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Scattering by curvatures, radiationless sources, transmission eigenfunctions, and inverse scattering problems. (English) Zbl 1479.35838

The authors of this paper consider several connected topics of interest in the theory of wave propagation: geometrical characterizations of radiationless sources, nonradiating incident waves, interior transmission eigenfunctions, and their applications to inverse scattering. Assume that \(f:\mathbb{R}^n \to\mathbb{C}\) be a function with compact support. The source \(f\) produces a scattered wave \(u \in H_{\mathrm{loc}}^2(\mathbb{R}^n)\) given by the unique solution to the following nonhomogeneous elliptic differential equation \((\triangle +k^2)u=f\), \(\lim\limits_{r\to\infty }r^{(n-1)/2} (\partial_r-ik)u=0\), where \(r=|x|\), \(x\in \mathbb{R}^n\). The limit here is known as the Sommerfeld radiation condition, which characterizes the outgoing radiating wave. There exist two types of scenarios. The first scenario is concerned with radiationless or nonradiating monochromatic sources. The second one is concerned with nonradiating waves that impinge against a certain given scatterer consisting of an inhomogeneous index of refraction. The geometrical properties of such invisible objects would be of interest. The investigation of these objects allows to classify radiating sources and incident waves that are always radiating for scatterers. Thus the authors establish unique determination results for a longstanding inverse scattering problem in certain scenarios of practical importance. First it is proved an explicit relationship between the intensity of such a source and the diameter of its support. It suggests that if the support of a generic source is sufficiently small in terms of the wavelength, then it must not be radiationless. It means that its radiating pattern cannot be identically zero. This result generalizes the classical result on radiationless sources which states that for a source supported in a ball with a constant intensity, if the radius of the ball is sufficiently small, then the source must be radiating. This leads to a discussion on both important properties localization and geometrization. It is concluded that a scatterer, which can be an active source or an inhomogeneous index of refraction, cannot be completely invisible if its support is small compared to the wavelength and scattering intensity. Next conclusion is the localization and geometrization of the “smallness” results to the case where there is a high-curvature point on the boundary of the scatterer’s support. It is derived explicit bounds between the intensity of an invisible scatterer and its diameter or its curvature at the aforementioned point. These results characterize radiationless sources or nonradiating waves near high-curvature points. Interior transmission eigenfunctions with some regularity must be nearly vanishing at a high-curvature point on the boundary. It is stated that ”the higher the curvature, the smaller the eigenfunction must be at the high-curvature point”. The behavior of nearly vanishing implies that as long as the shape of a scatterer possesses a highly curved part, then it scatters every incident wave field nontrivially unless the wave is vanishingly small at the highly curved part. The practical implication of this result indicates that even if a scatterer has a very smooth shape, nontrivial scattering can be caused due to the curvature of the shape.
An application is that it is possible to derive new intrinsic geometric properties of interior transmission eigenfunctions near high-curvature points. It is established unique determination results for the single-wave Schiffer’s problem in certain scenarios of practical interest. Meanwhile, the authors state here that at their point of view ”it is the first result for Schiffer’s problem with generic smooth scatterers”.

MSC:

35Q60 PDEs in connection with optics and electromagnetic theory
78A46 Inverse problems (including inverse scattering) in optics and electromagnetic theory
35P25 Scattering theory for PDEs
78A05 Geometric optics
81U40 Inverse scattering problems in quantum theory

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