Geometric Properties of the Sections of Solutions to the Monge-ampère Equation
نویسندگان
چکیده
In this paper we establish several geometric properties of the cross sections of generalized solutions φ to the Monge-Ampère equation detD2φ = μ, when the measure μ satisfies a doubling property. A main result is a characterization of the doubling measures μ in terms of a geometric property of the cross sections of φ. This is used to obtain estimates of the shape and invariance properties of the cross sections that are valid under appropriate normalizations.
منابع مشابه
Properties of the Solutions to the Monge-ampère Equation
We consider solutions to the equation detDφ = 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 C-estimates proved by L. Caffarelli is presented, settling one of the open questions posed by C. Villani in [11].
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