We establish a priori Lipschitz estimates for equations with mixed local and nonlocal diffusion, coercive gradient terms and unbounded right-hand side in Lebesgue spaces through an integral refinement of the Bernstein method. This relies on a nonlinear, nonlocal and variational version of the Bochner identity that involves the adjoint equation of the linearization of the initial problem.
|Original language||English (US)|
|Journal||Bolletino dell Unione Matematica Italiana|
|State||Published - Nov 5 2022|
Bibliographical noteKAUST Repository Item: Exported on 2022-12-07
Acknowledged KAUST grant number(s): CRG2021-4674
Acknowledgements: Open access funding provided by Università degli Studi di Padova within the CRUI-CARE Agreement. The author was partially supported by the INdAM-GNAMPA Project 2022 “Proprietà quantitative e qualitative per EDP non lineari con termini di gradiente” and by the King Abdullah University of Science and Technology (KAUST) project CRG2021-4674 “Mean-Field Games: models, theory and computational aspects". The author wishes to thank Prof. Barles for discussions on gradient estimates via viscosity solutions’ methods and for providing many references on the subject.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.