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The Foldy-Lax approximation of the scattered waves by many small bodies for the Lamé system. (English) Zbl 1334.35338

The authors study a 3D elastic scattering problem with small rigid obstacles \(D_m\) (\(m=1,2,\dots,M\)) of arbitrary, Lipschitz regular, shapes. The Navier equation \((\Delta^e +\omega^2) U=0\), \(\Delta^e := \mu \Delta +(\lambda+\mu) \nabla\operatorname{div}\), is fulfilled in a multiply connected domain with the boundary conditions \(U=0\) on \(\cup_{m=1}^M\partial D_m\). The Kupradze radiation conditions hold at infinity. Let \(a\) denote the maximal diameter of obstacles and \(d\) the minimum distance between them. It is proved that there exist constants \(a_0\) and \(c_0\) depending only on the Lipschitz constants of \(\partial D_m\) such that the points Foldy-Lax approximation of the far field is valid under the conditions \(a \leq a_0\) and \(\sqrt{M-1} \frac ad \leq c_0\). Applications to inverse problems are discussed.

MSC:

35Q74 PDEs in connection with mechanics of deformable solids
74J20 Wave scattering in solid mechanics
35J40 Boundary value problems for higher-order elliptic equations
45Q05 Inverse problems for integral equations
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[1] Ahmad, The equivalent refraction index for the acoustic scattering by many small obstacles: with error estimates, J. Math. Anal. Appl. 424 (1) pp 563– (2015) · Zbl 1305.76094 · doi:10.1016/j.jmaa.2014.11.020
[2] F. Al D. P. Challa M. Sini Location and size estimation of small rigid bodies using elastic far-fields · Zbl 1348.35308
[3] Alves, On the far-field operator in elastic obstacle scattering, IMA J. Appl. Math. 67 (1) pp 1– (2002) · Zbl 1141.35429 · doi:10.1093/imamat/67.1.1
[4] Ammari, Localization, stability, and resolution of topological derivative based imaging functionals in elasticity, SIAM J. Imaging Sci. 6 (4) pp 2174– (2013) · Zbl 1281.35085 · doi:10.1137/120899303
[5] Ammari, Direct elastic imaging of a small inclusion, SIAM J. Imaging Sci. 1 (2) pp 169– (2008) · Zbl 1179.35341 · doi:10.1137/070696076
[6] Ammari, Mathematical and Statistical Methods for Multistatic Imaging, Lecture Notes in Mathematics 2098 (2013) · Zbl 1288.35001
[7] Ammari, Polarization and Moment Tensors, Applied Mathematical Sciences 162 (2007) · Zbl 1220.35001
[8] Ammari, Reconstruction of closely spaced small inclusions, SIAM J. Numer. Anal. 42 (6) pp 2408– (2005) · Zbl 1081.35133 · doi:10.1137/S0036142903422752
[9] Ammari, Asymptotic expansions for eigenvalues of the Lamé system in the presence of small inclusions, Comm. Partial Differential Equations 32 (10-12) pp 1715– (2007) · Zbl 1136.35058 · doi:10.1080/03605300600910266
[10] Ammari, Effective parameters of elastic composites, Indiana Univ. Math. J. 55 (3) pp 903– (2006) · Zbl 1210.74037 · doi:10.1512/iumj.2006.55.2681
[11] Ammari, Complete asymptotic expansions of solutions of the system of elastostatics in the presence of an inclusion of small diameter and detection of an inclusion, J. Elasticity 67 (2) pp 97– (2002) · Zbl 1089.74576 · doi:10.1023/A:1023940025757
[12] Ammari, An accurate formula for the reconstruction of conductivity inhomogeneities, Adv. in Appl. Math. 30 (4) pp 679– (2003) · Zbl 1040.78008 · doi:10.1016/S0196-8858(02)00557-2
[13] Bensoussan, Asymptotic Analysis for Periodic Structures, Studies in Mathematics and its Applications 5 (1978) · Zbl 0404.35001
[14] Cassier, Multiple scattering of acoustic waves by small sound-soft obstacles in two dimensions: mathematical justification of the Foldy-Lax model, Wave Motion 50 (1) pp 18– (2013) · Zbl 1360.35047 · doi:10.1016/j.wavemoti.2012.06.001
