A duality approach to the fractional Laplacian with measure data

TL;DR

This paper introduces a duality method to establish existence and uniqueness of solutions for fractional Laplacian equations with measure data, expanding the theoretical understanding of nonlocal PDEs with singular sources.

Contribution

The paper develops a novel duality approach to solve fractional Laplacian problems with measure data, providing new existence and uniqueness results for these nonlocal equations.

Findings

Proves existence of solutions for fractional Laplacian with measure data.

Establishes uniqueness of solutions under given conditions.

Extends the theory of nonlocal PDEs with singular measures.

Abstract

We describe a duality method to prove both existence and uniqueness of solutions to nonlocal problems like $$ (-\Delta)^s v = \mu \quad \text{in}\ \mathbb{R}^N, $$ with vanishing conditions at infinity. Here $\mu$ is a bounded Radon measure whose support is compactly contained in $\mathbb{R}^N$, $N\geq2$, and $-(\Delta)^s$ is the fractional Laplace operator of order $s\in (1/2,1)$.