Partial Legendre transforms of non-linear equations
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- by Pengfei Guan and D. H. Phong PDF
- Proc. Amer. Math. Soc. 140 (2012), 3831-3842 Request permission
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.References
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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
- MR Author ID: 139200
- ORCID: 0000-0003-4997-4300
- Email: phong@math.columbia.edu
- 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
- © Copyright 2012 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 140 (2012), 3831-3842
- MSC (2010): Primary 35Hxx, 35Jxx; Secondary 58Jxx
- DOI: https://doi.org/10.1090/S0002-9939-2012-11210-9
- MathSciNet review: 2944724