Uniqueness in inverse elastic scattering from unbounded rigid surfaces of rectangular type

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February 2015, 9(1): 127-141. doi: 10.3934/ipi.2015.9.127

Uniqueness in inverse elastic scattering from unbounded rigid surfaces of rectangular type

Johannes Elschner ^1, , Guanghui Hu ^2, and Masahiro Yamamoto ^3,

1.	Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin
2.	Weierstrass Institute, Mohrenstr. 39, 10117 Berlin
3.	Graduate School of Mathematical Sciences University of Tokyo, 3-8-1 Komaba, Tokyo, 153-8914

Received June 2014 Revised September 2014 Published January 2015

Abstract
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Consider the two-dimensional inverse elastic scattering problem of recovering a piecewise linear rigid rough or periodic surface of rectangular type for which the neighboring line segments are always perpendicular. We prove the global uniqueness with at most two incident elastic plane waves by using near-field data. If the Lamé constants satisfy a certain condition, then the data of a single plane wave is sufficient to imply the uniqueness. Our proof is based on a transcendental equation for the Navier equation, which is derived from the expansion of analytic solutions to the Helmholtz equation. The uniqueness results apply also to an inverse scattering problem for non-convex bounded rigid bodies of rectangular type.

Keywords: Inverse scattering, rough surface, Navier equation, uniqueness, diffraction grating., linear elasticity, Dirichlet boundary condition.

Mathematics Subject Classification: Primary: 74J25; Secondary: 74J20, 35R30, 35Q7.

Citation: Johannes Elschner, Guanghui Hu, Masahiro Yamamoto. Uniqueness in inverse elastic scattering from unbounded rigid surfaces of rectangular type. Inverse Problems & Imaging, 2015, 9 (1) : 127-141. doi: 10.3934/ipi.2015.9.127

References:

[1]	G. Bao, H. Zhang and J. Zou, Unique determination of periodic polyhedral structures by scattered electromagnetic fields II: The resonance case,, Trans. Amer. Math. Soc., 366* (2014), 1333. doi: 10.1090/S0002-9947-2013-05761-3.*
[2]	A. Charalambopoulos, D. Gintides and K. Kiriaki, On the uniqueness of the inverse elastic scattering problem for periodic structures,, Inverse Problems, 17 (2001), 1923. doi: 10.1088/0266-5611/17/6/323.
[3]	M. Costabel, M. Dauge and Y. Lafranche, Fast semi-analytic computation of elastic edge singularities,, Computer Methods in Applied Mechanics and Engineering, 190* (2001), 2111. doi: 10.1016/S0045-7825(00)00226-7.*
[4]	D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory,, Springer, (1998). doi: 10.1007/978-3-662-03537-5.
[5]	J. Elschner and G. Hu, Variational approach to scattering of plane elastic waves by diffraction gratings,, Math. Meth. Appl. Sci., 33 (2010), 1924. doi: 10.1002/mma.1305.
[6]	J. Elschner and G. Hu, Global uniqueness in determining polygonal periodic structures with a minimal number of incident plane waves,, Inverse Problems, 26 (2010). doi: 10.1088/0266-5611/26/11/115002.
[7]	J. Elschner and G. Hu, Inverse scattering of elastic waves by periodic structures: Uniqueness under the third or fourth kind boundary conditions,, Meth. Appl. Anal., 18 (2011), 215. doi: 10.4310/MAA.2011.v18.n2.a6.
[8]	J. Elschner and G. Hu, An optimization method in inverse elastic scattering for one-dimensional grating profiles,, Commun. Comput. Phys., 12 (2012), 1434.
[9]	J. Elschner and G. Hu, Elastic scattering by unbounded rough surfaces: Solvability in weighted Sobolev spaces,, Appl. Anal., 94 (2015), 251. doi: 10.1080/00036811.2014.887695.
[10]	J. Elschner, G. Schmidt and M. Yamamoto, Global uniqueness in determining rectangular periodic structures by scattering data with a single wave number,, J. Inverse Ill-Posed Probl., 11 (2003), 235. doi: 10.1515/156939403769237024.
[11]	F. Hettlich and A. Kirsch, Schiffer's theorem in inverse scattering for periodic structures,, Inverse Problems, 13 (1997), 351. doi: 10.1088/0266-5611/13/2/010.
[12]	G. Hu, Inverse wave scattering by unbounded obstacles: Uniqueness for the two-dimensional Helmholtz equation,, Appl. Anal., 91 (2012), 703. doi: 10.1080/00036811.2011.587811.
[13]	G. Hu, Y. Lu and B. Zhang, The factorization method for inverse elastic scattering from periodic structures,, Inverse Problems, 29 (2013). doi: 10.1088/0266-5611/29/11/115005.
[14]	P. Grisvard, Singularités en élasticité,, Arch. Rational Mech. Anal., 107* (1989), 157. doi: 10.1007/BF00286498.*
[15]	A. Kirsch, Diffraction by periodic structures,, In 'Proc. Lapland Conf. Inverse Problems' (ed. L. Päivärinta et al), 422* (1993), 87. doi: 10.1007/3-540-57195-7_11.*
[16]	C. B. Morrey, Jr. and L. Nirenberg, On the analyticity of the solutions of linear elliptic systems of partial differential equations,, Communications on Pure and Applied Mathematics, 10 (1957), 271. doi: 10.1002/cpa.3160100204.
[17]	S. Nakagawa, K. T. Nihei, L. R. Myer and E. L. Majer, Three-dimensional elastic wave scattering by a layer containing vertical periodic fractures,, J. Acoust. Soc. Am., 113* (2003), 3012. doi: 10.1121/1.1572139.*
[18]	A. Rössle, Corner singularities and regularity of weak solution for the two-dimensional Lamé equations on domains with angular corners,, Journal of Elasticity, 60 (2000), 57. doi: 10.1023/A:1007639413619.

