Uniqueness and non-uniqueness pairs for the fractional Laplacian

TL;DR

This paper characterizes conditions under which discrete sets in Euclidean space guarantee uniqueness or non-uniqueness of solutions to the fractional Laplacian equation, extending to other multiplier operators.

Contribution

It provides new sufficient conditions for uniqueness and non-uniqueness pairs for the fractional Laplacian on discrete sets, with broader applicability to multiplier operators.

Findings

Identifies conditions for sets to ensure solution uniqueness.

Provides examples demonstrating non-uniqueness.

Extends ideas to a broader class of multiplier operators.

Abstract

We establish sufficient conditions on discrete subsets of $\mathbb{R}^d$ for them to form a uniqueness or a non-uniqueness pair for the fractional Laplacian. Specifically, assuming that $f=0$ on $\Lambda$ and that $(-\Delta)^sf=0$ on $M$, where $\Lambda, M \subset \mathbb{R}^d$ are discrete, we find sufficient conditions on these sets that force $f$ to vanish identically, and we provide examples in which non-uniqueness occurs. Some of the ideas used in the proofs also extend to a broader class of multiplier operators.