Application of partial regularity to fully nonlinear elliptic PDEs

Analysis of Fluids and Related Topics
  • Grigory Seregin
  • Oxford University
  • Application of partial regularity to fully nonlinear elliptic PDEs
Mar 21, 2024
3 - 4 pm
Fine Hall 314

We discuss a new approach to local estimates for fully nonlinear elliptic PDEs.  The idea is to use partial regularity to upgrade incomplete maximum principles to complete estimates. We were able to use this to solve the regularity problem for the quadratic Hessian PDEs in dimension four, which are related to finding a Euclidean hypersurface of prescribed scalar curvature.  Similar ideas give new proofs for the Monge-Ampere equation, special Lagrangian equation, and singularity removal.  The main work is joint with Y. Yuan, and the method draws from works of Chaudhuri-Trudinger, P.Guan-G.Qiu, and Savin.