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

Tadele Mengesha, Armin Schikorra, Adisak Seesanea, Sasikarn Yeepo

Abstract

We extend the Calderón-Zygmund theory for nonlocal equations to strongly coupled system of linear nonlocal equations $\mathcal{L}^{s}_{A} u = f$, where the operator $\mathcal{L}^{s}_{A}$ 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 $f\in L^{p}_{loc},$ for $p\geq 2$, the solution vector $u\in H^{2s-δ,p}_{loc}$ for some $δ\in (0, s)$.

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 , where the operator is formally given by For and 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 is uniformly Holder continuous and , then for for , the solution vector for some .
Paper Structure (7 sections, 10 theorems, 129 equations)

This paper contains 7 sections, 10 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Statement of the main result
  4. Notation and some preliminaries
  5. Commutator estimates
  6. The weighted fractional Lamé system
  7. The regularity theorem: Proof of Theorem \ref{['th:main']}

Key Result

Theorem 1.1

Let $s\in(0, 1)$ and $s \leq t<\min\{ 2s, 1\}$. Let $\Omega\subset \mathbb{R}^{n}$ be an open bounded set. If for $2\leq q<\infty$, $f_1,f_2 \in L^q(\Omega,\mathbb{R}^n)\cap L^2(\mathbb{R}^n,\mathbb{R}^n)$, and $u\in H^{s}(\mathbb{R}^n,\mathbb{R}^n)$ is a distributional solution of $\mathcal{L}^{s} with $\mathcal{L}^{s}_{A}$ corresponding to $A\in \mathcal{A}(\alpha, \lambda, \Lambda)$ for some g

Theorems & Definitions (17)

  • Theorem 1.1
  • Lemma 1.2
  • Lemma 1.3: MSY21
  • Proposition 1.4
  • proof
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 3.1
  • ...and 7 more