Global C1,α Regularity for Monge–Ampère Equation and Convex Envelope

In this paper we establish the global C 1 , α regularity for solutions to the Dirichlet problem of the Monge–Ampère equation det D 2 u = f . By examples we show that our conditions are optimal. Our proof allows the degenerate case f ≥ 0 , including the special case f ≡ 0 . We also prove the global C...

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Bibliographic Details
Published in:Archive for rational mechanics and analysis 2022-04, Vol.244 (1), p.127-155
Main Authors: Caffarelli, Luis A., Tang, Lan, Wang, Xu-Jia
Format: Article
Language:English
Subjects:
Classical Mechanics
Complex Systems
Digital Object Identifier
Dirichlet problem
Euclidean space
Fluid- and Aerodynamics
Mathematical and Computational Physics
Monge-Ampere equation
Partial differential equations
Physics
Physics and Astronomy
Regularity
Theoretical
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Summary:In this paper we establish the global C 1 , α regularity for solutions to the Dirichlet problem of the Monge–Ampère equation det D 2 u = f . By examples we show that our conditions are optimal. Our proof allows the degenerate case f ≥ 0 , including the special case f ≡ 0 . We also prove the global C 1 , α regularity for the convex envelope of a given function under optimal conditions.
ISSN:0003-9527
1432-0673
DOI:10.1007/s00205-022-01757-5