Rafe Mazzeo

Stanford University

Boundary regularity for fractional Laplacians: an unorthodox perspective

We consider the problem of understanding the sharp boundary regularity of solutions to the fractional Laplace equation \(\Delta^s u = f\) on a smoothly bounded domain in Euclidean space. This fractional Laplacian is defined via the usual extension method, i.e., as a sort of Dirichlet-to-Neumann operator for a degenerate elliptic problem in a half-space of one higher dimension. Detailed and sharp results about this problem are known through a set of important papers by Gerd Grubb. Our approach is based on techniques from geometric microlocal analysis, and provides new and hopefully useful ways of analyzing the regularity of solutions, with particular emphasis on the case where \(f\) is Hölder. This is a report on joint work in progress with Heiko Gimperlein and Nikoletta Louca.