show all references

References:

[1]	G. Bao, H. Zhang and J. Zou, Unique determination of periodic polyhedral structures by scattered electromagnetic fields II: The resonance case,, Trans. Amer. Math. Soc., 366* (2014), 1333. doi: 10.1090/S0002-9947-2013-05761-3.*
[2]	A. Charalambopoulos, D. Gintides and K. Kiriaki, On the uniqueness of the inverse elastic scattering problem for periodic structures,, Inverse Problems, 17 (2001), 1923. doi: 10.1088/0266-5611/17/6/323.
[3]	M. Costabel, M. Dauge and Y. Lafranche, Fast semi-analytic computation of elastic edge singularities,, Computer Methods in Applied Mechanics and Engineering, 190* (2001), 2111. doi: 10.1016/S0045-7825(00)00226-7.*
[4]	D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory,, Springer, (1998). doi: 10.1007/978-3-662-03537-5.
[5]	J. Elschner and G. Hu, Variational approach to scattering of plane elastic waves by diffraction gratings,, Math. Meth. Appl. Sci., 33 (2010), 1924. doi: 10.1002/mma.1305.
[6]	J. Elschner and G. Hu, Global uniqueness in determining polygonal periodic structures with a minimal number of incident plane waves,, Inverse Problems, 26 (2010). doi: 10.1088/0266-5611/26/11/115002.
[7]	J. Elschner and G. Hu, Inverse scattering of elastic waves by periodic structures: Uniqueness under the third or fourth kind boundary conditions,, Meth. Appl. Anal., 18 (2011), 215. doi: 10.4310/MAA.2011.v18.n2.a6.
[8]	J. Elschner and G. Hu, An optimization method in inverse elastic scattering for one-dimensional grating profiles,, Commun. Comput. Phys., 12 (2012), 1434.
[9]	J. Elschner and G. Hu, Elastic scattering by unbounded rough surfaces: Solvability in weighted Sobolev spaces,, Appl. Anal., 94 (2015), 251. doi: 10.1080/00036811.2014.887695.
[10]	J. Elschner, G. Schmidt and M. Yamamoto, Global uniqueness in determining rectangular periodic structures by scattering data with a single wave number,, J. Inverse Ill-Posed Probl., 11 (2003), 235. doi: 10.1515/156939403769237024.
[11]	F. Hettlich and A. Kirsch, Schiffer's theorem in inverse scattering for periodic structures,, Inverse Problems, 13 (1997), 351. doi: 10.1088/0266-5611/13/2/010.
[12]	G. Hu, Inverse wave scattering by unbounded obstacles: Uniqueness for the two-dimensional Helmholtz equation,, Appl. Anal., 91 (2012), 703. doi: 10.1080/00036811.2011.587811.
[13]	G. Hu, Y. Lu and B. Zhang, The factorization method for inverse elastic scattering from periodic structures,, Inverse Problems, 29 (2013). doi: 10.1088/0266-5611/29/11/115005.
[14]	P. Grisvard, Singularités en élasticité,, Arch. Rational Mech. Anal., 107* (1989), 157. doi: 10.1007/BF00286498.*
[15]	A. Kirsch, Diffraction by periodic structures,, In 'Proc. Lapland Conf. Inverse Problems' (ed. L. Päivärinta et al), 422* (1993), 87. doi: 10.1007/3-540-57195-7_11.*
[16]	C. B. Morrey, Jr. and L. Nirenberg, On the analyticity of the solutions of linear elliptic systems of partial differential equations,, Communications on Pure and Applied Mathematics, 10 (1957), 271. doi: 10.1002/cpa.3160100204.
[17]	S. Nakagawa, K. T. Nihei, L. R. Myer and E. L. Majer, Three-dimensional elastic wave scattering by a layer containing vertical periodic fractures,, J. Acoust. Soc. Am., 113* (2003), 3012. doi: 10.1121/1.1572139.*
[18]	A. Rössle, Corner singularities and regularity of weak solution for the two-dimensional Lamé equations on domains with angular corners,, Journal of Elasticity, 60 (2000), 57. doi: 10.1023/A:1007639413619.

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