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

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.