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Proyecciones (Antofagasta)

versión impresa ISSN 0716-0917

Proyecciones (Antofagasta) v.25 n.2 Antofagasta ago. 2006

http://dx.doi.org/10.4067/S0716-09172006000200006 

Proyecciones Journal of Mathematics
Vol. 25, No 2, pp. 205-229, August 2006.
Universidad Católica del Norte
Antofagasta - Chile

ON OUTGOING SOLUTIONS FOR A SYSTEM OF TIME—HARMONIC ELASTIC WAVE IN THE EXTERIOR OF A STAR—SHAPED DOMAIN

 

LUIS CORTÉS
Universidad de Antofagasta, Chile

CLAUDIO FERNÁNDEZ
Pontificia Universidad Católica de Chile, Chile

and

GUSTAVO PERLA
Universidade Federal de Río de Janeiro, Brasil


Abstract


In this work we consider the propagation of time—harmonic elastic waves outside of a star—shaped domain with a “linear velocity at the boundary”. We describe a new approach to investigate results of existence and uniqueness for this exterior problem. To this end, we used a method similar to the one discussed in [11, 12] which has its genesis in [13] and relies on a stationary approach of resonances. The fundamental step of our approach is to reduce the unbounded nature of the problem to a bounded domain introducing an auxiliary boundary condition of Dirichlet type. In particular, we find a large region in the complex plane which is “free” of resonances.


AMS Subject Classification : 35Q99, 35P99, 74B99


Key words and phrases : Existence and uniqueness of outgoing solutions, linear elastic wave equation, star-shaped domain, linear velocity boundary type conditions, resonances


Acknowledgements


L.C.V was partially supported by the Chile Science Foundation Conicyt-Fondecyt Grant 1040067 and Fondecyt-Incentivo a la Cooperación Internacional Grants 7040021 and 7050021. C.F. was partially supported by the
Chile Science Foundation Conicyt-Fondecyt Grants 1040067 and 1040839. G. P.M. thanks the partial support given by a Grant of CNPq, 300.948/92- 1 (Brazil), PRONEX (MCT, LNCC, Brazil) and Fondecyt-Incentivo a la Cooperación Internacional, Grants 7040021 and 7050021 of Conicyt-Chile, which made possible his visit to the Departamento de Matemticas of the Universidad de Antofagasta (UA-Chile), where part of this research was done. Part of this paper it was presented in the “XV Capricornio Mathematical Congress,” Antofagasta, Chile, August 3—6, 2005 and “Workshop On Partial Differential Equations.To celebrate the 60th anniversary of Professor Gustavo Perla Menzala,” Petrpolis—Rio de Janeiro, Brazil, August 10—12, 2005.


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Received : June 2006. Accepted : July 2006


Luis Cortés Vega
Universidad de Antofagasta,
Facultad de Ciencias Básicas
Departamento de Matemáticas
Casilla 170
Antofagasta
Chile
e-mail : lcortes@uantof.cl

Claudio Fernández
P. Universidad Católica de Chile
Facultad de Matemáticas
Casilla 306
Correo 22
Santiago
Chile
e-mail : cfernand@mat.puc.cl


and


Gustavo Perla Menzala
National Laboratory of Scientific Computation
LNCC/MCT
Rua Getulio Vargas 333
Quitandinha
Petrópolis
RJ, CEP 25651-070
Brazil
and
Institute of Mathematics
Universidade Federal de Ro de Janeiro
P.O. Box 68530
Rio de Janeiro, RJ
Brazil
e-mail : perla@lncc.br