Seminar Analysis: Yash Jhaveri (University of Texas at Austin)

Nonlinear Bounds in Hölder Spaces for the Monge-Ampère Equation

It is well known that for many second-order PDEs the solution v gains two derivatives with respect to the right-hand side g in Hölder spaces. Often, however, it is useful to have a quantitative understanding of regularity. In ’89, Caffarelli proved interior a priori estimates for fully nonlinear, uniformly elliptic equations. Specifically, he showed that ‖v‖_{C^2,α(B_{1/2})}≤C(‖v‖_{L^∞(B_{1})}+‖g‖_{C^α(B_{1})}) and C∼1/α as α→0. The natural question to ask is then: Can one extend such quantitative estimates to other equations? An equation that appears frequently in analysis, geometry, and applications is the Monge-Ampére equation det(D²u) = f. The Monge-Ampère equation enjoys the same qualitative regularity gains as its linear counterpart the Poisson equation in the appropriate setting, and so we ask whether or not the quantitative picture is also the same. This is not the case. In this talk, we will first review Caffarelli’s interior a priori estimates. Then, we will move to the Monge-Ampère equation and see a different picture.

(Joint work with Alessio Figalli and Connor Mooney)

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