Smooth Local Solutions to Degenerate Hyperbolic Monge-Ampère Equations

In this paper, we prove the existence of smooth solutions near 0 of the degenerate hyperbolic Monge-Ampère equation det ( D 2 u ) = K ( x ) h ( x , u , D u ) , where K ( 0 ) = 0 , K ≤ 0 , h ( 0 , 0 , 0 ) > 0 . We also assume that, the zero set of small perturbation of D n K has a simple structure...

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Bibliographic Details
Published in:Annals of PDE 2019-06, Vol.5 (1), p.1, Article 1
Main Author: Chen, Tiancong
Format: Article
Language:English
Subjects:
Cauchy problems
Decomposition
Linearization
Mathematical Methods in Physics
Monge-Ampere equation
Neighborhoods
Nonlinear equations
Partial Differential Equations
Physics
Physics and Astronomy
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Summary:In this paper, we prove the existence of smooth solutions near 0 of the degenerate hyperbolic Monge-Ampère equation det ( D 2 u ) = K ( x ) h ( x , u , D u ) , where K ( 0 ) = 0 , K ≤ 0 , h ( 0 , 0 , 0 ) > 0 . We also assume that, the zero set of small perturbation of D n K has a simple structure. For the proof, we first transform the linearized equation into a simple form by a suitable change of variables. Then we proceed to derive a priori estimates for the linearized equation, which is degenerately hyperbolic. Finally we use Nash-Moser iteration to prove the existence of local solutions.
ISSN:2524-5317
2199-2576
DOI:10.1007/s40818-018-0055-y