Wave enhancement through optimization of boundary conditions. (English) Zbl 1430.35171

Summary: In this paper, we present a new and efficient approach for optimizing the transmission signal between two points in a cavity at a given frequency, by changing boundary conditions. The proposed approach makes use of recent results on the monotonicity of the eigenvalues of the mixed boundary value problem and on the sensitivity of Green’s function to small changes in the boundary conditions. The switching of the boundary condition from Dirichlet to Neumann can be performed through the use of the recently modeled concept of metasurfaces which are comprised of coupled pairs of Helmholtz resonators. A variety of numerical experiments are presented to show the applicability and the accuracy of the proposed new methodology.


35P15 Estimates of eigenvalues in context of PDEs
35R30 Inverse problems for PDEs
35C20 Asymptotic expansions of solutions to PDEs
Full Text: DOI arXiv


[1] E. Akhmetgaliyev and O. P. Bruno, Regularized integral formulation of mixed Dirichlet-Neumann problems, J. Integral Equations Appl., 29 (2017), pp. 493-529, https://doi.org/10.1216/JIE-2017-29-4-493. · Zbl 1387.35116
[2] E. Akhmetgaliyev, O. P. Bruno, and N. Nigam, A boundary integral algorithm for the Laplace Dirichlet-Neumann mixed eigenvalue problem, J. Comput. Phys., 298 (2015), pp. 1-28, https://doi.org/10.1016/j.jcp.2015.05.016. · Zbl 1349.65600
[3] H. Ammari, B. Fitzpatrick, H. Kang, M. Ruiz, S. Yu, and H. Zhang, Mathematical and Computational Methods in Photonics and phononics, Math. Surveys Monogr. 235, American Mathematical Society, Providence, RI, 2018, https://doi.org/10.1090/surv/235. · Zbl 1420.78001
[4] H. Ammari, K. Imeri, and W. Wu, A mathematical framework for tunable metasurfaces. Part I, Asymptotic Anal., 114 (2019), pp. 129-179. · Zbl 1443.35153
[5] H. Ammari, K. Imeri, and W. Wu, A mathematical framework for tunable metasurfaces. Part II, Asymptotic Anal., 114 (2019), pp. 181-209. · Zbl 1442.35443
[6] H. Ammari, K. Kalimeris, H. Kang, and H. Lee, Layer potential techniques for the narrow escape problem, J. Math. Pures Appl., 97 (2012), pp. 66-84. · Zbl 1252.35116
[7] H. Ammari, H. Kang, and H. Lee, Layer potential techniques in spectral analysis, Math. Surveys Monogr. 153, American Mathematical Society, Providence, RI, 2009, https://doi.org/10.1090/surv/153. · Zbl 1167.47001
[8] M. Sh. Birman and M. Z. Solomjak, Spectral Theory of Self-Adjoint Operators in Hilbert Space, D. Reidel Publishing Co., Dordrecht, 1987. Translated from the 1980 Russian original by S. Khrushchëv and V. Peller.
[9] A. Dabrowski, Explicit terms in the small volume expansion of the shift of Neumann Laplacian eigenvalues due to a grounded inclusion in two dimensions, J. Math. Anal. Appl., 456 (2017), pp. 731-744, https://doi.org/10.1016/j.jmaa.2017.07.027. · Zbl 06864214
[10] N. Filonov, On an inequality for the eigenvalues of the Dirichlet and Neumann problems for the Laplace operator, Algebra i Analiz, 16 (2004), pp. 172-176, https://doi.org/10.1090/S1061-0022-05-00857-5.
[11] D. S. Grebenkov and B.-T. Nguyen, Geometrical structure of Laplacian eigenfunctions, SIAM Rev., 55 (2013), pp. 601-667, https://doi.org/10.1137/120880173. · Zbl 1290.35157
[12] E. M. Harrell, Geometric lower bounds for the spectrum of elliptic PDEs with Dirichlet conditions in part, J. Comput. Appl. Math., 194 (2006), pp. 26-35, https://doi.org/10.1016/j.cam.2005.06.012. · Zbl 1091.35046
[13] N. Kaina, M. Dupré, G. Lerosey, and M. Fink, Shaping complex microwave fields in reverberating media with binary tunable metasurfaces, Sci. Rep., 4 (2014), 6693.
[14] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition. · Zbl 0836.47009
[15] A. Laptev, A. Peicheva, and A. Shlapunov, Finding eigenvalues and eigenfunctions of the zaremba problem for the circle, Complex Anal. Oper. Theory, 11 (2017), pp. 895-926. · Zbl 1367.47052
[16] V. Lotoreichik and J. Rohleder, Eigenvalue inequalities for the Laplacian with mixed boundary conditions, J. Differential Equations, 263 (2017), pp. 491-508, https://doi.org/10.1016/j.jde.2017.02.043. · Zbl 1366.35106
[17] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, UK, 2000. · Zbl 0948.35001
[18] S. Ozawa, Asymptotic property of an eigenfunction of the laplacian under singular variation of domains-the neumann condition, Osaka J. Math., 22 (1985), pp. 639-655. · Zbl 0579.35065
[19] J. Saranen and G. Vainikko, Periodic Integral and Pseudodifferential Equations with Numerical Approximation, Springer Monogr. Math., Springer-Verlag, Berlin, 2002, https://doi.org/10.1007/978-3-662-04796-5. · Zbl 0991.65125
[20] K. Schmüdgen, Unbounded Self-Adjoint Operators on Hilbert Space, Grad. Texts in Math. 265, Springer, Dordrecht, 2012, https://doi.org/10.1007/978-94-007-4753-1. · Zbl 1257.47001
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.