Abstract
In the theory of second-order, nonlinear elliptic and parabolic equations, obtaining local or global gradient bounds is often a key step for proving the existence of solutions but it may be even more useful in many applications, for example to singular perturbations problems. The classical Bernstein’s method is a well-known tool to obtain these bounds but, in most cases, it has the defect of providing only a priori estimates. The “weak Bernstein’s method”, based on viscosity solutions’ theory, is an alternative way to prove the global Lipschitz regularity of solutions together with some estimates but it is not so easy to perform in the case of local bounds. The aim of this paper is to provide an extension of the “weak Bernstein’s method” which allows to prove local gradient bounds with reasonable technicalities.
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Acknowledgements
The author would like to thank the anonymous referees whose remarks led to significant improvements of the readability of this article. This work was partially supported by the project ANR MFG (ANR-16-CE40-0015-01) funded by the French National Research Agency.
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This article is part of the topical collection “Viscosity solutions–Dedicated to Hitoshi Ishii on the award of the 1st Kodaira Kunihiko Prize” edited by Kazuhiro Ishige, Shigeaki Koike, Tohru Ozawa, and Senjo Shimizu.
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Barles, G. Local gradient estimates for second-order nonlinear elliptic and parabolic equations by the weak Bernstein’s method . Partial Differ. Equ. Appl. 2, 71 (2021). https://doi.org/10.1007/s42985-021-00130-7
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DOI: https://doi.org/10.1007/s42985-021-00130-7
Keywords
- Second-order elliptic and parabolic equations
- Gradient bounds
- Weak Bernstein’s method
- Viscosity solutions