Abstract
This paper is concerned with the Bernstein estimates of viscosity solutions of the Cauchy problems for linear parabolic equations. The techniques of viscosity solution method given by H. Ishii and P.L. Lions in [1] allow us to deduce the estimates without differentiating the equation, which is in a way completely different from the classical one. We mainly get the estimate of 〈Du〉 /(α) x,Q under the corresponding assumptions on the smoothness of solutions and the known functions in the equation.
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References
H. Ishii and P.L. Lions. Viscosity Solutions of Fully Nonlinear Second-order Elliptic Partial Differential Equations.J. Differential Equations, 1990, 83: 26–78.
P.L. Lions. Optimal Control of Diffusion Process and Hamilton-Jacobi-Bellman Equations, Part II: Viscosity Solutions and Uniqueness.Comm. Partial Differential Equations, 1983, 8: 1229–1276.
H. Ishii. On Uniqueness and Existence of Viscosity Solutions of Fully Nonlinear Second Order Elliptic PDE's.Comm. Pure Appl. Math., 1989, 42: 15–45.
Bian Baojun and Dong Guangchang. Uniqueness of Viscosity Solutions of Fully Nonlinear Second Order Parabolic PDE's.Chin. Ann of Math., 1990, 11B(2): 156–170.
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This work is supported by the National Natural Science Foundation of China.
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Zhan, Y. The Bernstein estimates of viscosity solutions of linear parabolic equations. Acta Mathematicae Applicatae Sinica 11, 255–262 (1995). https://doi.org/10.1007/BF02011190
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DOI: https://doi.org/10.1007/BF02011190