Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/098a463efaf113120f72a85b52a04920453d3bc9
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In Applied Mathematics
- P. Hasto
- Matteo Bonforte
- Petteri Harjulehto
- Xavier Ros-Oton
- Juan Luis V'azquez
- Alessio Figalli
- Matti Nuortio
- Ut V. Le
In Computational Theory and Mathematics
- P. Hasto
- Matteo Bonforte
- Juan Luis V'azquez
- Petteri Harjulehto
- Xavier Ros-Oton
- Alessio Figalli
- Matti Nuortio
- Ut V. Le
In Mathematical Physics
- Matteo Bonforte
- P. Hasto
- Alessio Figalli
- Petteri Harjulehto
- Juan Luis V'azquez
- Matti Nuortio
- Ut V. Le
- Xavier Ros-Oton
Citations in the corpus
- Infinite Speed of Propagation and Regularity of Solutions to the Fractional Porous Medium Equation in General Domains (2015, background): “Let us mention that, while the case s = 1 (corresponding to the classical Porous Medium Equation) has been extensively studied in the last 30 years by many authors [1, 2, 3, 16, 17, 21, 22, 28, 29, 31, 32, 34, 39], see also the books [30, 33, 35, 66], not many results are currently known when s ∈ (0”
- Overview of differential equations with non-standard growth (2010, methodology): “Some references containing methods developed from De Georgi’s ideas in the constant p case are [3, 4, 25].”
- Forward, backward and elliptic Harnack inequalities for non-negative solutions to certain singular parabolic partial differential equations (2010, background): “2 can be derived as in [7] by almost identical arguments.”