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

Abstract

We prove that the zero set of a nonnegative plurisubharmonic function that solves $det(∂∂̄u) ≥ 1$ in $ℂn$ and is in $W2,n(n−k)∕k$ 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

Source
Anal. PDE, Volume 10, Number 8 (2017), 2031-2041.

Dates
Received: 23 March 2017
Revised: 18 June 2017
Accepted: 17 July 2017
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1510843590

Digital Object Identifier
doi:10.2140/apde.2017.10.2031

Mathematical Reviews number (MathSciNet)
MR3694014

Zentralblatt MATH identifier
06774334

Citation

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

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