The completeness of the generalized eigenfunctions and an upper bound for the counting function of the transmission eigenvalue problem for Maxwell equations. (English) Zbl 1481.35295
Summary: Cakoni and Nguyen recently proposed very general conditions on the coefficients of Maxwell equations for which they established the discreteness of the set of eigenvalues of the transmission eigenvalue problem and studied their locations. In this paper, we establish the completeness of the generalized eigenfunctions and derive an optimal upper bound for the counting function under these conditions, assuming additionally that the coefficients are twice continuously differentiable. The approach is based on the spectral theory of Hilbert-Schmidt operators.
MSC:
| 35P10 | Completeness of eigenfunctions and eigenfunction expansions in context of PDEs |
| 35Q61 | Maxwell equations |
Keywords:
optimal upper bound for the counting function; spectral theory of Hilbert-Schmidt operatorsReferences:
| [1] | Agmon, S.: Lectures on elliptic boundary value problems, Prepared for publication by B. Frank Jones, Jr. with the assistance of George W. Batten, Jr. Van Nostrand Mathematical Studies, No. 2, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London (1965) · Zbl 0142.37401 |
| [2] | Agmon, S.; Douglis, A.; Nirenberg, L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Commun. Pure Appl. Math., 12, 623-727 (1959) · Zbl 0093.10401 · doi:10.1002/cpa.3160120405 |
| [3] | Agmon, S.; Douglis, A.; Nirenberg, L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II., Commun. Pure Appl. Math., 17, 35-92 (1964) · Zbl 0123.28706 · doi:10.1002/cpa.3160170104 |
| [4] | Brezis, H., Functional Analysis, Sobolev Spaces and Partial Differential Equations (2011), New York: Universitext, Springer, New York · Zbl 1220.46002 · doi:10.1007/978-0-387-70914-7 |
| [5] | Cakoni, F.; Gintides, D.; Haddar, H., The existence of an infinite discrete set of transmission eigenvalues, SIAM J. Math. Anal., 42, 1, 237-255 (2010) · Zbl 1210.35282 · doi:10.1137/090769338 |
| [6] | Cakoni, F.; Haddar, H.; Meng, S., Boundary integral equations for the transmission eigenvalue problem for Maxwell‘s equations, J. Integr. Equ. Appl., 27, 3, 375-406 (2015) · Zbl 1332.35247 · doi:10.1216/JIE-2015-27-3-375 |
| [7] | Cakoni, F.; Colton, D.; Haddar, H., Inverse Scattering Theory and Transmission Eigenvalues, CBMS-NSF Regional Conference Series in Applied Mathematics (2016), Philadelphia: Society for Industrial and Applied Mathematics (SIAM), Philadelphia · Zbl 1366.35001 · doi:10.1137/1.9781611974461 |
| [8] | Cakoni, F.; Nguyen, H-M, On the discreteness of transmission eigenvalues for the Maxwell equations, SIAM J. Math. Anal., 53, 1, 888-913 (2021) · Zbl 1458.35401 · doi:10.1137/20M1335121 |
| [9] | Chesnel, L., Interior transmission eigenvalue problem for Maxwell‘s equations: the \(T\)-coercivity as an alternative approach, Inverse Probl., 28, 6, 065005 (2012) · Zbl 1243.78024 · doi:10.1088/0266-5611/28/6/065005 |
| [10] | Colton, D.; Monk, P., The inverse scattering problem for time-harmonic acoustic waves in an inhomogeneous medium, Quart. J. Mech. Appl. Math., 41, 1, 97-125 (1988) · Zbl 0637.73026 · doi:10.1093/qjmam/41.1.97 |
| [11] | Dhia, A-SB-B; Chesnel, L.; Haddar, H., On the use of \(T\)-coercivity to study the interior transmission eigenvalue problem, C. R. Math. Acad. Sci. Paris, 349, 11-12, 647-651 (2011) · Zbl 1244.35099 · doi:10.1016/j.crma.2011.05.008 |
| [12] | Gagliardo, E., Ulteriori proprietà di alcune classi di funzioni in più variabili, Ricerche Mat., 8, 24-51 (1959) · Zbl 0199.44701 |
| [13] | Girault, V.; Raviart, P-A, Finite Element Methods for Navier-Stokes Equations, Springer Series in Computational Mathematics, Theory and algorithms (1986), Berlin: Springer, Berlin · Zbl 0585.65077 · doi:10.1007/978-3-642-61623-5 |
| [14] | Haddar, H., The interior transmission problem for anisotropic Maxwell‘s equations and its applications to the inverse problem, Math. Methods Appl. Sci., 27, 18, 2111-2129 (2004) · Zbl 1062.35168 · doi:10.1002/mma.465 |
