# The Neumann problem and the fractional p-Laplacian in measure metric spaces

# The Neumann problem and the fractional p-Laplacian in measure metric spaces

**In-Person Talk **

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. Following ideas of Caffarelli and Silvestre in and using recent progress in hyperbolic fillings, we define fractional p-Laplacians on any compact, doubling metric measure space, and prove existence, regularity, harnack inequality and stability for the corresponding non-homogeneous non-local equation. These results, in turn, rest on the new existence, global Hölder regularity and stability theorems that we prove for the Neumann problem for p-Laplacians in bounded domains of measure metric spaces endowed with a doubling measure that supports a Poincaré inequality. Our work also includes as special cases many of the previous results by other authors in the Euclidean, Riemannian and Carnot group settings. Unlike other recent contributions in the metric measure space context, our work does not rely on the assumption that the space supports a Poincare inequality.