TY - JOUR
T1 - The Cauchy problem for the homogeneous Monge-Ampère equation, II. Legendre transform
AU - Rubinstein, Yanir A.
AU - Zelditch, Steve
N1 - Funding Information:
We thank the referees for their comments. This material is based upon work supported in part under National Science Foundation grants DMS-0603355, 0603850, 0904252. Y.A.R. was also supported by a National Science Foundation Postdoctoral Research Fellowship (at Johns Hopkins University during the academic year 2008–2009).
PY - 2011/12/20
Y1 - 2011/12/20
N2 - We continue our study of the Cauchy problem for the homogeneous (real and complex) Monge-Ampère equation (HRMA/HCMA). In the prequel (Y.A. Rubinstein and S. Zelditch [27]) a quantum mechanical approach for solving the HCMA was developed, and was shown to coincide with the well-known Legendre transform approach in the case of the HRMA. In this article-that uses tools of convex analysis and can be read independently-we prove that the candidate solution produced by these methods ceases to solve the HRMA, even in a weak sense, as soon as it ceases to be differentiable. At the same time, we show that it does solve the equation on its dense regular locus, and we derive an explicit a priori upper bound on its Monge-Ampère mass. The technique involves studying regularity of Legendre transforms of families of non-convex functions.
AB - We continue our study of the Cauchy problem for the homogeneous (real and complex) Monge-Ampère equation (HRMA/HCMA). In the prequel (Y.A. Rubinstein and S. Zelditch [27]) a quantum mechanical approach for solving the HCMA was developed, and was shown to coincide with the well-known Legendre transform approach in the case of the HRMA. In this article-that uses tools of convex analysis and can be read independently-we prove that the candidate solution produced by these methods ceases to solve the HRMA, even in a weak sense, as soon as it ceases to be differentiable. At the same time, we show that it does solve the equation on its dense regular locus, and we derive an explicit a priori upper bound on its Monge-Ampère mass. The technique involves studying regularity of Legendre transforms of families of non-convex functions.
KW - Biconjugate function
KW - Convexification
KW - Flat surfaces
KW - Legendre transform
KW - Monge-Ampère equation
KW - Partial subdifferential
KW - Regularity of convex envelopes
KW - Weak solutions
UR - http://www.scopus.com/inward/record.url?scp=80054682144&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=80054682144&partnerID=8YFLogxK
U2 - 10.1016/j.aim.2011.08.001
DO - 10.1016/j.aim.2011.08.001
M3 - Article
AN - SCOPUS:80054682144
SN - 0001-8708
VL - 228
SP - 2989
EP - 3025
JO - Advances in Mathematics
JF - Advances in Mathematics
IS - 6
ER -