| [15] | Haddar, H.; Meng, S., The spectral analysis of the interior transmission eigenvalue problem for Maxwell‘s equations, J. Math. Pures Appl., 120, 9, 1-32 (2018) · Zbl 1403.78009 · doi:10.1016/j.matpur.2018.10.004 |
| [16] | Kirsch, A., The denseness of the far field patterns for the transmission problem, IMA J. Appl. Math., 37, 3, 213-225 (1986) · Zbl 0652.35104 · doi:10.1093/imamat/37.3.213 |
| [17] | Lakshtanov, E.; Vainberg, B., Ellipticity in the interior transmission problem in anisotropic media, SIAM J. Math. Anal., 44, 2, 1165-1174 (2012) · Zbl 1245.35126 · doi:10.1137/11084738X |
| [18] | Nguyen, H-M, Asymptotic behavior of solutions to the Helmholtz equations with sign changing coefficients, Trans. Am. Math. Soc., 367, 9, 6581-6595 (2015) · Zbl 1321.35012 · doi:10.1090/S0002-9947-2014-06305-8 |
| [19] | Nguyen, H-M, Limiting absorption principle and well-posedness for the Helmholtz equation with sign changing coefficients, J. Math. Pures Appl., 106, 2, 342-374 (2016) · Zbl 1343.35074 · doi:10.1016/j.matpur.2016.02.013 |
| [20] | Nguyen, H-M, Superlensing using complementary media and reflecting complementary media for electromagnetic waves, Adv. Nonlinear Anal., 7, 4, 449-467 (2018) · Zbl 1404.35428 · doi:10.1515/anona-2017-0146 |
| [21] | Nguyen, H-M.: Cloaking property of a plasmonic structure in doubly complementary media and three-sphere inequalities with partial data (2019). Preprint arXiv:1912.09098 |
| [22] | Nguyen, H-M, Cloaking using complementary media for electromagnetic waves, ESAIM Control Optim. Calc. Var., 25, 29 (2019) · Zbl 1437.35649 · doi:10.1051/cocv/2017078 |
| [23] | Nguyen, H-M, The invisibility via anomalous localized resonance of a source for electromagnetic waves, Res. Math. Sci., 6, 4, 32 (2019) · Zbl 1425.35185 · doi:10.1007/s40687-019-0194-0 |
| [24] | Nguyen, H-M; Nguyen, Q-H, Discreteness of interior transmission eigenvalues revisited, Calc. Var. Partial Differ. Equ., 56, 2, 51 (2017) · Zbl 1375.35289 · doi:10.1007/s00526-017-1143-7 |
| [25] | Nguyen, H-M., Nguyen, Q-H.: The Weyl law of transmission eigenvalues and the completeness of generalized transmission eigenfunctions (2020). arXiv:2008.08540 |
| [26] | Nguyen, H-M; Sil, S., Limiting absorption principle and well-posedness for the time-harmonic maxwell equations with anisotropic sign-changing coefficients, Commun. Math. Phys., 379, 1, 145-176 (2020) · Zbl 1451.35207 · doi:10.1007/s00220-020-03805-1 |
| [27] | Nirenberg, L., On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 13, 3, 115-162 (1959) · Zbl 0088.07601 |
| [28] | Robbiano, L., Spectral analysis of the interior transmission eigenvalue problem, Inverse Probl., 29, 10, 104001 (2013) · Zbl 1296.35105 · doi:10.1088/0266-5611/29/10/104001 |
| [29] | Robbiano, L., Counting function for interior transmission eigenvalues, Math. Control Relat. Fields, 6, 1, 167-183 (2016) · Zbl 1332.35242 · doi:10.3934/mcrf.2016.6.167 |
| [30] | Shlëmovich, M., Filonov, N.D.: Weyl asymptotics of the spectrum of the Maxwell operator with non-smooth coefficients in Lipschitz domains, Nonlinear equations and spectral theory, Amer. Math. Soc. Transl. Ser. 2, vol. 220, Amer. Math. Soc., Providence, RI, pp. 27-44 (2007) · Zbl 1260.35096 |
| [31] | Sylvester, J., Discreteness of transmission eigenvalues via upper triangular compact operators, SIAM J. Math. Anal., 44, 1, 341-354 (2012) · Zbl 1238.81172 · doi:10.1137/110836420 |
| [32] | Vodev, G., Transmission eigenvalue-free regions, Commun. Math. Phys., 336, 3, 1141-1166 (2015) · Zbl 1323.35110 · doi:10.1007/s00220-015-2311-2 |
| [33] | Vodev, G., High-frequency approximation of the interior Dirichlet-to-Neumann map and applications to the transmission eigenvalues, Anal. PDE, 11, 1, 213-236 (2018) · Zbl 1386.35303 · doi:10.2140/apde.2018.11.213 |
| [34] | Vodev, G.: Semiclassical parametrix for the Maxwell equation and applications to the electromagnetic transmission eigenvalues (2021). arXiv:2102.08662 · Zbl 1477.35254 |
| [35] | Weyl, H., Über das Spektrum der Hohlraumstrahlung, J. Reine Angew. Math., 141, 163-181 (1912) · JFM 43.1063.02 |
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