On outgoing solutions for a system of time-harmonic elastic wave in the exterior of a star-shaped domain


  • Luis A. Cortés Vega Universidad de Antofagasta.
  • Claudio Fernández P. Universidad Católica de Chile.
  • Gustavo Perla Menzala National Laboratory of Scientific Computation; Universidade Federal de Rio de Janeiro.




Existence and uniqueness of outgoing solutions, linear elastic wave equation, star-shaped domain, linear velocity boundary type conditions, resonances, existencia y unicidad de soluciones externas, ecuación de onda elástica lineal.


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.

Author Biographies

Luis A. Cortés Vega, Universidad de Antofagasta.

Facultad de Ciencias Básicas,
Departamento de Matemáticas.

Claudio Fernández, P. Universidad Católica de Chile.

Facultad de Matemáticas.


[1] S. Agmon, Lectures on Elliptic Boundary Value Problems, D. Van Nostrand Company, Inc., Princeton, (1965).

[2] C. J. S. Alves, T. Ha Duong, F. Penzel, On the identification of conductive cracks. In: M.Tanaka, G.Dulikravich (Eds.), Inverse Problems in Engineering Mechanics II. Elsevier, pp. 213-218, (2000).

[3] H. Amann, Parabolic evolution equations and non linear boundary conditions. J. Differntial Equations. 72, pp. 201-269, (1988).

[4] K. T. Andrews, K. L. Kuttler and M. Shillor, Second order evolution equations with dynamic boundary conditions. J. Math. Anal. Appl. 197, pp. 781-795, (1996).

[5] M. A. Astaburuaga, R. Coimbra Chrao, C. Fernndez and G. Perla Menzala, Scattering frequencies for a perturbed system of elastic wave equations. J. Math. Anal. Appl. 219, pp. 52-75, (1998).

[6] H. Brezis and L.E. Fraenkel, A function with prescribed initial derivatives in different Banach spaces. J. Funct. Anal. 29, pp. 328-335, (1978).226

[7] R. Coimbra Charo and G. Perla Menzala, Scattering frequencies and a class of perturbed systems of elastic waves. Math. Meth. Appl. Sci. 19, pp. 699-716, (1996).

[8] J. Cooper and W.A. Strauss, Abstract scattering theory for the periodic systems with applications to electromagnetism. Indiana Math. J. 34, pp. 33-83, (1985).

[9] J. Cooper, G. Perla Menzala and W. A. Strauss, On the scattering frequencies of time-dependent potentials. Math. Meth. Appl. Sci 8, pp. 576-584, (1986).

[10] L. A. Cortés-Vega, Existence of solutions for a system of elastic wave equations, Proyecciones 20, pp. 305,321, (2001).

[11] L. A. Cortés-Vega, Resonant frequencies for a system of time-harmonic elastic wave. J. Math. Anal. Appl. 279, pp. 43-55, (2003).

[12] L. A. Cortés-Vega, A note on resonant frequencies for a system of elastic wave equations. Int. J. Math. Math. Sci. 64, pp. 3485-3498, (2004).

[13] L. A. Cortés-Vega, Frequncias de Espalhamento e a Propagaco de Ondas Elsticas no Exterior de um Corpo Tridimensional, Ph.D. Thesis. University of Sao Paulo (2000).

[14] M. Costabel and E. P. Stephan, Integral equations for transmission problems in linear elasticity, J. Integral Equations Appl 2, pp. 211-223, (1990).

[15] G. Duvaut and J. L. Lions, Les inequations in mecanique et physique. Dunond, paris (1972).

[16] J. Escher, Quasilinear parabolic sytems with dynamical boundary conditions. Comm. Part. Diff. Equations 18, pp. 1309-1364, (1993).

[17] H. O. Fattorini, Second Order Linear Differential Equations in Banach Spaces. Elsevier Science Publishers B. V. Amsterdam (1985).

[18] C. Fernández and G. Perla Menzala, Resonances of an elastic resonator. Appl. Anal. 76, pp. 41-49, (2000).

