Hölder estimates in space-time for viscosity solutions of Hamilton-Jacobi equations
Cannarsa, Piermarco; Cardaliaguet, Pierre (2010), Hölder estimates in space-time for viscosity solutions of Hamilton-Jacobi equations, Communications on Pure and Applied Mathematics, 63, 5, p. 590-629. http://dx.doi.org/10.1002/cpa.20315
TypeArticle accepté pour publication ou publié
External document linkhttp://hal.archives-ouvertes.fr/hal-00371681/fr/
Journal nameCommunications on Pure and Applied Mathematics
MetadataShow full item record
Abstract (EN)It is well-known that solutions to the basic problem in the calculus of variations may fail to be Lipschitz-continuous when the Lagrangian depends on t. Similarly, for viscosity solutions to time-dependent Hamilton-Jacobi equations one cannot expect Lipschitz bounds to hold uniformly with respect to the regularity of coefficients. This phenomenon raises the question whether such solutions satisfy uniform estimates in some weaker norm. We will show that this is the case for a suitable Hölder norm, obtaining uniform estimates in (x,t) for solutions to first- and second-order Hamilton-Jacobi equations. Our results apply to degenerate parabolic equations and require superlinear growth at infinity, in the gradient variables, of the Hamiltonian. Proofs are based on comparison arguments and representation formulas for viscosity solutions, as well as weak reverse Hölder inequalities.
Subjects / KeywordsHamilton-Jacobi equations; viscosity solutions; Hölder continuity; degenerate parabolic equations; reverse Hölder inequalities
Showing items related by title and author.
Hölder Regularity for Viscosity Solutions of Fully Nonlinear, Local or Nonlocal, Hamilton–Jacobi Equations with Superquadratic Growth in the Gradient Cardaliaguet, Pierre; Rainer, Catherine (2011) Article accepté pour publication ou publié
Error estimates and convergence rates for the stochastic homogenization of Hamilton-Jacobi equations Armstrong, Scott N.; Cardaliaguet, Pierre; Souganidis, Panagiotis E. (2014) Article accepté pour publication ou publié
Viscosity solutions of Hamilton-Jacobi equations in infinite dimensions. V. Unbounded linear terms and B-continuous solutions Crandall, Michael G.; Lions, Pierre-Louis (1991) Article accepté pour publication ou publié
Viscosity solutions of Hamilton-Jacobi equations in infinite dimensions. IV. Hamiltonians with unbounded linear terms Crandall, Michael G.; Lions, Pierre-Louis (1990) Article accepté pour publication ou publié