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

Yanir A. Rubinstein; Steve Zelditch

doi:10.1515/crelle-2014-0084

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

Yanir A. Rubinstein und Steve Zelditch

Veröffentlicht/Copyright: 31. Oktober 2014

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Aus der Zeitschrift Journal für die reine und angewandte Mathematik (Crelles Journal) Band 2017 Heft 724

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 C3 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 C3, in the sense that there exists a dense set of C3 Cauchy data for which there exists no C3 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 C1 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 C1 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.

Funding source: National Science Foundation

Award Identifier / Grant number: DMS-0603850

Award Identifier / Grant number: DMS-0904252

Funding statement: This material is based upon work supported in part by an NSF Postdoctoral Research Fellowship and grants DMS-0603850, 0904252.

Acknowledgements

We are grateful to T. Darvas, Z. Zhang, and a referee for a very careful reading and pertinent comments and corrections.

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Received: 2012-5-30

Revised: 2014-6-11

Published Online: 2014-10-31

Published in Print: 2017-3-1

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https://doi.org/10.1515/crelle-2014-0084