A note on the $L^p$-solvability of a strongly-coupled nonlocal system of equations

Abstract

The goal of this paper is to study the $L^p$-solvability of the strongly-coupled nonlocal system \[ \mathbb{L} \mathbf{u} (\mathbf{x}) + λ\mathbf{u}(\mathbf{x})= \mathbf{f}(\mathbf{x}) \quad \text{in ${\mathbb{R}}^d$ } \] where $\mathbb{L}$ is a linear nonlocal coupled vector-valued operator associated with a kernel $K$ comparable to $|\mathbf{y}{|}^{-(d+2s)}$ for $s\in (0,1)$, satisfying certain ellipticity and cancellation conditions. For any $\mathbf{f}\in [L^p({\mathbb{R}}^d){]}^d$, $1<p<\infty$, the existence of a unique strong solution $\mathbf{u}\in [H^{2s,p}({\mathbb{R}}^d){]}^d$ is proved via the method of continuity. To apply this method, we establish the continuity of the operator $\mathbb{L}$ and the necessary \textit{a priori} estimates. These are obtained through the study of the corresponding parabolic system. The proof strategy follows and extends recent ideas developed for the scalar setting, combining commutator estimates, Sobolev embeddings, a level set estimates and a bootstrap argument.