Uniqueness and some related estimates for Dirichlet problem with fractional Laplacian

Abstract

AbstractThe non‐uniqueness of solutions of basic Dirichlet‐type problems is a surprising feature of the fractional Laplace equation. This paper establishes a somewhat sharp uniqueness condition for the fractional Laplace equation. We derive the ‐estimate for fractional Laplacian operators to better understand this phenomenon. We introduce several naturally weighted fractional Sobolev spaces and establish embedding relationships among them. These existence‐uniqueness conditions and the spaces we introduce here are intrinsically related to the fractional Laplacian and provide basic information for studying related problems.