The Cauchy problem for the homogeneous Monge-Ampere equation, II. Legendre transform

Details The Cauchy problem for the homogeneous Monge-Ampere equation, II. Legendre transform The Cauchy problem for the homogeneous Monge-Ampere equation, II. Legendre transform

2010-10-12

arxiv.org/abs/1010.2442v2

Adv. Math. 228 (2011), 2989-3025

Mathematics

Analysis of PDEs

v2: added references

Scientific paper

10.1016/j.aim.2011.08.001

We continue our study of the Cauchy problem for the homogeneous (real and complex) Monge-Ampere equation (HRMA/HCMA). In the prequel 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.

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