Calderon-Zygmund theory for strongly coupled linear system of nonlocal equations with Holder-regular coefficient

Abstract

We extend the Calderón-Zygmund theory for nonlocal equations to strongly coupled system of linear nonlocal equations ${\mathcal{L}}_A^{s}u=f$, where the operator ${\mathcal{L}}_A^s$ is formally given by \[ \mathcal{L}^s_{A}u = \int_{\mathbb{R}^n}\frac{A(x, y)}{\vert x-y\vert ^{n+2s}} \frac{(x-y)\otimes (x-y)}{\vert x-y\vert ^2}(u(x)-u(y))dy. \] For $0<s<1$ and $A:{\mathbb{R}}^n\times {\mathbb{R}}^n\to \mathbb{R}$ taken to be symmetric and serving as a variable coefficient for the operator, the system under consideration is the fractional version of the classical Navier-Lamé linearized elasticity system. The study of the coupled system of nonlocal equations is motivated by its appearance in nonlocal mechanics, primarily in peridynamics. Our regularity result states that if $A(\cdot ,y)$ is uniformly Holder continuous and ${\inf }_{x\in {\mathbb{R}}^n}A(x,x)>0$, then for $f\in L_{loc}^p,$ for $p\ge 2$, the solution vector $u\in H_{loc}^{2s-\delta ,p}$ for some $\delta \in (0,s)$.