Existence of solutions for a system of elastic wave equations
DOI:
https://doi.org/10.4067/S0716-09172001000300004Keywords:
Elastic waves, Exterior domain, Resonant frequencies.Abstract
A simple and short proof of the existence of solutions for the direct scattering problem associated with the system of elastic wave equations is shown.
References
[1] J. F Ahner, The exterior time harmonic elasticity problem with prescribed displacement vector on the boundary, Arch. Math. 27, pp. 106–111, (1976).
[2] J.F Ahner and G.C. Hsiao, On the two dimensional exterior boundary value problems of elasticity, J. Apll. Math. 31, pp. 677-685, (1976).
[3] M.A. Astaburruaga and R. Coimbra Charao, C. Fernández and G. Perla Menzala, Scattering Frequencies for a perturbed system of elastic wave equations, J. Math. Anal Appl. 219, pp. 52–75, (1998).
[4] R. Coimbra Charao and G. Perla Menzala, Scattering Frequencies and a class of perturbed systems of elastic waves, Math. Meth. In the Appl. Sci. 19, pp. 699–716, (1996).
[5] L. A. Cortés–Vega, The exerior time-harmonic elasticity problem with prescribed stress- traction operator on the boundary, J. Math. Anal. Appl. (submitted).
[6] L. A. Cortés–Vega, “ Frequências de espalhamento e a propagacao de ondas elásticas no exterior de um corpo tridimensional ”. Ph. D. Thesis, (In portuguesse). University of Sao Paulo, Outubro– 2000.
[7] G. Duvaut and J. L. Lions, “Les inequations in Mecanique et en Physique ”, Dunond, Paris, (1972).
[8] G. Fichera, “Existence theorems in elasticity ”, Handbuch der Physik, Springer-Verlag, Berlin, Heidelberg, New York, (1973).
[9] P. Hähner and G.C Hsiao, Uniqueness theorems in inverse obstacle scattering of elastic waves, Inverse Prob. 9, pp. 525–534, (1993).
[10] P. Hähner, A uniqueness theorem in inverse scattering of elastic waves, IMA J. Of Appl. Math. 51, pp. 201–215, (1993).
[11] V.D. Kupradze, “Three-dimensional problems of the mathematical theory of elasticity and thermoelasticity”, Amsterdam, North– Holand, (1973).
[12] J. T. Marti, “Introduction to Sobolev spaces and finite element solution of elliptic boundary value problems ”, Academic Press, New York, (1986).
[13] M. Costabel and E.P. Stephan, Integral equations for transmission problems in linear elasticity, J. Int. Eq. and Appl. 2, pp. 211–223, (1996).
[14] H. J–P. Morand and R. Ohayon, “ Fluid–Structure interactions ”. John Wiley–Sons, New York, (1995).
[15] J. Necas, “Les méthodos directes em théoria des équations elliptiques ”, Masson, Paris, (1967).
[16] Y–H. Pao, F. Santosa, W.W. Symes and C. Holland, eds., “ Inverse problems of acoustic and elastic waves ”. SIAM, (1984).
[17] O. Poisson, Calcul des pôles de résonance associés à la diffraction d’ondes acoustiques par un obstacle en dimension deux, C. R. Acad. Sci. Paris, I. 315, pp. 747–752, (1992).
[18] A. G. Ramm, “Scattering by obstacles”, Riedel, Dordrecht, (1986).
[19] R. T. Seeley, Integral equations depending analytycally on parameter, Indag. Math. 24, pp. 434–442, (1962).
[20] S. Steinberg, Meromorphic families of compact operators, Arch. Rational. Mech. Anal. 31, pp. 372–379, (1968).
[2] J.F Ahner and G.C. Hsiao, On the two dimensional exterior boundary value problems of elasticity, J. Apll. Math. 31, pp. 677-685, (1976).
[3] M.A. Astaburruaga and R. Coimbra Charao, C. Fernández and G. Perla Menzala, Scattering Frequencies for a perturbed system of elastic wave equations, J. Math. Anal Appl. 219, pp. 52–75, (1998).
[4] R. Coimbra Charao and G. Perla Menzala, Scattering Frequencies and a class of perturbed systems of elastic waves, Math. Meth. In the Appl. Sci. 19, pp. 699–716, (1996).
[5] L. A. Cortés–Vega, The exerior time-harmonic elasticity problem with prescribed stress- traction operator on the boundary, J. Math. Anal. Appl. (submitted).
[6] L. A. Cortés–Vega, “ Frequências de espalhamento e a propagacao de ondas elásticas no exterior de um corpo tridimensional ”. Ph. D. Thesis, (In portuguesse). University of Sao Paulo, Outubro– 2000.
[7] G. Duvaut and J. L. Lions, “Les inequations in Mecanique et en Physique ”, Dunond, Paris, (1972).
[8] G. Fichera, “Existence theorems in elasticity ”, Handbuch der Physik, Springer-Verlag, Berlin, Heidelberg, New York, (1973).
[9] P. Hähner and G.C Hsiao, Uniqueness theorems in inverse obstacle scattering of elastic waves, Inverse Prob. 9, pp. 525–534, (1993).
[10] P. Hähner, A uniqueness theorem in inverse scattering of elastic waves, IMA J. Of Appl. Math. 51, pp. 201–215, (1993).
[11] V.D. Kupradze, “Three-dimensional problems of the mathematical theory of elasticity and thermoelasticity”, Amsterdam, North– Holand, (1973).
[12] J. T. Marti, “Introduction to Sobolev spaces and finite element solution of elliptic boundary value problems ”, Academic Press, New York, (1986).
[13] M. Costabel and E.P. Stephan, Integral equations for transmission problems in linear elasticity, J. Int. Eq. and Appl. 2, pp. 211–223, (1996).
[14] H. J–P. Morand and R. Ohayon, “ Fluid–Structure interactions ”. John Wiley–Sons, New York, (1995).
[15] J. Necas, “Les méthodos directes em théoria des équations elliptiques ”, Masson, Paris, (1967).
[16] Y–H. Pao, F. Santosa, W.W. Symes and C. Holland, eds., “ Inverse problems of acoustic and elastic waves ”. SIAM, (1984).
[17] O. Poisson, Calcul des pôles de résonance associés à la diffraction d’ondes acoustiques par un obstacle en dimension deux, C. R. Acad. Sci. Paris, I. 315, pp. 747–752, (1992).
[18] A. G. Ramm, “Scattering by obstacles”, Riedel, Dordrecht, (1986).
[19] R. T. Seeley, Integral equations depending analytycally on parameter, Indag. Math. 24, pp. 434–442, (1962).
[20] S. Steinberg, Meromorphic families of compact operators, Arch. Rational. Mech. Anal. 31, pp. 372–379, (1968).
Published
2017-04-24
How to Cite
[1]
L. A. Cortés Vega, “Existence of solutions for a system of elastic wave equations”, Proyecciones (Antofagasta, On line), vol. 20, no. 3, pp. 305-321, Apr. 2017.
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