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Dimension of the
minimum set for the real and complex Monge–Ampère equations
in critical Sobolev spaces
Tristan C. Collins and Connor Mooney |
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Vol. 10 (2017), No. 8, 2031–2041
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Abstract |
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We prove that the zero set of a nonnegative plurisubharmonic function that solves in and is in contains no analytic subvariety of dimension or larger. Along the way we prove an analogous result for the real Monge–Ampère equation, which is also new. These results are sharp in view of well-known examples of Pogorelov and Błocki. As an application, in the real case we extend interior regularity results to the case that lies in a critical Sobolev space (or more generally, certain Sobolev–Orlicz spaces). |
Keywords
Monge–Ampère, regularity, viscosity solution, Sobolev
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Mathematical Subject Classification 2010
Primary: 32W20, 35J96
Secondary: 35B33, 35B65
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Milestones
Received: 23 March 2017
Revised: 18 June 2017
Accepted: 17 July 2017
Published: 18 August 2017
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Authors |
| Tristan C. Collins | |
| Department of Mathematics Harvard University Cambridge, MA % 02138 United States |
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| Connor Mooney | |
| Department of Mathematics ETH Zurich Zurich Switzerland |
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