Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userWeakly singular corners always scatter
57
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Assume that a bounded scatterer is embedded into an infinite homogeneous isotropic background medium in two dimensions. The refractive index function is supposed to be piecewise constant. If the scattering interface contains a weakly or strongly singular point, we prove that the scattered field cannot vanish identically. This particularly leads to the absence of non-scattering energies for piecewise analytic interfaces with a weakly singular point. Local uniqueness is obtained for shape identification problems in inverse medium scattering with a single far-field pattern.
rate research
Read More
It is proved that an inhomogeneous medium whose boundary contains a weakly singular point of arbitrary order scatters every incoming wave. Similarly, a compactly supported source term with weakly singular points on the boundary always radiates acoustic waves. These results imply the absence of non-scattering energies and non-radiating sources in a domain that is not $C^infty$-smooth. Local uniqueness results with a single far-field pattern are obtained for inverse source and inverse medium scattering problems. Our arguments provide a sufficient condition of the surface under which solutions to the Helmholtz equation admits no analytical continuation.
In this paper, we consider the transmission eigenvalue problem associated with a general conductive transmission condition and study the geometric structures of the transmission eigenfunctions. We prove that under a mild regularity condition in terms of the Herglotz approximations of one of the pair of the transmission eigenfunctions, the eigenfunctions must be vanishing around a corner on the boundary. The Herglotz approximation can be regarded as the Fourier transform of the transmission eigenfunction in terms of the plane waves, and the growth rate of the transformed function can be used to characterize the regularity of the underlying wave function. The geometric structures derived in this paper include the related results in [5,19] as special cases and verify that the vanishing around corners is a generic local geometric property of the transmission eigenfunctions.
We study the singular perturbation of an elastic energy with a singular weight. The minimization of this energy results in a multi-scale pattern formation. We derive an energy scaling law in terms of the perturbation parameter and prove that, although one cannot expect periodicity of minimizers, the energy of a minimizer is uniformly distributed across the sample. Finally, following the approach developed by Alberti and M{u}ller in 2001 we prove that a sequence of minimizers of the perturbed energies converges to a Young measure supported on functions of slope $pm 1$ and of period depending on the location in the domain and the weights in the energy.
We develop in this note the tools of regularity structures to deal with singular stochastic PDEs that involve non-translation invariant differential operators. We describe in particular the renormalised equation for a very large class of spacetime dependent renormalization schemes.
The notion of singular reduction operators, i.e., of singular operators of nonclassical (conditional) symmetry, of partial differential equations in two independent variables is introduced. All possible reductions of these equations to first-order ODEs are are exhaustively described. As examples, properties of singular reduction operators of (1+1)-dimensional evolution and wave equations are studied. It is shown how to favourably enhance the derivation of nonclassical symmetries for this class by an in-depth prior study of the corresponding singular vector fields.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
