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A Classification of Isolated Singularities of Elliptic Monge‐Ampére Equations in Dimension Two
Author(s) -
Gálvez José A.,
Jiménez Asun,
Mira Pablo
Publication year - 2015
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.21581
Subject(s) - mathematics , gravitational singularity , singularity , dimension (graph theory) , monge–ampère equation , elliptic curve , class (philosophy) , mathematical analysis , space (punctuation) , isolated singularity , pure mathematics , regular polygon , geometry , linguistics , philosophy , artificial intelligence , computer science
Let ℳ 1 denote the space of solutions z ( x , y ) to an elliptic, real analytic Monge‐Ampére equation det ( D 2 z ) = φ ( x , y , z , D z ) > 0 whose graphs have a non‐removable isolated singularity at the origin. We prove that ℳ 1 is in one‐to‐one correspondence with ℳ 2 × ℤ 2 , where ℳ 2 is a suitable subset of the class of regular, real analytic, strictly convex Jordan curves in ℝ 2 . We also describe the asymptotic behavior of solutions of the Monge‐Ampére equation in the C k ‐smooth case, and a general existence theorem for isolated singularities of analytic solutions of the more general equation det ( D 2 z + A ( x , y , z , D z ) ) = φ ( x , y , z , D z ) > 0 .© 2015 Wiley Periodicals, Inc.