On the Existence of Smooth Solutions for Fully Nonlinear Parabolic Equations with Measurable "Coefficients" without Convexity Assumptions

Hongjie Dong; N. V. Krylov

doi:10.1080/03605302.2012.756013

On the Existence of Smooth Solutions for Fully Nonlinear Parabolic Equations with Measurable "Coefficients" without Convexity Assumptions

Hongjie Dong, N. V. Krylov

Mathematics

Research output: Contribution to journal › Article › peer-review

5 Scopus citations

Abstract

We show that for any uniformly parabolic fully nonlinear second-order equation with bounded measurable "coefficients" and bounded "free" term in any cylindrical smooth domain with smooth boundary data one can find an approximating equation which has a unique continuous solution with the first derivatives bounded and the second spacial derivatives locally bounded. The approximating equation is constructed in such a way that it modifies the original one only for large values of the unknown function and its spacial derivatives.

Original language	English (US)
Pages (from-to)	1038-1068
Number of pages	31
Journal	Communications in Partial Differential Equations
Volume	38
Issue number	6
DOIs	https://doi.org/10.1080/03605302.2012.756013
State	Published - Jun 1 2013

Keywords

Bellman's equations
Finite differences
Fully nonlinear parabolic equations

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10.1080/03605302.2012.756013

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N2 - We show that for any uniformly parabolic fully nonlinear second-order equation with bounded measurable "coefficients" and bounded "free" term in any cylindrical smooth domain with smooth boundary data one can find an approximating equation which has a unique continuous solution with the first derivatives bounded and the second spacial derivatives locally bounded. The approximating equation is constructed in such a way that it modifies the original one only for large values of the unknown function and its spacial derivatives.

AB - We show that for any uniformly parabolic fully nonlinear second-order equation with bounded measurable "coefficients" and bounded "free" term in any cylindrical smooth domain with smooth boundary data one can find an approximating equation which has a unique continuous solution with the first derivatives bounded and the second spacial derivatives locally bounded. The approximating equation is constructed in such a way that it modifies the original one only for large values of the unknown function and its spacial derivatives.

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