Fourier methods for fractional-order operators

Abstract

This is a survey on the use of Fourier transformation methods in the treatment of boundary problems for the fractional Laplacian $(-\Delta)^a$ (0<a<1), and pseudodifferential generalizations P, over a bounded open set $\Omega$ in $R^n$. The presentation starts at an elementary level. Two points are explained in detail: 1) How the factor $d^a$, with $d(x)=dist(x,d\Omega)$, comes into the picture, related to the fact that the precise solution spaces for the homogeneous Dirichlet problem are so-called a-transmission spaces. 2) The natural definition of a local nonhomogeneous Dirichlet condition $\gamma_0(u/d^{a-1})=\varphi$. We also give brief accounts of some further developments: Evolution problems (for $d_t u - r^+Pu = f(x,t)$) and resolvent problems (for $Pu-\lambda u=f$), also with nonzero boundary conditions. Integration by parts, Green's formula.