## On vanishing near corners of transmission eigenfunctions.(English)Zbl 1387.35437

The paper investigates the following (interior) transmission eigenvalue problem for nontrivial $$v, w\in L^2(\Omega)$$, $\begin{cases} (\Delta+k^2)v=0\quad &\text{in}\;\;\Omega,\\ (\Delta+k^2(1+V))w=0\quad &\text{in}\;\;\Omega,\\ w-v\in H^2_0(\Omega),\;\;||v||_{L^2(\Omega)}&=1, \end{cases}$ where $$\Omega$$ is a bounded domain in $$\mathbb{R}^n$$, $$n\geq2$$, $$k\in \mathbb{R}_+$$ and $$V\in L^\infty(\Omega)$$ is a potential function.
The transmission eigenvalue problem is a type of non elliptic and non self-adjoint problem. The existing results are mainly concerned with the spectral properties of the transmission eigenvalues, including the existence, discreteness and infiniteness, and Weyl laws. In this paper the vanishing properties of interior transmission eigenfunctions is investigated. It is proved that $$v$$ and $$w$$ vanish near a corner point on $$\partial \Omega$$ in a generic situation where the corner possesses an interior angle less than $$\pi$$ and the potential function $$V$$ does not vanish at the corner point. This is proved by an indirect approach, connecting to the wave scattering theory. Note that the vanishing behavior carries geometric information of the support of the involved potential function $$V$$. Implications of the obtained resuls to inverse scattering theory and invisibility are also discussed at end of the paper.

### MSC:

 35P25 Scattering theory for PDEs 58J50 Spectral problems; spectral geometry; scattering theory on manifolds 35R30 Inverse problems for PDEs 81V80 Quantum optics
Full Text:

