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Nonlinear Analysis

A doubling approach for sigma-k PDEs

Ravi Shankar, Princeton University

Location:  Hill 705
Date & time: Tuesday, 26 September 2023 at 1:40PM - 2:40PM

Abstract: The interior regularity for viscosity solutions of the sigma-2 equation is the remaining case of the Monge-Ampere sigma-k family to be understood. The dimension two case was done in the 1950’s by Heinz, and the dimension three case was done in 2008 by Warren and Yuan. In joint works with Yu Yuan, we show regularity in dimension four for sigma-2. Our argument uses a doubling inequality for the Hessian. A similar argument gives a proof of interior regularity for strictly convex solutions of the Monge-Ampere (sigma-n) equation in all dimensions, using a geometric approach distinct from Pogorelov’s maximum principle of the 70’s.

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