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• W2035690811 abstract "We consider Laplacian eigenfunctions in circular, spherical and elliptical domains in order to discuss three kinds of high-frequency localization: whispering gallery modes, bouncing ball modes, and focusing modes. Although the existence of these modes was known for a class of convex domains, the separation of variables for above domains helps to better understand the mechanism of localization, i.e. how an eigenfunction is getting distributed in a small region of the domain, and decays rapidly outside this region. Using the properties of Bessel and Mathieu functions, we derive the inequalities which imply and clearly illustrate localization. Moreover, we provide an example of a non-convex domain (an elliptical annulus) for which the high-frequency localized modes are still present. At the same time, we show that there is no localization in most of rectangle-like domains. This observation leads us to formulating an open problem of localization in polygonal domains and, more generally, in piecewise smooth convex domains." @default.
• W2035690811 created "2016-06-24" @default.
• W2035690811 creator A5047730752 @default.
• W2035690811 creator A5077899527 @default.
• W2035690811 date "2013-01-01" @default.
• W2035690811 modified "2023-09-25" @default.
• W2035690811 title "Localization of Laplacian Eigenfunctions in Circular, Spherical, and Elliptical Domains" @default.
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• W2035690811 doi "https://doi.org/10.1137/120869857" @default.
• W2035690811 hasPublicationYear "2013" @default.
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