The Cauchy problem for the homogeneous Monge-Ampere equation, III. Lifespan

TL;DR

This paper investigates the lifespan, regularity, and uniqueness of solutions to the homogeneous Monge-Ampere equations, revealing conditions for existence, ill-posedness, and the role of Hamiltonian mechanics in complex and real domains.

Contribution

It provides new bounds on solution lifespan, characterizes complex solution continuation, and introduces leafwise subsolutions, advancing understanding of Monge-Ampere equations.

Findings

Solutions have finite lifespan depending on regularity conditions.

The C^3 problem is ill-posed with dense data leading to no solutions.

Weak solutions in the real case are smooth, with lifespan tied to Moser map invertibility.

Abstract

We prove several results on the lifespan, regularity, and uniqueness of solutions of the Cauchy problem for the homogeneous complex and real Monge-Ampere equations (HCMA/HRMA) under various a priori regularity conditions. We use methods of characteristics in both the real and complex settings to bound the lifespan of solutions with prescribed regularity. In the complex domain, we characterize the C^3 lifespan of the HCMA in terms of analytic continuation of Hamiltonian mechanics and intersection of complex time characteristics. We use a conservation law type argument to prove uniqueness of solutions of the Cauchy problem for the HCMA. We then prove that the Cauchy problem is ill-posed in C^3, in the sense that there exists a dense set of C^3 Cauchy data for which there exists no C^3 solution even for a short time. In the real domain we show that the HRMA is equivalent to a Hamilton--Jacobi equation, and use the equivalence to prove that any differentiable weak solution is smooth, so that the differentiable lifespan equals the convex lifespan determined in our previous articles. We further show that the only obstruction to C^1 solvability is the invertibility of the associated Moser maps. Thus, a smooth solution of the Cauchy problem for HRMA exists for a positive but generally finite time and cannot be continued even as a weak C^1 solution afterwards. Finally, we introduce the notion of a "leafwise subsolution" for the HCMA that generalizes that of a solution, and many of our aforementioned results are proved for this more general object.