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

TL;DR

This work resolves $L^{p}$-solvability for a strongly-coupled nonlocal elliptic system in $\mathbb{R}^d$ with a kernel $K$ comparable to $|\mathbf{y}|^{-(d+2s)}$ by establishing well-posedness in the vector-valued Bessel potential space $[H^{2s,p}(\mathbb{R}^d)]^d$ for $1<p<\infty$. The authors prove that the leading operator $\mathbb{L}$ extends continuously from $[H^{2s,p}]^d$ to $[L^p]^d$ and that the resolvent problem $\mathbb{L}\mathbf{u}+\lambda\mathbf{u}=\mathbf{f}$ has a unique solution with quantitative a priori estimates, where $\|(-\Delta)^s\mathbf{u}\|_{L^p}+\lambda\|\mathbf{u}\|_{L^p} \le N\|\mathbf{f}\|_{L^p}$. To achieve this, the paper adapts a real-analytic approach via the method of continuity, deriving continuity of $\mathbb{L}$ and $a$ priori bounds through a fractional parabolic auxiliary problem and a bootstrap argument, and then extending the $p=2$ theory to general $p$ using level-set and maximal-function techniques. The results substantially generalize known $p=2$ and scalar cases, providing robust $L^{p}$-solvability for a broad class of nonlocal, vector-valued kernels with potential implications for peridynamics and related nonlocal models.

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.

Theorems & Definitions (25)

• Theorem 1.1
• Remark 1.2
• Proposition 2.1
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• Proposition 3.1
• Theorem 4.1
• Corollary 4.2