Mathematics – Optimization and Control
Scientific paper
2009-03-30
Communications on Pure and Applied Mathematics 63, 5 (2010) 559?696
Mathematics
Optimization and Control
Scientific paper
10.1002/cpa.20315
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 H\"older 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 H\"older inequalities.
Cannarsa Piermarco
Cardaliaguet Pierre
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