Scattering by curvatures, radiationless sources, transmission eigenfunctions, and inverse scattering problems. (English) Zbl 1479.35838

Blåsten, Emilia L. K.; Liu, Hongyu

SIAM J. Math. Anal. 53, No. 4, 3801-3837 (2021).

The authors of this paper consider several connected topics of interest in the theory of wave propagation: geometrical characterizations of radiationless sources, nonradiating incident waves, interior transmission eigenfunctions, and their applications to inverse scattering. Assume that \(f:\mathbb{R}^n \to\mathbb{C}\) be a function with compact support. The source \(f\) produces a scattered wave \(u \in H_{\mathrm{loc}}^2(\mathbb{R}^n)\) given by the unique solution to the following nonhomogeneous elliptic differential equation \((\triangle +k^2)u=f\), \(\lim\limits_{r\to\infty }r^{(n-1)/2} (\partial_r-ik)u=0\), where \(r=|x|\), \(x\in \mathbb{R}^n\). The limit here is known as the Sommerfeld radiation condition, which characterizes the outgoing radiating wave. There exist two types of scenarios. The first scenario is concerned with radiationless or nonradiating monochromatic sources. The second one is concerned with nonradiating waves that impinge against a certain given scatterer consisting of an inhomogeneous index of refraction. The geometrical properties of such invisible objects would be of interest. The investigation of these objects allows to classify radiating sources and incident waves that are always radiating for scatterers. Thus the authors establish unique determination results for a longstanding inverse scattering problem in certain scenarios of practical importance. First it is proved an explicit relationship between the intensity of such a source and the diameter of its support. It suggests that if the support of a generic source is sufficiently small in terms of the wavelength, then it must not be radiationless. It means that its radiating pattern cannot be identically zero. This result generalizes the classical result on radiationless sources which states that for a source supported in a ball with a constant intensity, if the radius of the ball is sufficiently small, then the source must be radiating. This leads to a discussion on both important properties localization and geometrization. It is concluded that a scatterer, which can be an active source or an inhomogeneous index of refraction, cannot be completely invisible if its support is small compared to the wavelength and scattering intensity. Next conclusion is the localization and geometrization of the “smallness” results to the case where there is a high-curvature point on the boundary of the scatterer’s support. It is derived explicit bounds between the intensity of an invisible scatterer and its diameter or its curvature at the aforementioned point. These results characterize radiationless sources or nonradiating waves near high-curvature points. Interior transmission eigenfunctions with some regularity must be nearly vanishing at a high-curvature point on the boundary. It is stated that ”the higher the curvature, the smaller the eigenfunction must be at the high-curvature point”. The behavior of nearly vanishing implies that as long as the shape of a scatterer possesses a highly curved part, then it scatters every incident wave field nontrivially unless the wave is vanishingly small at the highly curved part. The practical implication of this result indicates that even if a scatterer has a very smooth shape, nontrivial scattering can be caused due to the curvature of the shape.
An application is that it is possible to derive new intrinsic geometric properties of interior transmission eigenfunctions near high-curvature points. It is established unique determination results for the single-wave Schiffer’s problem in certain scenarios of practical interest. Meanwhile, the authors state here that at their point of view ”it is the first result for Schiffer’s problem with generic smooth scatterers”.

Reviewer: Dimitar A. Kolev (Sofia)

MSC:

35Q60	PDEs in connection with optics and electromagnetic theory
78A46	Inverse problems (including inverse scattering) in optics and electromagnetic theory
35P25	Scattering theory for PDEs
78A05	Geometric optics
81U40	Inverse scattering problems in quantum theory

Keywords:

radiationless sources; invisible; transmission eigenfunctions; inverse shape problems; geometrical properties; single far-field pattern

Software:

DLMF

References:

