Regularity of spectral fractional Dirichlet and Neumann problems

Consider the fractional powers $(A_{\mathrm{Dir}}{)}^a$ and $(A_{\mathrm{Neu}}{)}^a$ of the Dirichlet and Neumann realizations of a second-order strongly elliptic differential operator $A$ on a smooth bounded subset $\mathrm{\Omega }$ of ${\mathbb{R}}^n$. Recalling the results on complex powers and complex interpolation of domains of elliptic boundary value problems by Seeley in the 1970’s, we demonstrate how they imply regularity properties in full scales of $H_p^s$-Sobolev spaces and H"older spaces, for the solutions of the associated equations. Extensions to nonsmooth situations for low values of $s$ are derived by use of recent results on $H^{\mathrm{\infty }}$-calculus. We also include an overview of the various Dirichlet- and Neumann-type boundary problems associated with the fractional Laplacian.