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Hardy decomposition of first order Lipschitz functions by Lamé-Navier solutions

Daniel Alfonso Santiesteban, Ricardo Abreu Blaya, Daniel Alpay

TL;DR

This work addresses whether a boundary function in $Lip(1+\alpha,\Gamma)$ on a Jordan boundary can be decomposed into the sum of traces of two Lamé-Navier solutions with a jump across $\Gamma$. It employs Clifford analysis to reformulate the Lamé-Navier system via the Dirac operator and develops a Cauchy-type integral $\mathcal{C}_{\mathcal{L}}^0$ and a boundary operator $\mathcal{S}_{\mathcal L}$, establishing a Plemelj–Privalov-type theory for $\mathcal{S}_{\mathcal L}$ and showing it preserves $Lip(1+\alpha,\Gamma)$ and is an involution. This yields a Hardy decomposition $Lip(1+\alpha,\Gamma)=Lip^{+}(1+\alpha,\Gamma)\oplus Lip^{-}(1+\alpha,\Gamma)$ with explicit trace characterizations in terms of Lamé-Navier boundary data, linking interior/exterior Lamé-Navier solutions through the operators $\mathcal{M}_{\underline{x}}$ and boundary kernels. The results provide a rigorous framework for boundary-value decompositions in linear elasticity in higher dimensions, analogous to Riemann-Hilbert-type decompositions in complex and Clifford-analytic settings.

Abstract

The Clifford algebra language allows us to rewrite the Lamé-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 $Γ$ of a Jordan domain $Ω\subset\mathbb{R}^m$ can be decomposed into a sum of the two boundary values of a solution of the Lamé-Navier system with jump across $Γ$. 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.

Hardy decomposition of first order Lipschitz functions by Lamé-Navier solutions

TL;DR

This work addresses whether a boundary function in on a Jordan boundary can be decomposed into the sum of traces of two Lamé-Navier solutions with a jump across . It employs Clifford analysis to reformulate the Lamé-Navier system via the Dirac operator and develops a Cauchy-type integral and a boundary operator , establishing a Plemelj–Privalov-type theory for and showing it preserves and is an involution. This yields a Hardy decomposition with explicit trace characterizations in terms of Lamé-Navier boundary data, linking interior/exterior Lamé-Navier solutions through the operators and boundary kernels. The results provide a rigorous framework for boundary-value decompositions in linear elasticity in higher dimensions, analogous to Riemann-Hilbert-type decompositions in complex and Clifford-analytic settings.

Abstract

The Clifford algebra language allows us to rewrite the Lamé-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 of a Jordan domain can be decomposed into a sum of the two boundary values of a solution of the Lamé-Navier system with jump across . 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.
Paper Structure (8 sections, 9 theorems, 133 equations, 1 figure)

This paper contains 8 sections, 9 theorems, 133 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminary notions and auxiliary results
  3. Representation of Lamé-Navier solutions
  4. The Cauchy type integral for Lamé-Navier solutions
  5. Involution property of ${\cal S}_{\cal L}$
  6. Plemelj-Privalov Theorem
  7. ${\cal S}_{\cal L}$ is a linear involution
  8. Hardy Decomposition

Key Result

Theorem 1

Let ${\bf{f}}=\{f^{j},\,0\le j\le m\}$ be an ${\mathbb R}_{0,m}$-valued collection in ${\hbox{Lip}}(1+\alpha,\Gamma)$. Then, there exists a compact supported function $\tilde{f}\in C^{1,\alpha}({\mathbb R}^{m},{\mathbb R}_{0,m})$ satisfying

Figures (1)

  • Figure 1: Surface decomposition

Theorems & Definitions (10)

  • Theorem 1
  • Theorem 2: Borel-Pompeiu formula
  • Corollary 1: Cauchy integral formula
  • Theorem 3
  • Theorem 4
  • Remark 1
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8