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Microlocal defect measures, application to the Lamé system. (Mesures de défaut de compacité, application au système de Lamé.) (French) Zbl 1043.35009
Many authors, including the second author of this paper, have studied the asymptotic propagation of the energy for the solutions of systems of PDEs. In this paper generalizations of some well-known results are obtained. The key is the proposed definition of the microlocal defect measures for boundary value systems satisfying the strong Lopatinski condition. The proof of the theorem of propagation given is interesting. The imposed condition implies naturally the limit condition of Lopatinski type. Applications for the solutions of the Lame system are obtained. A description of the theorem of propagation for the transversal wave when the longitudinal energy is negligible. Finally, the conjecture of G. Lebeau and E. Zuazua [Arch. Ration. Mech. Anal. 148, No. 3, 179–231 (1999; Zbl 0939.74016)] is proved.
MSC:
35A27Microlocal methods; sheaf-theoretic methods (PDE)
35A21Propagation of singularities (PDE)
74B20Nonlinear elasticity
35Q72Other PDE from mechanics (MSC2000)
35L20Second order hyperbolic equations, boundary value problems
58J15Relations with hyperfunctions (PDE on manifolds)
74F05Thermal effects in solid mechanics
74G99Equilibrium (steady-state) problems