Hadamard-type instability has been known for over a century as a cause of illposedness 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.
ASJC Scopus subject areas
- Applied Mathematics