#### Vol. 10, No. 8, 2017

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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

Vol. 10 (2017), No. 8, 2031–2041
##### Abstract

We prove that the zero set of a nonnegative plurisubharmonic function that solves $det\left(\partial \stackrel{̄}{\partial }u\right)\ge 1$ in ${ℂ}^{n}$ and is in ${W}^{2,n\left(n-k\right)∕k}$ contains no analytic subvariety of dimension $k$ 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 $u$ lies in a critical Sobolev space (or more generally, certain Sobolev–Orlicz spaces).

##### Keywords
Monge–Ampère, regularity, viscosity solution, Sobolev
##### Mathematical Subject Classification 2010
Primary: 32W20, 35J96
Secondary: 35B33, 35B65