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

Tristan C. Collins and Connor Mooney

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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).

Article information

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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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 32W20: Complex Monge-Ampère operators 35J96: Elliptic Monge-Ampère equations
Secondary: 35B33: Critical exponents 35B65: Smoothness and regularity of solutions

Monge–Ampère regularity viscosity solution Sobolev


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.

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