The Cauchy problem for the homogeneous Monge–Ampère equation, III. Lifespan

Abstract

Abstract We prove several results on the lifespan, regularity, and uniqueness of solutions of the Cauchy problem for the homogeneous complex and real Monge–Ampère 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 ${C^{3}}$ 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 ${C^{3}}$ , in the sense that there exists a dense set of C 3 ${C^{3}}$ Cauchy data for which there exists no C 3 ${C^{3}}$ 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 ${C^{1}}$ 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 ${C^{1}}$ 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.