The Neumann problem and the fractional laplacian in measure metric spaces

In this talk we will report on some recent joint work with Josh Kline, Riikka Korte, Marie Snipes and Nages Shanmugalingam, concerning the Neumann problem in PI spaces, and a new definition of fractional p-Laplacians in arbitrary doubling measure metric space. We extend some earlier work by Lukas Maly and Nages Shanmugalingam, proving well posedness in appropriate Lebesgue classes for the Neumann problem for p-Laplacians, and then leverage these results to prove existence, stability and regularity for the corresponding non homogeneous non-local PDE.