Integral equation method for a Robin-type traction problem in a periodic domain

TL;DR

This paper develops an integral equation method to solve a Robin-type traction problem in a periodically perforated elastic domain, proving uniqueness and representing solutions via layer potentials.

Contribution

It introduces a novel integral equation approach for Robin traction problems in periodic elastic domains, including solution representation and uniqueness proof.

Findings

Solution expressed as a sum of layer potential, constant, and linear function.

Unique solutions for the integral equation are identified.

Proved uniqueness of the elastic traction problem.

Abstract

In this note, we consider a Robin-type traction problem for a linearly elastic body occupying an infinite periodically perforated domain. After proving the uniqueness of the solution we use periodic elastic layer potentials to show that the solution can be written as the sum of a single layer potential, a constant function and a linear function of the space variable. The density of the periodic single layer potential and the constant are identified as the unique solutions of a certain integral equation.