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

We prove that the zero set of a nonnegative plurisubharmonic function that solves det(̄u) 1 in n and is in W2,n(nk)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).

Monge–Ampère, regularity, viscosity solution, Sobolev
Mathematical Subject Classification 2010
Primary: 32W20, 35J96
Secondary: 35B33, 35B65
Received: 23 March 2017
Revised: 18 June 2017
Accepted: 17 July 2017
Published: 18 August 2017
Tristan C. Collins
Department of Mathematics
Harvard University
Cambridge, MA
% 02138
United States
Connor Mooney
Department of Mathematics
ETH Zurich