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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.