[1]	R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2nd ed., Pure Appl. Math. 140 (Amsterdam), Elsevier/Academic Press, Amsterdam, 2003. · Zbl 1098.46001
[2]	G. Alessandrini and L. Rondi, Determining a sound-soft polyhedral scatterer by a single far-field measurement, Proc. Amer. Math. Soc., 133 (2005), pp. 1685-1691, https://doi.org/10.1090/S0002-9939-05-07810-X. · Zbl 1061.35160
[3]	H. Ammari, Y. T. Chow, and H. Liu, Localized Sensitivity Analysis at High-Curvature Boundary Points of Reconstructing Inclusions in Transmission Problems, https://arxiv.org/abs/1911.00820, 2021.
[4]	H. Ammari, H. Kang, H. Lee, and M. Lim, Enhancement of near-cloaking. Part II: The Helmholtz equation, Comm. Math. Phys., 317 (2013), pp. 485-502, https://doi.org/10.1007/s00220-012-1620-y. · Zbl 1260.35095
[5]	H. Ammari, H. Kang, H. Lee, and M. Lim, Enhancement of near cloaking using generalized polarization tensors vanishing structures. Part I: The conductivity problem, Comm. Math. Phys., 317 (2013), pp. 253-266, https://doi.org/10.1007/s00220-012-1615-8. · Zbl 1303.35108
[6]	E. Blåsten, Nonradiating sources and transmission eigenfunctions vanish at corners and edges, SIAM J. Math. Anal., 50 (2018), pp. 6255-6270, https://doi.org/10.1137/18M1182048. · Zbl 1409.35164
[7]	E. Blåsten, H. Li, H. Liu, and Y. Wang, Localization and geometrization in plasmon resonances and geometric structures of Neumann-Poincaré eigenfunctions, ESAIM Math. Model. Numer. Anal., 54 (2020), pp. 957-976, https://doi.org/10.1051/m2an/2019091. · Zbl 1437.35647
[8]	E. Blåsten, X. Li, H. Liu, and Y. Wang, On vanishing and localizing of transmission eigenfunctions near singular points: A numerical study, Inverse Problems, 33 (2017), 105001, https://doi.org/10.1088/1361-6420/aa8826. · Zbl 1442.65332
[9]	E. Blåsten and H. Liu, Addendum to: “On Vanishing Near Corners of Transmission Eigenfunctions,” https://arxiv.org/abs/1710.08089, 2017. · Zbl 1387.35437
[10]	E. Blåsten and H. Liu, On vanishing near corners of transmission eigenfunctions, J. Funct. Anal., 273 (2017), pp. 3616-3632, https://doi.org/10.1016/j.jfa.2017.08.023. · Zbl 1387.35437
[11]	E. Blåsten and H. Liu, On corners scattering stably, nearly non-scattering interrogating waves, and stable shape determination by a single far-field pattern, Indiana Univ. Math. J., 70 (2021), pp. 907-947. · Zbl 1479.35648
[12]	E. Blåsten and H. Liu, Recovering piecewise constant refractive indices by a single far-field pattern, Inverse Problems, 36 (2020), 085005, https://doi.org/10.1088/1361-6420/ab958f. · Zbl 1446.35262
[13]	E. Blåsten and L. Päivärinta, Completeness of generalized transmission eigenstates, Inverse Problems, 29 (2013), 104002, https://doi.org/10.1088/0266-5611/29/10/104002. · Zbl 1294.35042
[14]	E. Blåsten, L. Päivärinta, and J. Sylvester, Corners always scatter, Comm. Math. Phys., 331 (2014), pp. 725-753, https://doi.org/10.1007/s00220-014-2030-0. · Zbl 1298.35214
[15]	E. Blåsten and J. Sylvester, Translation-invariant estimates for operators with simple characteristics, J. Differential Equations, 263 (2017), pp. 5656-5695, https://doi.org/10.1016/j.jde.2017.06.033. · Zbl 1405.35019