[19] G. Fichera, Existence theorems in elasticity. Handbuch der Physik, Springer-Verlag, Berlin, Heidelberg, New York (1973).

[20] G. Fichera, Existence theorems in elasticity. Unilateral constraints in elasticity, S. Flgge, (Ed.), Handbuch der Physik, Springer, Berlin (1972) 347-424.

[21] T. Hintermann, Evolution equations with dynamic boundary conditions. Proc. Royal Soc. Edinburgh A. 113, pp. 43-60, (1989).

[22] V. D. Kupradze, Three-Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity. North-Holland, Amsterdam (1973).

[23] V. D. Kupradze, Progress in Solid Mechanics, Vol III, in Dynamical Problems in Elasticity. North-Holland, Amsterdam (1963).

[24] C. Labreuche, Generalization of the Schwarz reflection principle in scattering theory for dissipative systems: apllication to purely imaginary resonant frequencies. SIAM J. Math. Appl. 30, pp. 848-878, (1999).

[25] J. Lagnese, Decay of solutions of wave equations in a bounded region with boundary dissipation. J. Differential Equations. 50, pp. 163-182, (1983).

[26] P. Lax and R. S. Phillips, Scattering theory for dissipative hyperbolic systems. J. Funct. Anal. 14, pp. 172-235, (1973).

[27] P. Lax and R. S. Phillips, On the scattering frequencies of the Laplace operator for exterior domains. Comm. Pure Appl. Math. 25, pp. 85-101, (1972).

[28] J. L. Lions and E. Magenes, Non-homogeneus Boundary Value Problems and Applications, vol. I and II, Springer-Verlag, New york, 1972.

[29] W. J. Liu and E. Zuazua, Uniform stabilization of the higherdimensional system of thermoelasticity with a nonlinear boundary feedback, Quarterly Appl. Math. 59, pp. 269-314 (2001).

[30] B. Loe, A pole-free strip for potential scattering, J. Differential Equations. 99, pp. 112-138, (1992).

[31] J. T. Marti, Introduction to Sobolev spaces and finite element solutions of elliptic boundary value problems, Acad. Press, N.Y. (1986).

[32] J. E. Muñoz Rivera, E. C. Lapa and R. Barreto, Decay rates for viscoelastic plates with memory. J. Elasticity. 44, pp. 61-87, (1996).

[33] J. Necas, Les Mthodos Directes em Thoria des quations Elliptiques, Masson, Paris. 1967.

[34] Y. H. Pao, F. Santosa, W. W. Symes and C. Holland, (eds.), Inverse problems of acoustic and elastic waves. SIAM, (1984).

[35] O. Poisson, Calcul des ples de rsonance associs la diffraction d’ondes acoustiques par un obstacles en dimension deux, C.R. Acad. Sci. Paris I. 315, pp. 747-752, (1992).

[36] A. G. Ramm, Mathematical foundations of the singularity and eigenmode expansion methods (SEM and EEM), J. Math. Anal. Appl. 86, pp. 562-591, (1979).

[37] A. S Barreto and M. Zworski, Existence of resonances in three dimensions, Comm. Math. Phys. 173, pp. 401-415, (1995).

[38] P. Stefanov and G. Vodev, Distribution of resonances for the Neumann problem in linear elasticity outside a ball, Ann. Inst. H. Poincar Phys. Thor. 60, pp. 303-321, (1994).

[39] P. Stefanov and G. Vodev, Distribution of resonances for the Neumann problem in linear elasticity outside a strictly convex body, Duke Math. J. 78, pp. 677-714, (1995).

[40] S. Steinberg, Meromorphic families of compact operators. Arch. Rational. Mech. Anal. 31, pp. 372-379, (1968).



How to Cite

L. A. Cortés Vega, C. Fernández, and G. Perla Menzala, “On outgoing solutions for a system of time-harmonic elastic wave in the exterior of a star-shaped domain”, Proyecciones (Antofagasta, On line), vol. 25, no. 2, pp. 205-229, May 2017.