On geometric characterizations for Monge–Ampère doubling measures

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].

Geometric Properties of the Sections of Solutions to the Monge-ampère Equation

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 th...

Monge-ampère Measures for Convex Bodies and Bernstein-markov Type Inequalities

We use geometric methods to calculate a formula for the complex Monge-Ampère measure (ddVK) n, for K Rn ⊂ Cn a convex body and VK its Siciak-Zaharjuta extremal function. Bedford and Taylor had computed this for symmetric convex bodies K. We apply this to show that two methods for deriving Bernstein-Markov type inequalities, i.e., pointwise estimates of gradients of polynomials, yield the same r...

Boundary Regularity for Solutions to the Linearized Monge-ampère Equations

We obtain boundary Hölder gradient estimates and regularity for solutions to the linearized Monge-Ampère equations under natural assumptions on the domain, Monge-Ampère measures and boundary data. Our results are affine invariant analogues of the boundary Hölder gradient estimates of Krylov.

Team Member: Bella Gu School: International School of Boston City: Cambridge, MA Supervisor: David Xianfeng Gu Title: Geometric Compression Based on Ricci Flow and Monge-Ampère Equation