[16]	N. Bleistein and J. K. Cohen, Nonuniqueness in the inverse source problem in acoustics and electromagnetics, J. Math. Phys., 18 (1977), pp. 194-201, https://doi.org/10.1063/1.523256. · Zbl 0379.76076
[17]	D. Bohm and M. Weinstein, The self-oscillations of a charged particle, Phys. Rev., 74 (1948), pp. 1789-1798, https://doi.org/10.1103/PhysRev.74.1789. · Zbl 0031.37805
[18]	F. Cakoni, D. Gintides, and H. Haddar, The existence of an infinite discrete set of transmission eigenvalues, SIAM J. Math. Anal., 42 (2010), pp. 237-255, https://doi.org/10.1137/090769338. · Zbl 1210.35282
[19]	F. Cakoni and H. Haddar, Transmission eigenvalues in inverse scattering theory, in Inverse Problems and Applications: Inside Out. II, Math. Sci. Res. Inst. Publ. 60, Cambridge University Press, Cambridge, UK, 2013, pp. 529-580. · Zbl 1316.35297
[20]	J. Cheng and M. Yamamoto, Uniqueness in an inverse scattering problem within non-trapping polygonal obstacles with at most two incoming waves, Inverse Problems, 19 (2003), pp. 1361-1384, https://doi.org/10.1088/0266-5611/19/6/008. · Zbl 1041.35078
[21]	Y.-T. Chow, Y. Deng, Y. He, H. Liu, and X. Wang, Surface-localized transmission eigenstates, super-resolution imaging and pseudo surface plasmon modes, SIAM J. Imaging Sci., to appear.
[22]	D. Colton, A. Kirsch, and L. Päivärinta, Far-field patterns for acoustic waves in an inhomogeneous medium, SIAM J. Math. Anal., 20 (1989), pp. 1472-1483, https://doi.org/10.1137/0520096. · Zbl 0681.76084
[23]	D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd ed., Appl. Math. Sci. 93, Springer-Verlag, Berlin, 1998, https://doi.org/10.1007/978-3-662-03537-5. · Zbl 0893.35138
[24]	D. Colton and P. Monk, The inverse scattering problem for time-harmonic acoustic waves in an inhomogeneous medium, Quart. J. Mech. Appl. Math., 41 (1988), pp. 97-125, https://doi.org/10.1093/qjmam/41.1.97. · Zbl 0637.73026
[25]	D. Colton and B. D. Sleeman, Uniqueness theorems for the inverse problem of acoustic scattering, IMA J. Appl. Math., 31 (1983), pp. 253-259, https://doi.org/10.1093/imamat/31.3.253. · Zbl 0539.76086
[26]	Y. Deng, H. Liu, and G. Uhlmann, On regularized full- and partial-cloaks in acoustic scattering, Comm. Partial Differential Equations, 42 (2017), pp. 821-851, https://doi.org/10.1080/03605302.2017.1286673. · Zbl 1375.35518
[27]	A. Devaney and E. Wolf, Non-radiating stochastic scalar sources, in Coherence and Quantum Optics V, L. Mandel and E. Wolf, eds., Plenum Press, New York, 1984, pp. 417-421.
[28]	A. J. Devaney and E. Wolf, Radiating and nonradiating classical current distributions and the fields they generate, Phys. Rev. D, 8 (1973), pp. 1044-1047, https://doi.org/10.1103/PhysRevD.8.1044.
[29]	H. Diao, X. Cao, and H. Liu, On the geometric structures of transmission eigenfunctions with a conductive boundary condition and applications, Comm. Partial Differential Equations, 46 (2021), pp. 630-679. · Zbl 1475.35328
[30]	P. Ehrenfest, Ungleichförmige elektrizitätsbewegungen ohne magnet- und strahlungsfeld, Physik. Zeit., 11 (1910), pp. 708-709. · JFM 41.0951.05
[31]	J. Elschner and M. Yamamoto, Uniqueness in determining polyhedral sound-hard obstacles with a single incoming wave, Inverse Problems, 24 (2008), 035004, https://doi.org/10.1088/0266-5611/24/3/035004. · Zbl 1187.35276