### References:

 [1] Blåsten, E.; Liu, H., On corners scattering stably, nearly non-scattering interrogating waves, and stable shape determination by a single far-field pattern [2] Blåsten, E.; Liu, H., Recovering piecewise constant refractive indices by a single far-field pattern [3] Blåsten, E.; Li, X.; Liu, H.; Wang, Y., On vanishing and localizing of transmission eigenfunctions near singular points: a numerical study, Inverse Probl., 33, (2017) · Zbl 1442.65332 [4] Blåsten, E.; Päivärinta, L., Completeness of generalized transmission eigenstates, Inverse Probl., 29, 10, (2013) · Zbl 1294.35042 [5] Blåsten, E.; Päivärinta, L.; Sylvester, J., Corners always scatter, Comm. Math. Phys., 331, 725-753, (2014) · Zbl 1298.35214 [6] F. Cakoni, 2016, private discussion. [7] Cakoni, F.; Gintides, D.; Haddar, H., The existence of an infinite discrete set of transmission eigenvalues, SIAM J. Math. Anal., 42, 237-255, (2010) · Zbl 1210.35282 [8] F. Cakoni, H. Haddar, Transmission eigenvalues in inverse scattering theory, in [35], 529-578. · Zbl 1316.35297 [9] Chen, G.; Morris, P.; Zhou, J., Visualization of special eigenmodes shapes of a vibrating elliptical membrane, SIAM Rev., 36, 453-469, (1994) · Zbl 0806.73032 [10] Colton, D.; Kirsch, A., A simple method for solving inverse scattering problems in the resonance region, Inverse Probl., 12, 4, 383-393, (1996) · Zbl 0859.35133 [11] Colton, D.; Kirsch, A.; Päivärinta, L., Far-field patterns for acoustic waves in an inhomogeneous medium, SIAM J. Math. Anal., 20, 1472-1482, (1989) · Zbl 0681.76084 [12] Colton, D.; Kress, R., Inverse acoustic and electromagnetic scattering theory, (1998), Springer New York · Zbl 0893.35138 [13] Colton, D.; Monk, P., The inverse scattering problem for time-harmonic acoustic waves in an inhomogeneous medium, Quart. J. Mech. Appl. Math., 41, 97-125, (1988) · Zbl 0637.73026 [14] Elschner, J.; Hu, G., Corners and edges always scatter, Inverse Probl., 31, (2015) · Zbl 1319.35140 [15] Elschner, J.; Hu, G., Acoustic scattering from corners, edges and circular cones · Zbl 1392.35224 [16] Gell-Redman, J.; Hassel, J., Potential scattering and the continuity of phase-shifts, Math. Res. Lett., 19, 719-729, (2012) · Zbl 1302.35283 [17] Grebenkov, D.; Ngyuen, B., Geometrical structure of Laplacian eigenfunctions, SIAM Rev., 55, 4, 601-667, (2013) · Zbl 1290.35157 [18] Greenleaf, A.; Kurylev, Y.; Lassas, M.; Uhlmann, G., Invisibility and inverse problems, Bull. Amer. Math. Soc., 46, 55-97, (2009) · Zbl 1159.35074 [19] Greenleaf, A.; Kurylev, Y.; Lassas, M.; Uhlmann, G., Cloaking devices, electromagnetic wormholes and transformation optics, SIAM Rev., 51, 3-33, (2009) · Zbl 1158.78004 [20] Heilman, S.; Strichartz, R., Localized eigenfunctions: here you see them, there you don’t, Notices Amer. Math. Soc., 5, (2010) · Zbl 1194.35283 [21] Hu, G.; Salo, M.; Vesalainen, E., Shape identification in inverse medium scattering, SIAM J. Math. Anal., 48, 152-165, (2016) · Zbl 1334.35427 [22] Isakov, V., Inverse problems for partial differential equations, (2006), Springer-Verlag New York · Zbl 1092.35001 [23] Keller, J.; Rubinow, S., Asymptotic solution of eigenvalue problems, Ann. Phys., 9, 24-75, (1960) · Zbl 0087.43002 [24] Kirsch, A.; Grinberg, N., The factorization method for inverse problems, Oxford Lecture Series in Mathematics and Its Applications, vol. 36, (2008), Oxford University Press Oxford · Zbl 1222.35001 [25] Lakshtanov, E.; Vainberg, B., Applications of elliptic operator theory to the isotropic interior transmission eigenvalue problem, Inverse Probl., 29, 10, (2013) · Zbl 1285.35059 [26] Nachman, A., Reconstructions from boundary measurements, Ann. of Math. (2), 128, 531-576, (1988) · Zbl 0675.35084 [27] Nazarov, S., Localization near the corner point of the principal eigenfunction of the Dirichlet problem in a domain with thin edging, Sib. Math. J., 52, 2, 274-290, (2011) · Zbl 1221.35257 [28] B. Nguyen, Localization of Laplacian eigenfunctions in simple and irregular domains, Mathematical Physics [math-ph], Ecole Polytechnique X (2012) English. . [29] Päivärinta, L.; Salo, M.; Vesalainen, E., Strictly convex corners scatter, Rev. Mat. Iberoam., (2017), in press · Zbl 1388.35138 [30] Päivärinta, L.; Sylvester, J., Transmission eigenvalues, SIAM J. Math. Anal., 40, 738-753, (2008) · Zbl 1159.81411 [31] Robbiano, L., Spectral analysis of the interior transmission eigenvalue problem, Inverse Probl., 29, 10, (2013) · Zbl 1296.35105 [32] Rynne, B.; Sleeman, B., The interior transmission problem and inverse scattering from inhomogeneous media, SIAM J. Math. Anal., 22, 1755-1762, (1991) · Zbl 0733.76065 [33] Sylvester, J.; Uhlmann, G., A global uniqueness theorem for an inverse boundary value problem, Ann. of Math. (2), 125, 153-169, (1987) · Zbl 0625.35078 [34] Uhlmann, G., Visibility and invisibility, ICIAM 07—6th international congress on industrial and applied mathematics, 381-408, (2009), Eur. Math. Soc. Zürich [35] (Uhlmann, G., Inverse Problems and Applications: Inside Out II, MSRI Publications, vol. 60, (2013), Cambridge University Press) · Zbl 1277.65002 [36] Weck, N., Approximation by Herglotz wave functions, Math. Methods Appl. Sci., 27, 155-162, (2004) · Zbl 1099.35007
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.