Fourier methods for fractional-order operators

TL;DR

This survey discusses Fourier transformation techniques for boundary problems involving the fractional Laplacian and pseudodifferential operators, explaining key concepts like solution spaces and boundary conditions, with insights into evolution and resolvent problems.

Contribution

It provides an elementary yet detailed overview of Fourier methods applied to fractional Laplacian boundary problems, including new explanations of solution spaces and boundary conditions.

Findings

Explanation of the role of the $d^a$ factor in solution spaces

Definition of local nonhomogeneous Dirichlet conditions

Brief overview of evolution and resolvent problems

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.