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Partial Legendre transforms of non-linear equations


Authors: Pengfei Guan and D. H. Phong
Journal: Proc. Amer. Math. Soc. 140 (2012), 3831-3842
MSC (2010): Primary 35Hxx, 35Jxx; Secondary 58Jxx
Published electronically: March 1, 2012
MathSciNet review: 2944724
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Abstract: The partial Legendre transform of a non-linear elliptic differential equation is shown to be another non-linear elliptic differential equation. In particular, the partial Legendre transform of the Monge-Ampère equation is another equation of Monge-Ampère type. In $ 1+1$ dimensions, this can be applied to obtain uniform estimates to all orders for the degenerate Monge-Ampère equation with boundary data satisfying a strict convexity condition.


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Additional Information

Pengfei Guan
Affiliation: Department of Mathematics, McGill University, Montreal, Quebec H3A 2K6, Canada
Email: guan@math.mcgill.ca

D. H. Phong
Affiliation: Department of Mathematics, Columbia University, New York, New York 10027
Email: phong@math.columbia.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11210-9
Received by editor(s): November 17, 2010
Received by editor(s) in revised form: April 20, 2011
Published electronically: March 1, 2012
Additional Notes: The research of the first author was supported in part by an NSERC Discovery Grant
The research of the second author was supported in part by National Science Foundation grant DMS-07-57372.
Communicated by: Chuu-Lian Terng
Article copyright: © Copyright 2012 American Mathematical Society