期刊
JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES
卷 143, 期 -, 页码 116-161出版社
ELSEVIER
DOI: 10.1016/j.matpur.2020.09.011
关键词
Laplacian eigenfunctions; Nodal and generalised singular lines; Impedance obstacle; A single far-field pattern
资金
- Fundamental Research Funds for the Central Universities [2412017FZ007]
- City University of Hong Kong
- Hong Kong RGC General Research Fund [14304517, 14306718]
In this paper, we present some novel and intriguing findings on the geometric structures of Laplacian eigenfunctions and their deep relationship to the quantitative behaviours of the eigenfunctions in two dimensions. We introduce a new notion of generalized singular lines of the Laplacian eigenfunctions, and carefully study these singular lines and the nodal lines. The studies reveal that the intersecting angle between two of those lines is closely related to the vanishing order of the eigenfunction at the intersecting point. We establish an accurate and comprehensive quantitative characterisation of the relationship. Roughly speaking, the vanishing order is generically infinite if the intersecting angle is irrational, and the vanishing order is finite if the intersecting angle is rational. In fact, in the latter case, the vanishing order is the degree of the rationality. The theoretical findings are original and of significant interest in spectral theory. Moreover, they are applied directly to some physical problems of great importance, including the inverse obstacle scattering problem and the inverse diffraction grating problem. It is shown in a certain polygonal setup that one can recover the support of the unknown scatterer as well as the surface impedance parameter by finitely many far-field patterns. Indeed, at most two far-field patterns are sufficient for some important applications. Unique identifiability by finitely many far-field patterns remains to be a highly challenging fundamental mathematical problem in the inverse scattering theory. (C) 2020 Elsevier Masson SAS. All rights reserved.
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