Analysis & PDE
- Anal. PDE
- Volume 10, Number 8 (2017), 2031-2041.
Dimension of the minimum set for the real and complex Monge–Ampère equations in critical Sobolev spaces
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).
Anal. PDE, Volume 10, Number 8 (2017), 2031-2041.
Received: 23 March 2017
Revised: 18 June 2017
Accepted: 17 July 2017
First available in Project Euclid: 16 November 2017
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Collins, Tristan C.; Mooney, Connor. Dimension of the minimum set for the real and complex Monge–Ampère equations in critical Sobolev spaces. Anal. PDE 10 (2017), no. 8, 2031--2041. doi:10.2140/apde.2017.10.2031. https://projecteuclid.org/euclid.apde/1510843590