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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.

MSC:
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
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[8] Deleted in proof.
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