A Potential Space Estimate for Solutions of Systems of Nonlocal Equations in Peridynamics

TL;DR

This paper establishes higher integrability of weak solutions to nonlocal linearized peridynamics equations, showing they belong to a potential space with fractional derivatives in L^p, extending Meyers' inequality to nonlocal systems.

Contribution

It introduces a nonlocal analogue of Meyers' inequality, demonstrating that solutions have fractional derivatives in L^p without extra assumptions, and characterizes their potential space membership.

Findings

Weak solutions belong to a potential space with higher integrability.

Solutions' fractional derivatives are in L^p for some p > 2.

The nonlocal coupling is captured by Marcinkiewicz-type integrals.

Abstract

We show that weak solutions to the strongly-coupled system of nonlocal equations of linearized peridynamics belong to a potential space with higher integrability. Specifically, we show a function that measures local fractional derivatives of weak solutions to a linear system belongs to $L^p$ for some $p > 2$ with no additional assumption other than measurability and ellipticity of coefficients. This is a nonlocal analogue of an inequality of Meyers for weak solutions to an elliptic system of equations. We also show that functions in $L^p$ whose Marcinkiewicz-type integrals are in $L^p$ in fact belong to the Bessel potential space $\mathcal{L}^{p}_s$. Thus the fractional analogue of higher integrability of the solution's gradient is displayed explicitly. The distinction here is that the Marcinkiewicz-type integral exhibits the coupling from the nonlocal model and does not resemble other classes of potential-type integrals found in the literature.