Some remarks on the duality method for Integro-Differential equations with measure data

TL;DR

This paper investigates the existence, uniqueness, and regularity of solutions to boundary value problems involving nonlocal fractional operators with measure data, extending the duality method to integro-differential equations.

Contribution

It extends the duality method to establish well-posedness and regularity results for fractional integro-differential equations with measure data.

Findings

Proved existence and uniqueness of solutions.

Established regularity properties of solutions.

Analyzed the impact of measure data on solution behavior.

Abstract

We deal with existence, uniqueness, and regularity for solutions of the boundary value problem $$ \begin{cases} {\mathcal L}^s u = \mu &\quad \text{in $\Omega$}, u(x)=0 \quad &\text{on} \ \ \mathbb{R}^N\backslash\Omega, \end{cases} $$ where $\Omega$ is a bounded domain of $\mathbb{R}^N$, $\mu$ is a bounded radon measure on $\Omega$, and ${\mathcal L}^s$ is a nonlocal operator of fractional order $s$ whose kernel $K$ is comparable with the one of the factional laplacian.