Some Remarks on the Duality Method for Integro-Differential Equations with Measure Data

Abstract

Abstract We deal with existence, uniqueness and regularity for solutions of the boundary value problem { ℒ s ⁢ u = μ in ⁢ Ω , u ⁢ ( x ) = 0 on ⁢ ℝ n \ Ω , $\left\{\begin{aligned} \displaystyle\mathcal{L}^{s}u&\displaystyle=\mu&&% \displaystyle\text{in }\Omega,\\ \displaystyle u(x)&\displaystyle=0&&\displaystyle\text{on }\mathbb{R}^{n}% \backslash\Omega,\end{aligned}\right.$ where Ω is a bounded domain of ℝ n ${\mathbb{R}^{n}}$ , μ is a bounded Radon measure on Ω, and ℒ s ${\mathcal{L}^{s}}$ is a non-local operator of fractional order s whose kernel K is comparable with the one of the fractional Laplacian.