[15] Challa, Inverse scattering by point-like scatterers in the Foldy regime, Inverse Problems 28 (12) pp 125006– (2012) · Zbl 1322.76055 · doi:10.1088/0266-5611/28/12/125006
[16] Challa, On the justification of the Foldy-Lax approximation for the acoustic scattering by small rigid bodies of arbitrary shapes, Multiscale Model. Simul. 12 (1) pp 55– (2014) · Zbl 1311.35070 · doi:10.1137/130919313
[17] Colton, Inverse Acoustic and Electromagnetic Scattering Theory, Applied Mathematical Sciences 93 (1998) · Zbl 0893.35138 · doi:10.1007/978-3-662-03537-5
[18] Colton, Integral equation methods in scattering theory, Pure and Applied Mathematics (New York) (1983) · Zbl 0522.35001
[19] Hu, Inverse elastic scattering for multiscale rigid bodies with a single far-field pattern, SIAM J. Imaging Sci. 7 (3) pp 1799– (2014) · Zbl 1305.74046 · doi:10.1137/130944187
[20] Hu, Elastic scattering by finitely many point-like obstacles, J. Math. Phys. 54 (4) pp 042901– (2013) · Zbl 1282.74043 · doi:10.1063/1.4799145
[21] Jikov, Homogenization of Differential Operators and Integral Functionals (1994) · doi:10.1007/978-3-642-84659-5
[22] Kupradze, Potential Methods in the Theory of Elasticity (1965)
[23] Kupradze, Three-Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity, North-Holland Series in Applied Mathematics and Mechanics 25 (1979)
[24] Marchenko, Homogenization of Partial Differential Equations, Progress in Mathematical Physics 46 (2006)
[25] Martin, Multiple Scattering of Encyclopedia of Mathematics and its Applications 107 (2006)
[26] Mayboroda, Integral Methods in Science and Engineering pp 137– (2006) · Zbl 1330.35441 · doi:10.1007/0-8176-4450-4_13
[27] Maz’ya, Asymptotic treatment of perforated domains without homogenization, Math. Nachr. 283 (1) pp 104– (2010) · Zbl 1185.35060 · doi:10.1002/mana.200910045
[28] Maz’ya, Uniform asymptotic formulae for Green’s tensors in elastic singularly perturbed domains, Asymptot. Anal. 52 (3-4) pp 173– (2007)
[29] Maz’ya, Mesoscale asymptotic approximations to solutions of mixed boundary value problems in perforated domains, Multiscale Model. Simul. 9 (1) pp 424– (2011) · Zbl 1223.35293 · doi:10.1137/100791294
[30] Maz’ya, Green’s Kernels and Meso-Scale Approximations in Perforated Domains, of Lecture Notes in Mathematics 2077 (2013) · Zbl 1273.35007 · doi:10.1007/978-3-319-00357-3
[31] Mendez, The Banach envelopes of Besov and Triebel-Lizorkin spaces and applications to partial differential equations, J. Fourier Anal. Appl. 6 (5) pp 503– (2000) · Zbl 0972.46017 · doi:10.1007/BF02511543
[32] Mitrea, The method of layer potentials for non-smooth domains with arbitrary topology, Integral Equations Operator Theory 29 (3) pp 320– (1997) · Zbl 0915.35032 · doi:10.1007/BF01320705
[33] Namias, A simple derivation of Stirling’s asymptotic series, Amer. Math. Monthly 93 (1) pp 25– (1986) · Zbl 0615.05010 · doi:10.2307/2322540
[34] Nazarov, Self-adjoint extensions for the Neumann Laplacian and applications, Acta Math. Sin. (Engl. Ser.) 22 (3) pp 879– (2006) · Zbl 1284.49048 · doi:10.1007/s10114-005-0652-z
[35] Ramm, Wave Scattering by Small Bodies of Arbitrary Shapes (2005) · Zbl 1081.78001 · doi:10.1142/5765
[36] Ramm, Many-body wave scattering by small bodies and applications, J. Math. Phys. 48 (10) pp 103511 29– (2007) · Zbl 1152.81588 · doi:10.1063/1.2799258
[37] Ramm, Wave scattering by small bodies and creating materials with a desired refraction coefficient, Afr. Mat. 22 (1) pp 33– (2011) · Zbl 1302.35138 · doi:10.1007/s13370-011-0004-3
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