[32]	G. Eskin, Lectures on Linear Partial Differential Equations, Grad. Stud. Math. 123, AMS, Providence, RI, 2011, https://doi.org/10.1090/gsm/123. · Zbl 1228.35001
[33]	F. G. Friedlander, An inverse problem for radiation fields, Proc. London Math. Soc. (3), 27 (1973), pp. 551-576, https://doi.org/10.1112/plms/s3-27.3.551. · Zbl 0286.35050
[34]	A. Gamliel, K. Kim, A. I. Nachman, and E. Wolf, A new method for specifying nonradiating, monochromatic, scalar sources and their fields, J. Opt. Soc. Am. A, 6 (1989), pp. 1388-1393, https://doi.org/10.1364/JOSAA.6.001388.
[35]	G. Gbur, Nonradiating Sources and the Inverse Source Problem, Doctoral thesis, University of Rochester, 2001.
[36]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Grundlehren Math. Wiss. 224, Springer-Verlag, Berlin, 1983, https://doi.org/10.1007/978-3-642-61798-0. · Zbl 0562.35001
[37]	G. H. Goedecke, Classically radiationless motions and possible implications for quantum theory, Phys. Rev., 135 (1964), pp. B281-B288, https://doi.org/10.1103/PhysRev.135.B281. · Zbl 0125.19802
[38]	L. Grafakos, Classical Fourier Analysis, 2nd ed., Grad. Texts in Math. 249, Springer, New York, 2008. · Zbl 1220.42001
[39]	A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, Cloaking devices, electromagnetic wormholes, and transformation optics, SIAM Rev., 51 (2009), pp. 3-33, https://doi.org/10.1137/080716827. · Zbl 1158.78004
[40]	A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, Invisibility and inverse problems, Bull. Amer. Math. Soc. (N.S.), 46 (2009), pp. 55-97, https://doi.org/10.1090/S0273-0979-08-01232-9. · Zbl 1159.35074
[41]	A. Greenleaf, M. Lassas, and G. Uhlmann, Anisotropic conductivities that cannot be detected by EIT, Physiolog. Meas., 24 (2003), pp. 413-419.
[42]	R. Griesmaier, M. Hanke, and T. Raasch, Inverse source problems for the Helmholtz equation and the windowed Fourier transform, SIAM J. Sci. Comput., 34 (2012), pp. A1544-A1562, https://doi.org/10.1137/110855880. · Zbl 1251.35187
[43]	R. Griesmaier, M. Hanke, and T. Raasch, Inverse source problems for the Helmholtz equation and the windowed Fourier transform II, SIAM J. Sci. Comput., 35 (2013), pp. A2188-A2206, https://doi.org/10.1137/130908658. · Zbl 1282.35412
[44]	R. Griesmaier, M. Hanke, and J. Sylvester, Far field splitting for the Helmholtz equation, SIAM J. Numer. Anal., 52 (2014), pp. 343-362, https://doi.org/10.1137/120891381. · Zbl 1295.35381
[45]	B. J. Hoenders and H. P. Baltes, The scalar theory of nonradiating partially coherent sources, Lett. Nuovo Cimento, 25 (1979), pp. 206-208, https://doi.org/10.1007/BF02776289.
[46]	N. Honda, G. Nakamura, and M. Sini, Analytic extension and reconstruction of obstacles from few measurements for elliptic second order operators, Math. Ann., 355 (2013), pp. 401-427, https://doi.org/10.1007/s00208-012-0786-0. · Zbl 1263.35082
[47]	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), pp. 152-165, https://doi.org/10.1137/15M1032958. · Zbl 1334.35427
[48]	M. Ikehata, Reconstruction of a source domain from the Cauchy data, Inverse Problems, 15 (1999), pp. 637-645, https://doi.org/10.1088/0266-5611/15/2/019. · Zbl 0926.35161
