Extension of the Günter Derivatives to the Lipschitz Domains and Application to the Boundary Potentials of Elastic Waves

A. Bendali, S. Tordeux, Yu M. Volchkov

Research output: Contribution to journalArticlepeer-review

Abstract

Regularization techniques for the trace and the traction of elastic waves potentials previously built for domains of the class C2 are 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 scalar Helmholtz equation without resorting to the more advanced theory for elliptic systems in the Lipschitz domains. Scalar Günter 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 ∇∂Ωu × n of this function in the canonical orthonormal basis of the ambient space. This, in particular, implies that these derivatives define bounded operators from Hs to Hs−1 (0 ≤ s ≤ 1) on the boundary of the Lipschitz domain and can easily be implemented in boundary element codes. Representations of the Guünter operator and potentials of single and double layers of elastic waves in the two-dimensional case are provided.

Original languageEnglish
Pages (from-to)139-156
Number of pages18
JournalJournal of Applied Mechanics and Technical Physics
Volume61
Issue number1
DOIs
Publication statusPublished - 1 Jan 2020

Keywords

  • boundary integral operators
  • elastic waves
  • Günter derivatives
  • layer potentials
  • Lipschitz domains
  • Gunter derivatives
  • INTEGRAL-EQUATIONS
  • MAXWELL
  • FORMULATION
  • RADIATION
  • SCATTERING

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