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

School of Mathematics

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

7 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 2013

Bibliographical note

Funding Information:
The authors are sincerely grateful to the referee for his very quick report and helpful comments on an earlier version of the manuscript. H. Dong was partially supported by NSF Grant DMS-1056737. N. V. Krylov was partially supported by NSF Grant DMS-1160569

Keywords

Bellman's equations
Finite differences
Fully nonlinear parabolic equations

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10.1080/03605302.2012.756013

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keywords = "Bellman's equations, Finite differences, Fully nonlinear parabolic equations",

author = "Hongjie Dong and Krylov, {N. V.}",

note = "Funding Information: The authors are sincerely grateful to the referee for his very quick report and helpful comments on an earlier version of the manuscript. H. Dong was partially supported by NSF Grant DMS-1056737. N. V. Krylov was partially supported by NSF Grant DMS-1160569",

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AU - Krylov, N. V.

N1 - Funding Information: The authors are sincerely grateful to the referee for his very quick report and helpful comments on an earlier version of the manuscript. H. Dong was partially supported by NSF Grant DMS-1056737. N. V. Krylov was partially supported by NSF Grant DMS-1160569

PY - 2013/6

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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.

KW - Bellman's equations

KW - Finite differences

KW - Fully nonlinear parabolic equations

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M3 - Article

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JO - Communications in Partial Differential Equations

JF - Communications in Partial Differential Equations

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ER -

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

Abstract

Bibliographical note

Keywords

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