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On corners scattering stably and stable shape determination by a single far-field pattern. (English) Zbl 1479.35648

Summary: In this paper, we establish two sharp quantitative results for the direct and inverse time-harmonic acoustic wave scattering. The first one is concerned with the recovery of the support of an inhomogeneous medium, independent of its contents, by a single far-field measurement. For this challenging inverse scattering problem, we establish a sharp stability estimate of logarithmic type when the medium support is a polyhedral domain in \(\mathbb{R}^n\), \(n=2,3\). The second one is concerned with the stability for corner scattering. More precisely, if an inhomogeneous scatterer, whose support has a corner, is probed by an incident plane-wave, we show that the energy of the scattered far-field possesses a positive lower bound depending only on the geometry of the corner and bounds on the refractive index of the medium there. This implies the impossibility of approximate invisibility cloaking by a device containing a corner and made of isotropic material. Our results sharply quantify the qualitative corner scattering results in the literature, and the corresponding proofs involve much more subtle analysis and technical arguments. As a significant byproduct of this study, we establish a quantitative Rellich’s theorem that continues smallness of the wave field from the far-field up to the interior of the inhomogeneity. The result is of significant mathematical interest for its own sake and is surprisingly not yet known in the literature.

MSC:

35Q35 PDEs in connection with fluid mechanics
76Q05 Hydro- and aero-acoustics
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35P25 Scattering theory for PDEs
74J20 Wave scattering in solid mechanics
35R30 Inverse problems for PDEs
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[1] G.ALESSANDRINIand L.RONDI,Determining a sound-soft polyhedral scatterer by a single farfield measurement, Proc. Amer. Math. Soc.133(2005), no. 6, 1685-1691.http://dx.doi.org/ 10.1090/S0002-9939-05-07810-X.MR2120253 · Zbl 1061.35160
[2] A.BEHZADANand M.HOLST,Multiplication in Sobolev Spaces, revisited(December 2015), preprint, available athttp://arxiv.org/abs/arXiv:1507.07895.
[3] E.BL˚ASTEN,Nonradiating sources and transmission eigenfunctions vanish at corners and edges, SIAM J. Math. Anal50(2018), no. 6, 6255-6270.https://doi-org.proxyiub.uits.iu. edu/10.1137/18M1182048.MR3885754 · Zbl 1409.35164
[4] E.BLASTEN˚, X.LI, H.LIU, and Y.WANG,On vanishing and localizing of transmission eigenfunctions near singular point: a numerical study, Inverse Problems33(2017), 105001, 24.https:// doi-org.proxyiub.uits.iu.edu/10.1088/1361-6420/aa8826.MR3706182 · Zbl 1442.65332
[5] E.BLASTEN˚and Y. H.LIN,Radiating and non-radiating sources in elasticity, Inverse Problems35 (2019), no. 1, 015005, 16.https://doi-org.proxyiub.uits.iu.edu/10.1088/1361-6420/ aae99e.MR3884602 · Zbl 1408.74032
[6] E.BLASTEN˚and H.LIU,On vanishing near corners of transmission eigenfunctions, J. Funct. Anal.273(2017), no. 11, 3616-3632.https://doi-org.proxyiub.uits.iu.edu/10.1016/ j.jfa.2017.08.023.MR3706612 · Zbl 1387.35437
[7] ,Recovering piecewise-constant refractive indices by a single far-field pattern, Inverse Problems36(2020), no. 8, 085005, 16.https://doi-org.proxyiub.uits.iu.edu/10.1088/ 1361-6420/ab958f.MR4149841 · Zbl 1446.35262
[8] ,Scattering by curvatures, radiationless sources, transmission eigenfunctions and inverse scattering problems, SIAM J. Math. Anal. (2021), in process. · Zbl 1479.35838
[9] E.BLASTEN˚, H.LIU, and J.XIAO,On an electromagnetic problem in a corner and its applications, Analysis and PDE (2020), in process.
[10] E.BLASTEN˚, L.P ¨AIVARINTA¨, and J.SYLVESTER,Corners always scatter, Comm. Math. Phys. 331(2014), no. 2, 725-753. http://dx.doi.org/10.1007/s00220-014-2030-0.MR3238529 · Zbl 1298.35214
[11] F.CAKONIand J.XIAO,On corner scattering for operators of divergence form and applications to inverse scattering(May 2019), preprint, available athttp://arxiv.org/abs/arXiv:arXiv: 1905.02558.
[12] X.CAO, H.DIAO, and H.LIU,Determining a piecewise conductive medium body by a single far-field measurement, CSIAM Trans. Appl. Math.1(2020), 740-765.
