Hardy decomposition of first order Lipschitz functions by Lam\'e-Navier solutions

TL;DR

This paper explores a Hardy decomposition approach for first order Lipschitz functions on domain boundaries using Lamé-Navier solutions and Clifford algebra, revealing new insights into boundary value problems in elasticity.

Contribution

It introduces a Hardy projection method based on Clifford analysis to decompose Lipschitz functions into Lamé-Navier boundary values, advancing boundary value problem techniques.

Findings

Hardy projections act as involutions on Lipschitz classes

Decomposition of boundary functions into Lamé-Navier solutions is possible

Clifford algebra simplifies the Lamé-Navier system analysis

Abstract

The Clifford algebra language allows us to rewrite the Lam\'e-Navier system in terms of the Euclidean Dirac operator. In this paper, the main question we shall be concerned with is whether or not a higher order Lipschitz function on the boundary $\Gamma$ of a Jordan domain $\Omega\subset\mathbb{R}^m$ can be decomposed into a sum of the two boundary values of a solution of the Lam\'e-Navier system with jump across $\Gamma$. Our main tool are the Hardy projections related to a singular integral operator arising in the context of Clifford analysis, which turns out to be an involution operator on the first order Lipschitz classes.