Smoothness of radial solutions to Monge-Ampère equations

Rios, Cristian; Sawyer, Eric

doi:10.1090/S0002-9939-08-09694-9

Smoothness of radial solutions to Monge-Ampère equations
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by Cristian Rios and Eric T. Sawyer

Proc. Amer. Math. Soc. 137 (2009), 1373-1379

DOI: https://doi.org/10.1090/S0002-9939-08-09694-9

Published electronically: November 5, 2008

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Abstract:

We prove that generalized convex radial solutions to the generalized Monge-Ampère equation $\det D^2u = f(|x|^2/2,u,|\nabla u|^2/2)$ with $f$ smooth are always smooth away from the origin. Moreover, we characterize the global smoothness of these solutions in terms of the order of vanishing of $f$ at the origin.

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