Jump problem for generalized Lamé-Navier systems in $\mathbb{R}^m$
Daniel Alfonso Santiesteban, Ricardo Abreu Blaya, Daniel Alpay
TL;DR
The paper develops a Clifford-analysis framework for generalized Lamé-Navier elastic systems in $\mathbb{R}^m$, formulating a two-operator Lamé-Navier operator $^{\varphi,\psi}\mathcal{L}_{\underline{x}}$ from Dirac operators associated with structural sets $\varphi$ and $\psi$. It derives Cauchy and Teodorescu transforms tailored to these operators, establishes a Clifford-version of the Borel–Pompeiu formula, and constructs a fundamental solution $K_{\varphi,\psi}$ with the relation $^{\psi\partial_{\underline{x}}}K_{\varphi,\psi}=K_{\varphi}$, enabling explicit jump representations. The jump problem for the generalized Lamé-Navier system is solved on smooth domains, yielding a solution expressed via combinations of Cauchy/Teodorescu transforms and boundary data, and is extended to fractal boundaries through $d$-summable conditions, Whitney extension, and Kats/Dolzhenko-type techniques to obtain a constructive solution $F(\underline{x})=\tilde f(\underline{x})\chi_{\Omega}(\underline{x})-\mathcal{T}_{\varphi,\psi}^{\dagger}(^{\varphi,\psi}\mathcal{L}_{\underline{x}}\tilde f)(\underline{x})$) with corresponding regularity and uniqueness statements.
Abstract
This paper is devoted to study a fundamental system of equations in Linear Elasticity Theory: the famous Lamé-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.