Sharp regularity results on second derivatives of solutions to the Monge‐Ampère equation with VMO type data | Zendy

Huang Qingbo | Zendy

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Sharp regularity results on second derivatives of solutions to the Monge‐Ampère equation with VMO type data

Author(s) -

Huang Qingbo

Publication year - 2009

Publication title -

communications on pure and applied mathematics

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 3.12

H-Index - 115

eISSN - 1097-0312

pISSN - 0010-3640

DOI - 10.1002/cpa.20272

Subject(s) - mathematics , monge–ampère equation , type (biology) , bounded function , oscillation (cell signaling) , mathematical analysis , zero (linguistics) , value (mathematics) , pure mathematics , boundary values , boundary value problem , statistics , ecology , linguistics , philosophy , genetics , biology

We establish interior estimates for L p ‐norms, Orlicz norms, and mean oscillation of second derivatives of solutions to the Monge‐Ampère equation det D 2 u = f ( x ) with zero boundary value, where f ( x ) is strictly positive, bounded, and satisfies a VMO‐type condition. These estimates develop the regularity theory of the Monge‐Ampère equation in VMO‐type spaces. Our Orlicz estimates also sharpen Caffarelli's celebrated W 2, p ‐estimates. © 2008 Wiley Periodicals, Inc.

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