Space–time semiclassical measures for three-dimensional elastodynamics: Boundary conditions for the hyperbolic set

Article type: Research Article

Authors: Akian, Jean-Luc

Affiliations: ONERA – The French Aerospace Lab, F-92322 Châtillon, France. Tel.: +33 1 46 73 46 41; Fax: +33 1 46 73 41 43; E-mail: [email protected]

Abstract: This paper is devoted to the linear elastodynamics equations in an open bounded set with a smooth boundary in the high-frequency limit. The boundary conditions are of Dirichlet or Neumann type. Semiclassical (or Wigner) measures enable to estimate the energy density related to these equations for high frequency phenomena. We determine here the boundary conditions verified by the space–time semiclassical measures related to the elastodynamics equations for the doubly hyperbolic set (under a non-interference hypothesis on the incident semiclassical measures) and for the hyperbolic–elliptic set. The non-interference hypothesis is similar to that of L. Miller in the case of transmission problems for the wave equation.

Keywords: high frequency, semiclassical measure, elastodynamics, Dirichlet, Neumann

DOI: 10.3233/ASY-2011-1084

Journal: Asymptotic Analysis, vol. 78, no. 1-2, pp. 37-83, 2012