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Search: MSC category 35J96 ( Elliptic Monge-Ampere equations )

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1. CJM Online first

Jiang, Feida; Trudinger, Neil S; Xiang, Ni
On the Neumann problem for Monge-Ampère type equations
In this paper, we study the global regularity for regular Monge-Ampère type equations associated with semilinear Neumann boundary conditions. By establishing a priori estimates for second order derivatives, the classical solvability of the Neumann boundary value problem is proved under natural conditions. The techniques build upon the delicate and intricate treatment of the standard Monge-Ampère case by Lions, Trudinger and Urbas in 1986 and the recent barrier constructions and second derivative bounds by Jiang, Trudinger and Yang for the Dirichlet problem. We also consider more general oblique boundary value problems in the strictly regular case.

Keywords:semilinear Neumann problem, Monge-Ampère type equation, second derivative estimates
Categories:35J66, 35J96

2. CJM 2011 (vol 64 pp. 924)

McCann, Robert J.; Pass, Brendan; Warren, Micah
Rectifiability of Optimal Transportation Plans
The regularity of solutions to optimal transportation problems has become a hot topic in current research. It is well known by now that the optimal measure may not be concentrated on the graph of a continuous mapping unless both the transportation cost and the masses transported satisfy very restrictive hypotheses (including sign conditions on the mixed fourth-order derivatives of the cost function). The purpose of this note is to show that in spite of this, the optimal measure is supported on a Lipschitz manifold, provided only that the cost is $C^{2}$ with non-singular mixed second derivative. We use this result to provide a simple proof that solutions to Monge's optimal transportation problem satisfy a change of variables equation almost everywhere.

Categories:49K20, 49K60, 35J96, 58C07

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