×

Distribution of resonances for the Neumann problem in linear elasticity outside a strictly convex body. (English) Zbl 0846.35139

Let \(\Delta_e\) be the elasticity operator defined by \[ \Delta_e v= \mu_0 \Delta v+ (\lambda_0+ \mu_0) \nabla (\nabla v), \] \(v= (v_1, v_2, v_3)\). \(\lambda_0\), \(\mu_0\) are the Lamé constants with \(\mu_0> 0\), \(3\lambda_0+ 2\mu_0> 0\). The authors consider the elasticity operator under Neumann boundary conditions of the form \[ \sum^3_{j= 1} \sigma_{ij}(v) \nu_j|_\Gamma= 0,\quad i= 1, 2, 3, \] where \(\sigma_{ij}\) is the stress tensor and \(\nu\) is the outer normal to \(\Gamma= \partial \Omega\), \(\Omega= \mathbb{R}^3\backslash {\mathcal O}\) and \(\mathcal O\) is a strictly convex compact in \(\mathbb{R}^3\) with \(C^\infty\)-smooth boundary \(\Gamma\). The operator \(-\Delta_e\) can be extended to a selfadjoint operator \(L\) on \(L^2(\Omega; C^3)\). If the cut-off resolvent \(R_\chi(\lambda)= \chi(L- \lambda^2)^{- 1} \chi\), \(\chi\in \mathbb{C}^\infty_0\) is extended to the whole complex plane then the poles are called resonances. The main result is the following:
(a) For any \(C_1> 0\), there exists a \(C_2> 0\), such that for any \(N> 0\) there are no resonances in the domain \(C_N|\lambda|^{- N}< \text{Im } \lambda< C_1\ln|\lambda|,\;|\text{Re } \lambda|> C_2\), with some \(C_N> 0\).
(b) There exist 2 infinite sequences \(\{\lambda_j\}\), \(\{- \overline\lambda_j\}\) of distinct resonances of \(L\), such that for any \(N> 0\), \(0< \text{Im } \lambda_j\leq C_N|\lambda_j|^{- N}\).
Reviewer: B.Dittmar (Halle)

MSC:

