Nonlinear Partial Differential Equations

Date: November 29, 2022

Time: 3:00PM - 4:00PM

Location: BLOC 302

Speaker: Mary Vaughan, University of Texas at Austin

Title: Regularity theory for fractional elliptic equations in nondivergence form

Abstract: In this talk, we will define fractional powers of nondivergence form elliptic operators in bounded domains under minimal regularity assumptions and highlight several applications. We will characterize a Poisson problem driven by such operators with a degenerate/singular extension problem. We then develop the method of sliding paraboloids in the Monge–Ampère geometry to prove Harnack inequality and Hölder regularity for classical solutions to the extension equation. This in turn implies Harnack inequality and Hölder regularity for solutions to the fractional Poisson problem. This work is joint with Pablo Raúl Stinga (Iowa State University).