[49]	V. Isakov, On uniqueness in the inverse transmission scattering problem, Comm. Partial Differential Equations, 15 (1990), pp. 1565-1587, https://doi.org/10.1080/03605309908820737. · Zbl 0728.35148
[50]	V. Isakov, Inverse Problems for Partial Differential Equations, 2nd ed., Appl. Math. Sci. 127, Springer, New York, 2006. · Zbl 1092.35001
[51]	K. Kim and E. Wolf, Non-radiating monochromatic sources and their fields, Opt. Commun., 59 (1986), pp. 1-6.
[52]	A. Kirsch, The denseness of the far field patterns for the transmission problem, IMA J. Appl. Math., 37 (1986), pp. 213-225, https://doi.org/10.1093/imamat/37.3.213. · Zbl 0652.35104
[53]	A. Kirsch and R. Kress, Uniqueness in inverse obstacle scattering, Inverse Problems, 9 (1993), pp. 285-299, http://stacks.iop.org/0266-5611/9/285. · Zbl 0787.35119
[54]	R. V. Kohn, D. Onofrei, M. S. Vogelius, and M. I. Weinstein, Cloaking via change of variables for the Helmholtz equation, Comm. Pure Appl. Math., 63 (2010), pp. 973-1016, https://doi.org/10.1002/cpa.20326. · Zbl 1194.35505
[55]	S. Kusiak and J. Sylvester, The scattering support, Comm. Pure Appl. Math., 56 (2003), pp. 1525-1548, https://doi.org/10.1002/cpa.3038. · Zbl 1037.35099
[56]	S. Kusiak and J. Sylvester, The convex scattering support in a background medium, SIAM J. Math. Anal., 36 (2005), pp. 1142-1158, https://doi.org/10.1137/S0036141003433577. · Zbl 1072.81060
[57]	E. Lakshtanov and B. Vainberg, Applications of elliptic operator theory to the isotropic interior transmission eigenvalue problem, Inverse Problems, 29 (2013), 104003, https://doi.org/10.1088/0266-5611/29/10/104003. · Zbl 1285.35059
[58]	P. D. Lax and R. S. Phillips, Scattering theory, Pure Appl. Math. 26, Academic Press, New York, 1967. · Zbl 0186.16301
[59]	U. Leonhardt, Optical conformal mapping, Science, 312 (2006), pp. 1777-1780, https://doi.org/10.1126/science.1126493. · Zbl 1226.78001
[60]	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), pp. 671-712, https://doi.org/10.1007/s00220-015-2318-8. · Zbl 1325.35209
[61]	H. Liu, Virtual reshaping and invisibility in obstacle scattering, Inverse Problems, 25 (2009), 045006, https://doi.org/10.1088/0266-5611/25/4/045006. · Zbl 1169.35392
[62]	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 Equations, 262 (2017), pp. 1631-1670, https://doi.org/10.1016/j.jde.2016.10.021. · Zbl 1352.74148
[63]	H. Liu, L. Rondi, and J. Xiao, Mosco convergence for \(H(curl)\) spaces, higher integrability for Maxwell’s equations, and stability in direct and inverse EM scattering problems, J. Eur. Math. Soc. (JEMS), 21 (2019), pp. 2945-2993, https://doi.org/10.4171/JEMS/895. · Zbl 1502.78020
[64]	H. Liu and G. Uhlmann, Regularized transformation-optics cloaking in acoustic and electromagnetic scattering, in Inverse Problems and Imaging, Panor. Synthèses 44, Société Mathématique de France, Paris, 2015, pp. 111-136. · Zbl 1346.35191
[65]	H. Liu and J. Zou, Uniqueness in an inverse acoustic obstacle scattering problem for both sound-hard and sound-soft polyhedral scatterers, Inverse Problems, 22 (2006), pp. 515-524, https://doi.org/10.1088/0266-5611/22/2/008. · Zbl 1094.35142
[66]	E. A. Marengo and R. W. Ziolkowski, On the radiating and nonradiating components of scalar, electromagnetic, and weak gravitational sources, Phys. Rev. Lett., 83 (1999), pp. 3345-3349, https://doi.org/10.1103/PhysRevLett.83.3345.
