Revista Matemática Iberoamericana


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Volume 35, Issue 5, 2019, pp. 1415–1449
DOI: 10.4171/rmi/1093

Published online: 2019-07-17

A priori Lipschitz estimates for solutions of local and nonlocal Hamilton–Jacobi equations with Ornstein–Uhlenbeck operator

Emmanuel Chasseigne[1], Olivier Ley[2] and Thi Tuyen Nguyen[3]

(1) Université de Tours, France
(2) Institut National des Sciences Appliquées de Rennes and Université de Rennes, France
(3) Università di Padova, Italy

We establish a priori Lipschitz estimates for unbounded solutions of second-order Hamilton–Jacobi equations in $\mathbb R^N$ in presence of an Ornstein–Uhlenbeck drift. We generalize the results obtained by Fujita, Ishii and Loreti (2006) in several directions. The first one is to consider more general operators. We first replace the Laplacian by a general diffusion matrix and then consider a nonlocal integro-differential operator of fractional Laplacian type. The second kind of extension is to deal with more general Hamiltonians which are merely sublinear. These results are obtained for both degenerate and nondegenerate equations.

Keywords: Nonlinear partial differential equations, Lipschitz estimates, elliptic equations, integro-partial differential equations, Ornstein–Uhlenbeck operator, Hamilton–Jacobi equations

Chasseigne Emmanuel, Ley Olivier, Nguyen Thi Tuyen: A priori Lipschitz estimates for solutions of local and nonlocal Hamilton–Jacobi equations with Ornstein–Uhlenbeck operator. Rev. Mat. Iberoam. 35 (2019), 1415-1449. doi: 10.4171/rmi/1093