Extension of the GГјnter Derivatives to the Lipschitz Domains and Application to the Boundary Potentials of Elastic Waves
A. Bendali1, S. Tordeux2, Yu. M. Volchkov3,4
1UniversitГ©de Toulouse, Institut MathГ©matique de Toulouse, Toulouse, France
2UniversitГ© de Pau et des Pays de l'Adour, Pau, France
3Lavrent'ev Institute for Hydrodynamics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090 Russia
4Novosibirsk State University, Novosibirsk, 630090 Russia
Keywords: граничные интегральные операторы, производные Гюнтера, упругие волны, поверхностные потенциалы, липшицевы области, boundary integral operators, GГјnter derivatives, elastic waves, layer potentials, Lipschitz domains
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 Δ δ 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 Günter operator and potentials of single and double layers of elastic waves in the two-dimensional case are provided.
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