A nonlocal biharmonic operator and its connection with the classical analogue

TL;DR

This paper introduces a nonlocal biharmonic operator within the nonlocal calculus framework, connecting it to peridynamics, and demonstrates existence, uniqueness, regularity, and convergence of solutions to classical boundary value problems.

Contribution

It presents a novel nonlocal biharmonic operator linked to peridynamics and establishes fundamental analytical properties and convergence results for boundary value problems.

Findings

Existence and uniqueness of solutions for nonlocal biharmonic equations.

Regularity results for solutions under nonlocal boundary conditions.

Convergence of nonlocal solutions to classical biharmonic solutions.

Abstract

We introduce here a nonlocal operator as a natural generalization to the biharmonic operator that appears in plate theory. This operator is built in the nonlocal calculus framework defined by Du et al. and its connected with the recent theory of peridynamics. For the steady state equation coupled with different boundary conditions we show existence and uniqueness of solutions, as well as regularity of solutions. The boundary conditions considered are nonlocal counterparts of the classical clamped and hinged boundary conditions. For each system we show convergence of the nonlocal solutions to their local equivalents using compactness arguments developed by Bourgain, Brezis and Mironescu.