Vol. 9, No. 4, 2016

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Some counterexamples to Sobolev regularity for degenerate Monge–Ampère equations

Connor Mooney

Vol. 9 (2016), No. 4, 881–891
Abstract

We construct a counterexample to W2,1 regularity for convex solutions to

detD2u 1,u| Ω =  const.

in two dimensions. We also prove a result on the propagation of singularities of the form |x2||logx2| in two dimensions. This generalizes a classical result of Alexandrov and is optimal by example.

Keywords
degenerate Monge–Ampère, Sobolev regularity
Mathematical Subject Classification 2010
Primary: 35B65, 35J96
Milestones
Received: 23 September 2015
Accepted: 11 March 2016
Published: 3 July 2016
Authors
Connor Mooney
Department of Mathematics
University of Texas at Austin
Austin, TX 78712
United States