[Paper Review] Extension of the G{ü}nter derivatives to Lipschitz domains and application to the boundary potentials of elastic waves

A. Bendali, Sébastien Tordeux|arXiv (Cornell University)|Nov 14, 2016

Advanced Numerical Methods in Computational Mathematics19 references3 citations

TL;DR

This paper extends Günter derivatives to Lipschitz domains, proving they define bounded operators from $H^s(dyOmega)$ to $H^{s-1}(dyOmega)$ for $0 \leq s \leq 1$, enabling regularization of elastic wave boundary potentials. The approach establishes mapping properties of elastic layer potentials by reducing them to Helmholtz operators, avoiding advanced elliptic system theory, and provides explicit formulas for single- and double-layer potentials and their tractions in 3D and 2D settings.

ABSTRACT

The scalar G{ü}nter derivatives of a function defined on the boundary of a three-dimensional domain are expressed as components (or their opposites) of the tangential vector rotational of this function in the canonical orthonormal basis of the ambient space. This in particular implies that these derivatives define bounded operators from H s into H s--1 for 0 $\le$ s $\le$ 1 on the boundary of a Lipschitz domain, and can easily be implemented in boundary element codes. Regularization techniques for the trace and the traction of elastic waves potentials, previously built for a domain of class C 2 , can thus be extended to the Lipschitz case. In particular, this yields an elementary way to establish the mapping properties of elastic wave potentials from those of the Helmholtz equation without resorting to the more advanced theory for elliptic systems. Some attention is finally paid to the two-dimensional case.

Motivation & Objective

• To extend the theory of Günter derivatives to Lipschitz boundaries, where classical regularity assumptions fail.
• To establish boundedness of Günter derivatives as operators from $H^s$ to $H^{s-1}$ on Lipschitz domains for $0 \leq s \leq 1$, restoring lost regularity.
• To generalize regularization techniques for elastic wave potentials from $\mathcal{C}^2$ domains to Lipschitz domains.
• To provide a direct, elementary derivation of mapping properties of elastic layer potentials by reducing them to the Helmholtz equation framework.
• To present explicit expressions for single- and double-layer potentials and their tractions in both 3D and 2D cases on Lipschitz boundaries.

Proposed method

• Define Günter derivatives $\mathcal{M}_{ij}^{(\boldsymbol{n})}u = n_j\partial_{x_i}u - n_i\partial_{x_j}u$ as tangential derivatives via the normal vector $\boldsymbol{n}$ on the boundary $\partial\Omega$.
• Use the fact that $\mathcal{M}_{ij}^{(\boldsymbol{n})}u = \boldsymbol{\nabla}u \cdot \boldsymbol{\tau}_{ij}$, where $\boldsymbol{\tau}_{ij} = n_j\boldsymbol{e}_i - n_i\boldsymbol{e}_j$, to interpret them as directional derivatives in the tangent plane.
• Prove boundedness of $\mathcal{M}_{ij}^{(\boldsymbol{n})}$ from $H^s(\partial\Omega)$ to $H^{s-1}(\partial\Omega)$ for $0 \leq s \leq 1$ on Lipschitz domains using trace and normal trace properties.
• Apply the extended Günter derivatives to regularize the trace and traction of elastic wave potentials, reducing their analysis to known properties of Helmholtz layer potentials.
• Derive explicit component-wise formulas for the single-layer potential $S\boldsymbol{p}$, double-layer potential $K\boldsymbol{\psi}$, and their tractions $T^{(\boldsymbol{n})}S\boldsymbol{p}$, $T^{(\boldsymbol{n})}K\boldsymbol{\psi}$ in terms of scalar potentials $V_{\kappa_p}, V_{\kappa_s}, N_{\kappa_s}$, and $\mathcal{M}_{\bot}^{(\boldsymbol{n})}$.
• Provide alternative expressions for the traction of the double-layer potential using mixed derivatives and normal derivatives, confirming consistency with known integral identities.

Experimental results

Research questions

• RQ1 Can Günter derivatives be meaningfully extended to Lipschitz domains, where classical regularity is lost?
• RQ2 Do Günter derivatives remain bounded operators from $H^s(\partial\Omega)$ to $H^{s-1}(\partial\Omega)$ for $0 \leq s \leq 1$ on Lipschitz boundaries?
• RQ3 Can regularization techniques for elastic wave potentials on $\mathcal{C}^2$ domains be extended to Lipschitz domains using Günter derivatives?
• RQ4 Can the mapping properties of elastic layer potentials be derived from those of the Helmholtz equation without invoking advanced elliptic system theory?
• RQ5 What are the explicit component-wise representations of elastic layer potentials and their tractions on Lipschitz boundaries in 3D and 2D?

Key findings

• Günter derivatives $\mathcal{M}_{ij}^{(\boldsymbol{n})}$ are bounded operators from $H^s(\partial\Omega)$ to $H^{s-1}(\partial\Omega)$ for $0 \leq s \leq 1$ on Lipschitz domains, restoring the lost half-order of regularity.
• The tangential derivative structure $\mathcal{M}_{ij}^{(\boldsymbol{n})}u = \boldsymbol{\nabla}u \cdot \boldsymbol{\tau}_{ij}$ ensures that Günter derivatives are well-defined and bounded in $L^2(\partial\Omega)$ even for Lipschitz boundaries.
• Regularization of elastic wave potentials on Lipschitz domains is achieved via Günter derivatives, enabling the extension of $\mathcal{C}^2$-based techniques to less regular domains.
• The traction of the single-layer potential $T^{(\boldsymbol{n})}S\boldsymbol{p}$ is expressed as a sum of normal derivatives and tangential derivatives involving $V_{\kappa_p}, V_{\kappa_s}, S_\bot$, and $\mathcal{M}_{\bot}^{(\boldsymbol{n})}$, with explicit component decomposition.
• The traction of the double-layer potential $T^{(\boldsymbol{n})}K\boldsymbol{\psi}$ admits two equivalent formulations: one involving $\mathcal{M}_{\bot}^{(\boldsymbol{n})}$ and normal derivatives, and another using second-order tangential derivatives and normal derivatives of $V_{\kappa_s}$.
• Explicit formulas for the 2D case are provided, showing that the theory generalizes beyond 3D and applies to planar elastic wave problems with Lipschitz boundaries.