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On the shielding effect of the Helmholtz equation
Author(s) -
Chen Yu
Publication year - 2008
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20211
Subject(s) - helmholtz equation , mathematics , mathematical analysis , instability , hadamard transform , laplace's equation , cauchy distribution , well posed problem , cauchy problem , inverse scattering problem , inverse problem , initial value problem , partial differential equation , physics , boundary value problem , quantum mechanics
Hadamard‐type instability has been known for over a century as a cause of ill‐posedness of the Cauchy problem for elliptic PDEs. This ill‐posedness manifests itself as evanescent modes growing exponentially when propagated in the reverse direction. Since every oscillating mode of the Laplace equation is evanescent, the ill‐posedness of its Cauchy problem is solely due to Hadamard‐type instability. The presence of the propagating modes and beams for the Helmholtz equation gives rise to an entirely different type of ill‐posedness, hitherto unknown to the practice, and untreated by the theory, of inverse scattering. We will present this fundamental phenomenon of ill‐posedness for the Helmholtz equation. © 2007 Wiley Periodicals, Inc.