Jump problem for generalized Lam\'e-Navier systems in $\mathbb{R}^m$

TL;DR

This paper addresses the jump problem in generalized Lamé-Navier systems within linear elasticity, utilizing Clifford algebra and Dirac operators to derive explicit solutions applicable to complex regions, including fractal boundaries.

Contribution

It introduces a generalized Teodorescu transform and extends the jump problem solution to a broad class of elastic systems and regions with fractal boundaries.

Findings

Explicit solutions for the jump problem in generalized Lamé-Navier systems.

Application of Clifford algebra and Dirac operators to elasticity equations.

Solution applicability to regions with fractal boundaries.

Abstract

This paper is devoted to study a fundamental system of equations in Linear Elasticity Theory: the famous Lam\'e-Navier system. The Clifford algebra language allows us to rewrite this system in terms of the Euclidean Dirac operator, which at the same time suggests a very natural generalization involving the so-called structural sets. Our interest lies mainly in the jump problem for these elastic systems. A generalized Teodorescu transform, to be introduced here, provides the means for obtaining the explicit solution of the jump problem for a very wide classes of regions, including those with a fractal boundary.