Solvability of $L^p$ Regularity problem for parabolic operators with rough coefficients
Dr. Linhan Li
2025-05-27 10:30-11:30
MCM110
Speaker: Dr. Linhan Li (University of Edinburgh)
Time: 10:30-11:30 May 27, 2025 (Tuesday)
Venue: MCM110
Title: Solvability of $L^p$ Regularity problem for parabolic operators with rough coefficients
Abstract: The $L^p$ Regularity problem is a Dirichlet problem with data in the homogeneous Sobolev space $\dot W^{1,p}$, or in the parabolic Sobolev space $\dot L^p_{1,1/2}$ for parabolic operators. The $L^p$ Regularity problem is solvable if the $L^p$ norm of the non-tangential maximal function of the derivatives of solutions can be controlled. Recently, Dindos, Pipher and I have obtained optimal solvability of the $L^p$ Regularity problem for a class of parabolic operators with non-smooth, time-dependent coefficients that satisfy a well-studied minimal smoothness assumption on cylindrical Lipschitz domains. Our method is inspired by the recent breakthrough on the problem in the elliptic case but requires many new ideas and techniques due to the presence of the half derivative in time, which is a non-local operator.
