×

Spectral patterns of elastic transmission eigenfunctions: boundary localization, surface resonance, and stress concentration. (English) Zbl 07780345

Summary: We present a comprehensive study of new discoveries on the spectral patterns of elastic transmission eigenfunctions, including boundary localization, surface resonance, and stress concentration. In the case where the domain is radial and the underlying parameters are constant, we give rigorous justifications and derive a thorough understanding of those intriguing geometric and physical patterns. We also present numerical examples to verify that the same results hold in general geometric and parameter setups.

MSC:

35Q74 PDEs in connection with mechanics of deformable solids
74J15 Surface waves in solid mechanics
74J25 Inverse problems for waves in solid mechanics
74J20 Wave scattering in solid mechanics
35P15 Estimates of eigenvalues in context of PDEs
47A40 Scattering theory of linear operators
35B36 Pattern formations in context of PDEs
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Abramowitz, M. and Stegun, I. A., Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, U.S. Department of Commerce, Washington, DC, 1972. · Zbl 0543.33001
[2] Blåsten, E., Nonradiating sources and transmission eigenfunctions vanish at corners and edges, SIAM J. Math. Anal., 50 (2018), pp. 6255-6270, doi:10.1137/18M1182048. · Zbl 1409.35164
[3] Blåsten, E. and Liu, H., On vanishing near corners of transmission eigenfunctions, J. Funct. Anal., 273 (2017), pp. 3616-3632. · Zbl 1387.35437
[4] Blåsten, E. and Liu, H., Scattering by curvatures, radiationless sources, transmission eigenfunctions, and inverse scattering problems, SIAM J. Math. Anal., 53 (2021), pp. 3801-3837, doi:10.1137/20M1384002. · Zbl 1479.35838
[5] Blåsten, E., Liu, H., and Xiao, J., On an electromagnetic problem in a corner and its applications, Anal. PDE, 14 (2021), pp. 2207-2224. · Zbl 1480.78011
[6] Cakoni, F., Colton, D., and Haddar, H., Transmission eigenvalues, Notices Amer. Math. Soc., 68 (2021), pp. 1499-1510. · Zbl 1479.35589
[7] Chow, Y. T., Deng, Y., He, Y., Liu, H., and Wang, X., Surface-localized transmission eigenstates, super-resolution imaging and pseudo surface plasmon modes, SIAM J. Imaging Sci., 14 (2021), pp. 946-975, doi:10.1137/20M1388498. · Zbl 1478.35159
[8] Chow, Y. T., Deng, Y., Liu, H., and Sunkula, M., Surface concentration of transmission eigenfunctions, Arch. Ration. Mech. Anal., 274 (2023), 54. · Zbl 1517.35141
[9] Deng, Y., Liu, H., Wang, X., and Wu, W., On geometrical properties of electromagnetic transmission eigenfunctions and artificial mirage, SIAM J. Appl. Math., 82 (2022), pp. 1-24, doi:10.1137/21M1413547. · Zbl 1482.35149
[10] Deng, Y., Jiang, Y., Liu, H., and Zhang, K., On new surface-localized transmission eigenmodes, Inverse Probl. Imaging, 16 (2022), pp. 595-611. · Zbl 1487.35266
[11] Diao, H., Cao, X., and Liu, H., 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
[12] Diao, H., Li, H., Liu, H., and Tang, J., Spectral properties of an acoustic-elastic transmission eigenvalue problem with applications, J. Differential Equations, 371 (2023), pp. 629-659. · Zbl 1520.35109
[13] Diao, H., Liu, H., and Sun, B., On a local geometric property of the generalized elastic transmission eigenfunctions and application, Inverse Problems, 37 (2023), 105015. · Zbl 1479.35286
[14] Jiang, Y., Liu, H., Zhang, J., and Zhang, K., Boundary localization of transmission eigenfunctions in spherically stratified media, Asymptot. Anal., 132 (2023), pp. 285-303. · Zbl 07702838
[15] Jiang, Y., Liu, H., Zhang, J., and Zhang, K., Spectral Patterns of Elastic Transmission Eigenfunctions: Boundary Localisation, Surface Resonance and Stress Concentration, preprint, https://arxiv.org/abs/2211.16729, 2022.
[16] Krasikov, I., Uniform bounds for Bessel function, J. Appl. Anal., 12 (2006), pp. 83-91. · Zbl 1108.33004
[17] Liu, H., On local and global structures of transmission eigenfunctions and beyond, J. Inverse Ill-Posed Probl., 30 (2022), pp. 287-305. · Zbl 1486.35320
[18] Korenev, B. G., Bessel functions and their applications, Integral Transforms Spec. Funct., 25 (2002), pp. 272-282. · Zbl 1285.44001
[19] Meng, Q., Bai, Z., Diao, H., and Liu, H., Effective medium theory for embedded obstacles in elasticity with applications to inverse problems, SIAM J. Appl. Math., 82 (2022), pp. 720-749, doi:10.1137/21M1431369. · Zbl 1495.35024
[20] Paris, R., An inequality for the Bessel function \(J_{\nu } (\nu x)\), SIAM J. Math. Anal., 15 (1984), pp. 203-205, doi:10.1137/0515016.
[21] Wong, R. and Qu, C., Best possible upper and lower bounds for the zeros of the Bessel function \(J_{\nu }(x)\), Trans. Amer. Math. Soc., 35 (1999), pp. 2833-2859. · Zbl 0930.41020
[22] Yang, Y., Wang, S., and Bi, H., The finite element method for the elastic transmission eigenvalue problem with different elastic tensors, J. Sci. Comput., 93 (2022), pp. 65-84. · Zbl 1503.65284
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.