[67]	F. J. W. Olver, D. W. Lozier, R. F. Boisvert, and C. W. e. Clark, NIST Handbook of Mathematical Functions, Cambridge University Press, Cambridge, UK, 2010. · Zbl 1198.00002
[68]	F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds., NIST Digital Library of Mathematical Functions, Release 1.1.1, http://dlmf.nist.gov/, 2021.
[69]	L. Päivärinta, M. Salo, and E. V. Vesalainen, Strictly convex corners scatter, Rev. Mat. Iberoam., 33 (2017), pp. 1369-1396, https://doi.org/10.4171/RMI/975. · Zbl 1388.35138
[70]	L. Päivärinta and J. Sylvester, Transmission eigenvalues, SIAM J. Math. Anal., 40 (2008), pp. 738-753, https://doi.org/10.1137/070697525. · Zbl 1159.81411
[71]	P. Pearle, Classical electron models, in Electromagnetism Paths to Research, Plenum Press, New York, 1982, pp. 211-295.
[72]	J. B. Pendry, D. Schurig, and D. R. Smith, Controlling electromagnetic fields, Science, 312 (2006), pp. 1780-1782, https://doi.org/10.1126/science.1125907. · Zbl 1226.78003
[73]	L. Robbiano, Spectral analysis of the interior transmission eigenvalue problem, Inverse Problems, 29 (2013), 104001, https://doi.org/10.1088/0266-5611/29/10/104001. · Zbl 1296.35105
[74]	L. Rondi, Stable determination of sound-soft polyhedral scatterers by a single measurement, Indiana Univ. Math. J., 57 (2008), pp. 1377-1408, https://doi.org/10.1512/iumj.2008.57.3217. · Zbl 1152.35114
[75]	B. P. Rynne and B. D. Sleeman, The interior transmission problem and inverse scattering from inhomogeneous media, SIAM J. Math. Anal., 22 (1991), pp. 1755-1762, https://doi.org/10.1137/0522109. · Zbl 0733.76065
[76]	G. A. Schott, The electromagnetic field of a moving uniformly and rigidly electrified sphere and its radiationless orbits, London Edinburgh Dublin Philos. Magazine J. Sci., 15 (1933), pp. 752-761, https://doi.org/10.1080/14786443309462219. · Zbl 0006.38005
[77]	G. A. Schott, The uniform circular motion with invariable normal spin of a rigidly and uniformly electrified sphere, IV, Proc. A, 159 (1937), pp. 570-591, https://doi.org/10.1098/rspa.1937.0089. · Zbl 0016.33605
[78]	M. A. Shubin, ed., Partial Differential Equations. I, Foundations of the Classical Theory, Encyclopaedia Math. Sci. 30, Springer-Verlag, Berlin, 1992. · Zbl 0741.00024
[79]	A. Sommerfeld, Zur Elektronentheorie. I. Allgemeine Untersuchung des Feldes eines beliebig bewegten Elektrons, Akad. der Wiss. (Gött.), Math. Phys. Klasse, Nach., (1904), pp. 99-130, http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002499991. · JFM 35.0866.01
[80]	A. Sommerfeld, Zur Elektronentheorie. II. Grundlagen für eine allgemeine Dynamik des Elektrons, Akad. der Wiss. (Gött.), Math. Phys. Klasse, Nach., (1904), pp. 363-439, http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002500167. · JFM 35.0866.01
[81]	J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math. (2), 125 (1987), pp. 153-169, https://doi.org/10.2307/1971291. · Zbl 0625.35078
[82]	G. Uhlmann, ed., Inside Out: Inverse Problems and Applications, Math. Sci. Res. Inst. Publ. 47, Cambridge University Press, Cambridge, UK, 2003, https://doi.org/10.1090/conm/333. · Zbl 1034.78003