zbMATH — the first resource for mathematics

Continuous dependence estimates for viscosity solutions of fully nonlinear degenerate parabolic equations. (English) Zbl 1086.35061
Using the maximum principle for semicontinuous functions, we establish a general “continuous dependence on the nonlinearities” estimate for viscosity solutions of fully nonlinear degenerate parabolic equations with time- and space-dependent nonlinearities. Our result generalizes a result by P. E. Souganidis [J. Differ. Equations 56, 345–390 (1985; Zbl 0506.35020)] for first-order Hamilton-Jacobi equations and a recent result by B. Cockburn, G. Gripenberg and S.-O. Londen [J. Differ. Equations 170, No. 1, 180–187 (2001; Zbl 0973.35107)] for a class of degenerate parabolic second-order equations. We apply this result to a rather general class of equations and obtain: (i) Explicit continuous dependence estimates. (ii) \(L^\infty\) and Hölder regularity estimates. (iii) A rate of convergence for the vanishing viscosity method. Finally, we illustrate results (i)–(iii) on the Hamilton-Jacobi-Bellman partial differential equation associated with optimal control of a degenerate diffusion process over a finite horizon. For this equation such results are usually derived via probabilistic arguments, which we avoid entirely here.

35K65 Degenerate parabolic equations
35K55 Nonlinear parabolic equations
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
Full Text: DOI
[1] Cockburn, B.; Gripenberg, G.; Londen, S.-O., Continuous dependence on the nonlinearity of viscosity solutions of parabolic equations, J. differential equations, 170, 180-187, (2001) · Zbl 0973.35107
[2] Crandall, M.G.; Ishii, H., The maximum principle for semicontinuous functions, Differential integral equations, 3, 1001-1014, (1990) · Zbl 0723.35015
[3] Crandall, M.G.; Lions, P.-L., Viscosity solutions of hamilton – jacobi equations, Trans. amer. math. soc., 277, 1-42, (1983) · Zbl 0599.35024
[4] Crandall, M.G.; Ishii, H.; Lions, P.-L., User’s guide to viscosity solutions of second order partial differential equations, Bull. amer. math. soc. (N.S.), 27, 1-67, (1992) · Zbl 0755.35015
[5] Fleming, W.H.; Soner, H.M., Controlled Markov processes and viscosity solutions, (1993), Springer-Verlag New York · Zbl 0773.60070
[6] Ishii, H., A boundary value problem of the Dirichlet type for hamilton – jacobi equations, Ann. scuola norm. sup. Pisa cl. sci. (4), 6, 105-135, (1989) · Zbl 0701.35052
[7] Ishii, H., On uniqueness and existence of viscosity solutions of fully nonlinear second-order elliptic pdes, Comm. pure appl. math., 42, 15-45, (1989) · Zbl 0645.35025
[8] Deleted in proof.
[9] Krylov, N.V., Controlled diffusion processes, (1980), Springer-Verlag New York · Zbl 0459.93002
[10] Souganidis, P.E., Existence of viscosity solutions of hamilton – jacobi equations, J. differential equations, 56, 345-390, (1985) · Zbl 0506.35020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.