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

TL;DR

This paper investigates the limitations of the Legendre transform approach to solving the homogeneous Monge-Ampere equation, showing it fails when the solution becomes non-differentiable, thus highlighting the method's constraints.

Contribution

It demonstrates that the Legendre transform-based solution to the homogeneous Monge-Ampere equation fails at non-differentiable points, clarifying the method's applicability limits.

Findings

Legendre transform solution ceases to solve the HRMA at non-differentiable points

Quantum mechanical approach aligns with Legendre transform in the prequel

The method's validity is limited to differentiable solutions

Abstract

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.