160 research outputs found
Discreteness of interior transmission eigenvalues revisited
This paper is devoted to the discreteness of the transmission eigenvalue
problems. It is known that this problem is not self-adjoint and a priori
estimates are non-standard and do not hold in general. Two approaches are used.
The first one is based on the multiplier technique and the second one is based
on the Fourier analysis. The key point of the analysis is to establish the
compactness and the uniqueness for Cauchy problems under various conditions.
Using these approaches, we are able to rediscover quite a few known
discreteness results in the literature and obtain various new results for which
only the information near the boundary are required and there might be no
contrast of the coefficients on the boundary
Weyl asymptotics of the transmission eigenvalues for a constant index of refraction
We prove Weyl type of asymptotic formulas for the real and the complex
internal transmission eigenvalues when the domain is a ball and the index of
refraction is constant
Boundary Integral Equations for the Transmission Eigenvalue Problem for Maxwell’s Equations
International audienceIn this paper we consider the transmission eigenvalue problem for Maxwell’s equations corresponding to non-magnetic inhomogeneities with contrast in electric permittivity that changes sign inside its support. We formulate the transmission eigenvalue problem as an equivalent homogeneous system of boundary integral equa- tion, and assuming that the contrast is constant near the boundary of the support of the inhomogeneity, we prove that the operator associated with this system is Fredholm of index zero and depends analytically on the wave number. Then we show the existence of wave numbers that are not transmission eigenvalues which by an application of the analytic Fredholm theory implies that the set of transmission eigenvalues is discrete with positive infinity as the only accumulation point
Non-scattering wavenumbers and far field invisibility for a finite set of incident/scattering directions
We investigate a time harmonic acoustic scattering problem by a penetrable
inclusion with compact support embedded in the free space. We consider cases
where an observer can produce incident plane waves and measure the far field
pattern of the resulting scattered field only in a finite set of directions. In
this context, we say that a wavenumber is a non-scattering wavenumber if the
associated relative scattering matrix has a non trivial kernel. Under certain
assumptions on the physical coefficients of the inclusion, we show that the
non-scattering wavenumbers form a (possibly empty) discrete set. Then, in a
second step, for a given real wavenumber and a given domain D, we present a
constructive technique to prove that there exist inclusions supported in D for
which the corresponding relative scattering matrix is null. These inclusions
have the important property to be impossible to detect from far field
measurements. The approach leads to a numerical algorithm which is described at
the end of the paper and which allows to provide examples of (approximated)
invisible inclusions.Comment: 20 pages, 7 figure
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