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

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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 -