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Calderón–Zygmund theory for strongly coupled linear system of nonlocal equations with Hölder-regular coefficient

https://doi.org/10.1515/acv-2024-0005

Mengesha, Tadele; Schikorra, Armin; Seesanea, Adisak; Yeepo, Sasikarn (November 2024, Advances in Calculus of Variations)

Abstract We extend the Calderón–Zygmund theory for nonlocal equations tostrongly coupled system of linear nonlocal equations

ℒ_{A}^{s} u = f

{\mathcal{L}^{s}_{A}u=f}, where the operator

ℒ_{A}^{s}

{\mathcal{L}^{s}_{A}}is formally given by

ℒ_{A}^{s} u = \int_{ℝ^{n}} \frac{A (x, y)}{{| x - y |}^{n + 2 s}} \frac{(x - y) \otimes (x - y)}{{| x - y |}^{2}} (u (x) - u (y)) 𝑑 y .

\mathcal{L}^{s}_{A}u=\int_{\mathbb{R}^{n}}\frac{A(x,y)}{|x-y|^{n+2s}}\frac{(x-%y)\otimes(x-y)}{|x-y|^{2}}(u(x)-u(y))\,dy. For

0 < s < 1

{0~~<1}~~and $A : ℝ^{n} \times ℝ^{n} \to ℝ$ {A:\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}}taken to be symmetric and serving asa 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 $A (\cdot, y)$ {A(\,\cdot\,,y)}is uniformly Holder continuous and ${inf}_{x \in ℝ^{n}} A (x, x) > 0$ {\inf_{x\in\mathbb{R}^{n}}A(x,x)>0}, then for $f \in L_{loc}^{p}$ {f\in L^{p}_{\rm loc}}, for $p \geq 2$ {p\geq 2}, the solution vector $u \in H_{loc}^{2 s - δ, p}$ {u\in H^{2s-\delta,p}_{\rm loc}}for some $δ \in (0, s)$ {\delta\in(0,s)}.

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