35Q72 Other PDE from mechanics (MSC2000)
74B05 Classical linear elasticity
35J15 Second-order elliptic equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] J. D. Achenbach, Wave Propagation in Elastic Solids , North-Holland, New York, 1973. · Zbl 0268.73005
[2] C. Bardos, G. Lebeau, and J. Rauch, Scattering frequencies and Gevrey \(3\) singularities , Invent. Math. 90 (1987), no. 1, 77-114. · Zbl 0723.35058
[3] F. Cardoso and G. Popov, Rayleigh quasimodes in linear elasticity , Comm. Partial Differential Equations 17 (1992), no. 7-8, 1327-1367. · Zbl 0795.35067
[4] J. Duistermaat, Oscillatory integrals, Lagrange immersions and unfolding of singularities , Comm. Pure Appl. Math. 27 (1974), 207-281. · Zbl 0285.35010
[5] C. Gérard, Asymptotique des pôles de la matrice de scattering pour deux obstacles strictement convexes , Mém. Soc. Math. France (N.S.) 116 (1988), no. 31, 146. · Zbl 0654.35081
[6] I. Gohberg and M. Krein, Introduction to the theory of linear nonselfadjoint operators , Translated from the Russian by A. Feinstein. Translations of Mathematical Monographs, Vol. 18, American Mathematical Society, Providence, R.I., 1969. · Zbl 0181.13504
[7] R. Gregory, The propagation of Rayleigh waves over curved surfaces at high frequency , Proc. Cambridge Philos. Soc. 70 (1971), 103-121. · Zbl 0218.73036
[8] J. C. Guillot, Existence and uniqueness of a Rayleigh surface wave propagating along the free boundary of a transversely isotropic elastic half space , Math. Methods Appl. Sci. 8 (1986), no. 2, 289-310. · Zbl 0606.73024
[9] T. Harge and G. Lebeau, Diffraction par un convexe , Invent. Math. 118 (1994), no. 1, 161-196. · Zbl 0831.35121
[10] M. Ikawa, On the poles of the scattering matrix for two strictly convex obstacles , J. Math. Kyoto Univ. 23 (1983), no. 1, 127-194. · Zbl 0561.35060
[11] M. Ikawa, Precise informations on the poles of the scattering matrix for two strictly convex obstacles , J. Math. Kyoto Univ. 27 (1987), no. 1, 69-102. · Zbl 0637.35068
[12] M. Ikawa, Trapping obstacles with a sequence of poles of the scattering matrix converging to the real axis , Osaka J. Math. 22 (1985), no. 4, 657-689. · Zbl 0617.35102
[13] M. Ikehata and G. Nakamura, Decaying and nondecaying properties of the local energy of an elastic wave outside an obstacle , Japan J. Appl. Math. 6 (1989), no. 1, 83-95. · Zbl 0696.73017
[14] M. Kawashita, On the local-energy decay property for the elastic wave equation with the Neumann boundary condition , Duke Math. J. 67 (1992), no. 2, 333-351. · Zbl 0795.35061
[15] P. D. Lax and R. S. Phillips, Scattering theory for transport phenomena , Functional Analysis (Proc. Conf., Irvine, Calif., 1966), Academic Press, London, 1967, pp. 119-130. · Zbl 0214.12002
[16] P. D. Lax and R. S. Phillips, A logarithmic bound on the location of the poles of the scattering matrix , Arch. Rational Mech. Anal. 40 (1971), 268-280. · Zbl 0216.13002
[17] R. B. Melrose, Microlocal parametrices for diffractive boundary value problems , Duke Math. J. 42 (1975), no. 4, 605-635. · Zbl 0368.35055
[18] R. B. Melrose and J. Sjöstrand, Singularities of boundary value problems. I , Comm. Pure Appl. Math. 31 (1978), no. 5, 593-617. · Zbl 0368.35020
[19] R. B. Melrose and J. Sjöstrand, Singularities of boundary value problems. II , Comm. Pure Appl. Math. 35 (1982), no. 2, 129-168. · Zbl 0546.35083
[20] Lord Rayleigh, On waves propagated along plane surface of an elastic solid , Proc. London Math. Soc. (3) 17 (1885), 4-11. · JFM 17.0962.01
[21] M. A. Shubin, Pseudodifferential operators and spectral theory , Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1987. · Zbl 0616.47040
[22] J. Sjöstrand and M. Zworski, The complex scaling method for scattering by strictly convex obstacles , to appear in Ark. Mat. · Zbl 0839.35095
[23] P. Stefanov and G. Vodev, Distribution of resonances for the Neumann problem in linear elasticity outside a ball , Ann. Inst. H. Poincaré Phys. Théor. 60 (1994), no. 3, 303-321. · Zbl 0805.73016
[24] M. Taylor, Rayleigh waves in linear elasticity as a propagation of singularities phenomenon , Partial differential equations and geometry (Proc. Conf., Park City, Utah, 1977), Lecture Notes in Pure and Appl. Math., vol. 48, Dekker, New York, 1979, pp. 273-291. · Zbl 0432.73021
[25] M. Taylor, Pseudodifferential operators , Princeton Mathematical Series, vol. 34, Princeton University Press, Princeton, N.J., 1981. · Zbl 0453.47026
[26] E. C. Titchmarsh, The Theory of Functions , Oxford Univ. Press, Oxford, 1968. · Zbl 0005.21004
[27] K. Yamamoto, Singularities of solutions to the boundary value problems for elastic and Maxwell’s equations , Japan. J. Math. (N.S.) 14 (1988), no. 1, 119-163. · Zbl 0669.73017
[28] B. R. Vainberg, Asymptotic methods in equations of mathematical physics , Gordon & Breach Science Publishers, New York, 1989. · Zbl 0743.35001
[29] G. Vodev, Sharp bounds on the number of scattering poles for perturbations of the Laplacian , Comm. Math. Phys. 146 (1992), no. 1, 205-216. · Zbl 0766.35032
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.