[13] ,On the geometric structures of transmission eigenfunctions with a conductive boundary condition and applications, Comm. Partial Differential Equations46(2021), no. 4, 630-679. https://doi-org.proxyiub.uits.iu.edu/10.1080/03605302.2020. 1857397.MR4260457 · Zbl 1475.35328
[14] H.CHENand C. T.CHAN,Acoustic cloaking in three dimensions using acoustic metamaterials, Applied Physics Letters91(October 2007), no. 18, 183518.http://dx.doi.org/10.1063/1. 2803315.
[15] J.CHENGand M.YAMAMOTO,Uniqueness in an inverse scattering problem within non-trapping polygonal obstacles with at most two incoming waves, Inverse Problems19(2003), no. 6, 1361- 1384.http://dx.doi.org/10.1088/0266-5611/19/6/008.MR2036535 · Zbl 1041.35078
[16] D.COLTONand R.KRESS,Inverse Acoustic and Electromagnetic Scattering Theory, 2nd ed., Applied Mathematical Sciences, vol. 93, Springer-Verlag, Berlin, 1998.http://dx.doi.org/ 10.1007/978-3-662-03537-5.MR1635980 · Zbl 0893.35138
[17] D.GILBARGand N. S.TRUDINGER,Elliptic Partial Differential Equations of Second Order, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983.http://dx.doi.org/10.1007/ 978-3-642-61798-0.MR737190 · Zbl 0562.35001
[18] A.GREENLEAF, Y.KURYLEV, M.LASSAS, and G.UHLMANN,Cloaking devices, electromagnetic wormholes, and transformation optics, SIAM Rev.51(2009), no. 1, 3-33.http://dx.doi. org/10.1137/080716827.MR2481110 · Zbl 1158.78004
[19] ,Invisibility and inverse problems, Bull. Amer. Math. Soc. (N.S.)46(2009), no. 1, 55-97. http://dx.doi.org/10.1090/S0273-0979-08-01232-9.MR2457072 · Zbl 1159.35074
[20] A.GREENLEAF, M.LASSAS, and G.UHLMANN,On nonuniqueness for Calder´on’s inverse problem, Math. Res. Lett.10(2003), no. 5-6, 685-693.http://dx.doi.org/10.4310/MRL.2003. v10.n5.a11.MR2024725 · Zbl 1054.35127
[21] L.H ¨ORMANDER,The Analysis of Linear Partial Differential Operators. II:Differential Operators with Constant Coefficients, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 257, Springer-Verlag, Berlin, 1983.http://dx.doi. org/10.1007/978-3-642-96750-4.MR705278 · Zbl 0521.35002
[22] G.HU, M.SALO, and E. V.VESALAINEN,Shape identification in inverse medium scattering problems with a single far-field pattern, SIAM J. Math. Anal.48(2016), no. 1, 152-165.http:// dx.doi.org/10.1137/15M1032958.MR3439763 · Zbl 1334.35427
[23] V.ISAKOV,Stability estimates for obstacles in inverse scattering, J. Comput. Appl. Math.42(1992), no. 1, 79-88.http://dx.doi.org/10.1016/0377-0427(92)90164-S.MR1181582 · Zbl 0767.65090
[24] ,New stability results for soft obstacles in inverse scattering, Inverse Problems9(1993), no. 5, 535-543.MR1242693 · Zbl 0789.35174
[25] ,Inverse Problems for Partial Differential Equations, 2nd ed., Applied Mathematical Sciences, vol. 127, Springer, New York, 2006.https://doi.org/10.1007/ 978-3-319-51658-5.MR2193218 · Zbl 1092.35001
[26] C. E.KENIG, A.RUIZ, and C. D.SOGGE,Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators, Duke Math. J.55(1987), no. 2, 329-347. http://dx.doi.org/10.1215/S0012-7094-87-05518-9.MR894584 · Zbl 0644.35012
[27] U.LEONHARDT,Optical conformal mapping, Science312(2006), no. 5781, 1777-1780. http://dx.doi.org/10.1126/science.1126493.MR2237569 · Zbl 1226.78001
[28] J.LI, H.LIU, L.RONDI, and G.UHLMANN,Regularized transformation-optics cloaking for the Helmholtz equation:From partial cloak to full cloak, Comm. Math. Phys.335(2015), no. 2, 671-712.http://dx.doi.org/10.1007/s00220-015-2318-8.MR3316643 · Zbl 1325.35209
[29] H.LIU,On local and global structures of transmission eigenfunctions and beyond, Journal of Inverse and Ill-posed Problems (2020), 000010151520200099.https://doi.org/10.1515/ jiip-2020-0099. · Zbl 1486.35320
[30] ,Virtual reshaping and invisibility in obstacle scattering, Inverse Problems25(2009), no. 4, 045006, 16.http://dx.doi.org/10.1088/0266-5611/25/4/045006.MR2482157 · Zbl 1169.35392
