Properties of the solutions to the Monge–Ampère equation

By Liliana Forzani, Diego Maldonado. 2004

59 citations recorded by the source corpus.

Abstract We consider solutions to the equation det D 2 ϕ=μ when μ has a doubling property. We prove new geometric characterizations for this doubling property (by means of the so-called engulfing property) and deduce the quantitative behaviour of ϕ. Also, a constructive approach to the celebrated C1,β-estimates proved by L. Caffarelli is presented, settling one of the open questions posed by Villani (Amer. Math. Soc. 58 (2003)).

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/ce4bb3b9c704229fb3aafac24bfe6646e70a43b4

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Five lectures on optimal transportation: Geometry, regularity and applications (2010, background): “For the quadratic cost c(x, y) = −x · y, the best known estimates [56] for the Hölder exponent α are much worse, and depend on bounds for log(f+/f−).”
Continuity and injectivity of optimal maps for non-negatively cross-curved costs (2009, background): “Explicit bounds on the exponent are much worse in the classical case c(x, y) = −〈x, y〉 [19], when such exponents do not even exist unless log f +(x) f−(y) ∈ L ∞ [8] [47].”
The Monge–Ampère equation and its link to optimal transportation (2013, background): “16 (which avoids any compactness argument) has been given by Forzani and Maldonado [61], allowing one to compute the explicit dependence of α on λ.”
Hölder Continuity and Injectivity of Optimal Maps (2011, methodology): “Then, the engulfing property allows us to apply the method of Forzani and Maldonado [24] to obtain local C1,α estimates (Theorem 9.”