Some counterexamples to Sobolev regularity for degenerate Monge-Amp\`ere equations
Abstract
We construct a counterexample to W2,1 regularity for convex solutions to D2u ≤ 1, u|∂ = 0 in two dimensions. We also prove a result on the propagation of singularities in two dimensions that are logarithmically slower than Lipschitz. This generalizes a classical result of Alexandrov and is optimal by example.
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