Boundary Problems of Fractional Laplacian

In this talk we shall present some new results concerning fractional Laplacian on a region and the associated reflected$\alpha$-stable processes. Among other things we have an
integral by parts formula for the regional fractional Laplacian which serves as the Gauss-Green formula for the classical Laplacian. Weshow also that the generator of a reflected $\alpha$-stable process can be described as a regional fractional Laplacian with Neumannboundary condition. By making use of these new resuts, we are able to discuss the boundary value problems of the fractional Laplacian with both Dirichlet and Neumann boundary conditions.
The talk is based on my recent joint work with G.Q.Yang.