Uniqueness and some related estimates for Dirichlet problem with fractional Laplacian

Abstract

The 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.

Maximum principle for stable operators

“…Note that their proof is rather different from ours since they use explicit upper Poisson kernel estimates. This is similar to the proof in [35, Theorem 1.6] and [34, Theorem 3.1] by Li and Liu. In contrast, our result is true for the larger class of stable, nondegenerate operators.…”