Communications in Mathematical Sciences

Volume 16 (2018)

Number 6

De Giorgi techniques applied to Hamilton–Jacobi equations with unbounded right-hand side

Pages: 1465 – 1487

DOI: https://dx.doi.org/10.4310/CMS.2018.v16.n6.a1

Authors

Logan F. Stokols (Department of Mathematics, University of Texas, Austin, Tx., U.S.A.)

Alexis F. Vasseur (Department of Mathematics, University of Texas, Austin, Tx., U.S.A.)

Abstract

In this article we obtain Holder estimates for solutions to second-order Hamilton–Jacobi equations with super-quadratic growth in the gradient and unbounded source term. The estimates are uniform with respect to the smallness of the diffusion and the smoothness of the Hamiltonian. Our work is in the spirit of a result by P. Cardaliaguet and L. Silvestre [P. Cardaliaguet and L. Silvestre, Comm. Partial Differential Equations, 37(9):1668–1688, 2012]. We utilize De Giorgi’s method, which was introduced to this class of equations in [C.-H. Chan and A.F. Vasseur, ArXiv e-prints, November 2014].

Keywords

Hamilton–Jacobi equation, Hölder regularity, De Giorgi method

2010 Mathematics Subject Classification

35B65, 35G20

Received 7 March 2017

Accepted 25 July 2017

Published 7 February 2019