[31] H.LIU, M.PETRINI, L.RONDI, and J.XIAO,Stable determination of sound-hard polyhedral scatterers by a minimal number of scattering measurements, J. Differential Equations262(2017), no. 3, 1631-1670.http://dx.doi.org/10.1016/j.jde.2016.10.021.MR3582207 · Zbl 1352.74148
[32] H.LIUand H.SUN,Enhanced near-cloak by FSH lining, J. Math. Pures Appl. (9)99(2013), no. 1, 17-42 (English, with English and French summaries).http://dx.doi.org/10.1016/j. matpur.2012.06.001.MR3003281 · Zbl 1259.35223
[33] H.LIUand G.UHLMANN,Regularized transformation-optics cloaking in acoustic and electromagnetic scattering, Inverse Problems and Imaging, Panor. Synth‘eses, vol. 44, Soc. Math. France, Paris, 2015, pp. 111-136 (English, with English and French summaries).MR3497713 · Zbl 1346.35191
[34] H.LIUand J.ZOU,Uniqueness in determining multiple polygonal scatterers of mixed type, Discrete Contin. Dyn. Syst. Ser. B9(2008), no. 2, 375-396.http://dx.doi.org/10.3934/dcdsb. 2008.9.375.MR2373387 · Zbl 1156.76051
[35] ,Uniqueness in an inverse acoustic obstacle scattering problem for both sound-hard and soundsoft polyhedral scatterers, Inverse Problems22(2006), no. 2, 515-524.http://dx.doi.org/10. 1088/0266-5611/22/2/008.MR2216412 · Zbl 1094.35142
[36] A. I.NACHMAN,Reconstructions from boundary measurements, Ann. of Math. (2)128(1988), no. 3, 531-576.http://dx.doi.org/10.2307/1971435.MR970610 · Zbl 0675.35084
[37] L.P ¨AIVARINTA¨, M.SALO, andVESALAINENE. V.,Strictly convex corners scatter(April 2014), preprint, available athttp://arxiv.org/abs/arXiv:1404.2513.
[38] J. B.PENDRY, D.SCHURIG, and D. R.SMITH,Controlling electromagnetic fields, Science312 (2006), no. 5781, 1780-1782. http://dx.doi.org/10.1126/science.1125907.MR2237570 · Zbl 1226.78003
[39] RAKESHand G.UHLMANN,Uniqueness for the inverse backscattering problem for angularly controlled potentials, Inverse Problems30(2014), no. 6, 065005, 24.http://dx.doi.org/10. 1088/0266-5611/30/6/065005.MR3224125 · Zbl 1294.33016
[40] ,The point source inverse back-scattering problem, Analysis, Complex Geometry, and Mathematical Physics: In honor of Duong H. Phong, Contemp. Math., vol. 644, Amer. Math. Soc., Providence, RI, 2015, pp. 279-289.http://dx.doi.org/10.1090/conm/644/ 12784.MR3372472 · Zbl 1334.35433
[41] L.RONDI,Stable determination of sound-soft polyhedral scatterers by a single measurement, Indiana Univ. Math. J.57(2008), no. 3, 1377-1408.http://dx.doi.org/10.1512/iumj.2008.57. 3217.MR2429096 · Zbl 1152.35114
[42] L.RONDIand M.SINI,Stable determination of a scattered wave from its far-field pattern:The high frequency asymptotics, Arch. Ration. Mech. Anal.218(2015), no. 1, 1-54.http://dx.doi. org/10.1007/s00205-015-0855-0.MR3360734 · Zbl 1323.35151
[43] A.RUIZ,Harmonic Analysis and Inverse Problems:Lecture Notes, 2002, accessed 24.01.2018.
[44] J.SYLVESTERand G.UHLMANN,A global uniqueness theorem for an inverse boundary value problem, Ann. of Math. (2)125(1987), no. 1, 153-169.http://dx.doi.org/10.2307/ 1971291.MR873380 · Zbl 0625.35078
[45] H.TRIEBEL,Theory of Function Spaces, Monographs in Mathematics, vol. 78, Birkh¨auser Verlag, Basel, 1983.http://dx.doi.org/10.1007/978-3-0346-0416-1.MR781540 · Zbl 0546.46028
[46] ,Theory of Function Spaces. II, Monographs in Mathematics, vol. 84, Birkh¨auser Verlag, Basel, 1992.http://dx.doi.org/10.1007/978-3-0346-0419-2.MR1163193 · Zbl 0763.46025
[47] G.UHLMANN,Visibility and invisibility, ICIAM 07—6th International Congress on Industrial and Applied Mathematics, Eur. Math. Soc., Z¨urich, 2009, pp. 381-408.http://dx.doi.org/ 10.4171/056-1/18.MR2588602
[48] (ed.),Inverse Problems and Aplications: Inside Out. II, Mathematical Sciences Research Institute Publications, vol. 60, Cambridge University Press, Cambridge, 2013 · Zbl 1277.65002
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