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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

 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

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.
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), 1333-1361. doi: 10.1090/S0002-9947-2013-05761-3.  Google Scholar [2] A. Charalambopoulos, D. Gintides and K. Kiriaki, On the uniqueness of the inverse elastic scattering problem for periodic structures, Inverse Problems, 17 (2001), 1923-1935. doi: 10.1088/0266-5611/17/6/323.  Google Scholar [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), 2111-2134. doi: 10.1016/S0045-7825(00)00226-7.  Google Scholar [4] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Springer, Berlin, 1998. doi: 10.1007/978-3-662-03537-5.  Google Scholar [5] J. Elschner and G. Hu, Variational approach to scattering of plane elastic waves by diffraction gratings, Math. Meth. Appl. Sci., 33 (2010), 1924-1941. doi: 10.1002/mma.1305.  Google Scholar [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), 115002, 23pp. doi: 10.1088/0266-5611/26/11/115002.  Google Scholar [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), 215-243. doi: 10.4310/MAA.2011.v18.n2.a6.  Google Scholar [8] J. Elschner and G. Hu, An optimization method in inverse elastic scattering for one-dimensional grating profiles, Commun. Comput. Phys., 12 (2012), 1434-1460.  Google Scholar [9] J. Elschner and G. Hu, Elastic scattering by unbounded rough surfaces: Solvability in weighted Sobolev spaces, Appl. Anal., 94 (2015), 251-278. doi: 10.1080/00036811.2014.887695.  Google Scholar [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), 235-244. doi: 10.1515/156939403769237024.  Google Scholar [11] F. Hettlich and A. Kirsch, Schiffer's theorem in inverse scattering for periodic structures, Inverse Problems, 13 (1997), 351-361. doi: 10.1088/0266-5611/13/2/010.  Google Scholar [12] G. Hu, Inverse wave scattering by unbounded obstacles: Uniqueness for the two-dimensional Helmholtz equation, Appl. Anal., 91 (2012), 703-717. doi: 10.1080/00036811.2011.587811.  Google Scholar [13] G. Hu, Y. Lu and B. Zhang, The factorization method for inverse elastic scattering from periodic structures, Inverse Problems, 29 (2013), 115005, 25pp. doi: 10.1088/0266-5611/29/11/115005.  Google Scholar [14] P. Grisvard, Singularités en élasticité, Arch. Rational Mech. Anal., 107 (1989), 157-180. doi: 10.1007/BF00286498.  Google Scholar [15] A. Kirsch, Diffraction by periodic structures, In 'Proc. Lapland Conf. Inverse Problems' (ed. L. Päivärinta et al), Springer, Berlin, Lecture Notes in Phys., 422 (1993), 87-102. doi: 10.1007/3-540-57195-7_11.  Google Scholar [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), 271-290. doi: 10.1002/cpa.3160100204.  Google Scholar [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), 3012-3023. doi: 10.1121/1.1572139.  Google Scholar [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), 57-75. doi: 10.1023/A:1007639413619.  Google Scholar

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##### 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), 1333-1361. doi: 10.1090/S0002-9947-2013-05761-3.  Google Scholar [2] A. Charalambopoulos, D. Gintides and K. Kiriaki, On the uniqueness of the inverse elastic scattering problem for periodic structures, Inverse Problems, 17 (2001), 1923-1935. doi: 10.1088/0266-5611/17/6/323.  Google Scholar [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), 2111-2134. doi: 10.1016/S0045-7825(00)00226-7.  Google Scholar [4] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Springer, Berlin, 1998. doi: 10.1007/978-3-662-03537-5.  Google Scholar [5] J. Elschner and G. Hu, Variational approach to scattering of plane elastic waves by diffraction gratings, Math. Meth. Appl. Sci., 33 (2010), 1924-1941. doi: 10.1002/mma.1305.  Google Scholar [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), 115002, 23pp. doi: 10.1088/0266-5611/26/11/115002.  Google Scholar [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), 215-243. doi: 10.4310/MAA.2011.v18.n2.a6.  Google Scholar [8] J. Elschner and G. Hu, An optimization method in inverse elastic scattering for one-dimensional grating profiles, Commun. Comput. Phys., 12 (2012), 1434-1460.  Google Scholar [9] J. Elschner and G. Hu, Elastic scattering by unbounded rough surfaces: Solvability in weighted Sobolev spaces, Appl. Anal., 94 (2015), 251-278. doi: 10.1080/00036811.2014.887695.  Google Scholar [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), 235-244. doi: 10.1515/156939403769237024.  Google Scholar [11] F. Hettlich and A. Kirsch, Schiffer's theorem in inverse scattering for periodic structures, Inverse Problems, 13 (1997), 351-361. doi: 10.1088/0266-5611/13/2/010.  Google Scholar [12] G. Hu, Inverse wave scattering by unbounded obstacles: Uniqueness for the two-dimensional Helmholtz equation, Appl. Anal., 91 (2012), 703-717. doi: 10.1080/00036811.2011.587811.  Google Scholar [13] G. Hu, Y. Lu and B. Zhang, The factorization method for inverse elastic scattering from periodic structures, Inverse Problems, 29 (2013), 115005, 25pp. doi: 10.1088/0266-5611/29/11/115005.  Google Scholar [14] P. Grisvard, Singularités en élasticité, Arch. Rational Mech. Anal., 107 (1989), 157-180. doi: 10.1007/BF00286498.  Google Scholar [15] A. Kirsch, Diffraction by periodic structures, In 'Proc. Lapland Conf. Inverse Problems' (ed. L. Päivärinta et al), Springer, Berlin, Lecture Notes in Phys., 422 (1993), 87-102. doi: 10.1007/3-540-57195-7_11.  Google Scholar [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), 271-290. doi: 10.1002/cpa.3160100204.  Google Scholar [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), 3012-3023. doi: 10.1121/1.1572139.  Google Scholar [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), 57-75. doi: 10.1023/A:1007639413619.  Google Scholar
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