On the Mean Oscillation of the Hessian of Solutions to the Monge-Ampère Equation

Qingbo Huang

doi:10.1016/j.aim.2005.12.005

On the Mean Oscillation of the Hessian of Solutions to the Monge-Ampère Equation

Qingbo Huang

Mathematics and Statistics

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

Abstract

We give interior a priori estimates for the mean oscillation of second derivatives of solutions to the Monge-Ampère equation det D2 u = f (x) with zero boundary values, where f (x) is a non-Dini continuous function. If the modulus of continuity of f (x) is φ (r) such that limr → 0 φ (r) log (1 / r) = 0, then D2 u ∈ VMO. © 2005 Elsevier Inc. All rights reserved.

Original language	American English
Pages (from-to)	599
Number of pages	616
Journal	Advances in Mathematics
Volume	207
Issue number	2
DOIs	https://doi.org/10.1016/j.aim.2005.12.005 https://doi.org/10.1016/j.aim.2005.12.005
State	Published - Dec 20 2006

Keywords

Mean oscillation of Hessian, Monge-Ampère equation, Regularity

Disciplines

Applied Mathematics
Applied Statistics
Mathematics
Physical Sciences and Mathematics
Statistics and Probability

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10.1016/j.aim.2005.12.005License: CC BY
10.1016/j.aim.2005.12.005License: Other

https://www.sciencedirect.com/science/article/pii/S0001870805004172?via%3Dihub

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abstract = " We give interior a priori estimates for the mean oscillation of second derivatives of solutions to the Monge-Amp{\`e}re equation det D2 u = f (x) with zero boundary values, where f (x) is a non-Dini continuous function. If the modulus of continuity of f (x) is φ (r) such that limr → 0 φ (r) log (1 / r) = 0, then D2 u ∈ VMO. {\textcopyright} 2005 Elsevier Inc. All rights reserved.",

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N2 - We give interior a priori estimates for the mean oscillation of second derivatives of solutions to the Monge-Ampère equation det D2 u = f (x) with zero boundary values, where f (x) is a non-Dini continuous function. If the modulus of continuity of f (x) is φ (r) such that limr → 0 φ (r) log (1 / r) = 0, then D2 u ∈ VMO. © 2005 Elsevier Inc. All rights reserved.

AB - We give interior a priori estimates for the mean oscillation of second derivatives of solutions to the Monge-Ampère equation det D2 u = f (x) with zero boundary values, where f (x) is a non-Dini continuous function. If the modulus of continuity of f (x) is φ (r) such that limr → 0 φ (r) log (1 / r) = 0, then D2 u ∈ VMO. © 2005 Elsevier Inc. All rights reserved.

KW - Mean oscillation of Hessian, Monge-Ampère equation, Regularity

UR - https://corescholar.libraries.wright.edu/math/354

UR - https://www.sciencedirect.com/science/article/pii/S0001870805004172?via%3Dihub

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DO - 10.1016/j.aim.2005.12.005

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SP - 599

JO - Advances in Mathematics

JF - Advances in Mathematics

IS - 2

ER -

On the Mean Oscillation of the Hessian of Solutions to the Monge-Ampère Equation

Abstract

Keywords

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