378 research outputs found
Spectral properties of elliptic operator with double-contrast coefficients near a hyperplane
In this paper we study the asymptotic behaviour as of the
spectrum of the elliptic operator posed in a bounded domain
subject to Dirichlet boundary
conditions on . When both coefficients
and become high contrast in a small
neighborhood of a hyperplane intersecting . We prove that the
spectrum of converges to the spectrum of an operator
acting in and generated by the operation
in , the Dirichlet boundary conditions on
and certain interface conditions on containing the
spectral parameter in a nonlinear manner. The eigenvalues of this operator may
accumulate at a finite point. Then we study the same problem, when is
an infinite straight strip ("waveguide") and is parallel to its
boundary. We show that has at least one gap in the
spectrum when is small enough and describe the asymptotic
behaviour of this gap as . The proofs are based on methods of
homogenization theory.Comment: In the second version we added the case r=0, also the presentation is
essentially improved. The manuscript is submitted to a journa
A computer-assisted existence proof for Emden's equation on an unbounded L-shaped domain
We prove existence, non-degeneracy, and exponential decay at infinity of a
non-trivial solution to Emden's equation on an unbounded
-shaped domain, subject to Dirichlet boundary conditions. Besides the direct
value of this result, we also regard this solution as a building block for
solutions on expanding bounded domains with corners, to be established in
future work. Our proof makes heavy use of computer assistance: Starting from a
numerical approximate solution, we use a fixed-point argument to prove
existence of a near-by exact solution. The eigenvalue bounds established in the
course of this proof also imply non-degeneracy of the solution
Localized Modes of the Linear Periodic Schr\"{o}dinger Operator with a Nonlocal Perturbation
We consider the existence of localized modes corresponding to eigenvalues of
the periodic Schr\"{o}dinger operator with an interface.
The interface is modeled by a jump either in the value or the derivative of
and, in general, does not correspond to a localized perturbation of the
perfectly periodic operator. The periodic potentials on each side of the
interface can, moreover, be different. As we show, eigenvalues can only occur
in spectral gaps. We pose the eigenvalue problem as a gluing problem for
the fundamental solutions (Bloch functions) of the second order ODEs on each
side of the interface. The problem is thus reduced to finding matchings of the
ratio functions , where are
those Bloch functions that decay on the respective half-lines. These ratio
functions are analyzed with the help of the Pr\"{u}fer transformation. The
limit values of at band edges depend on the ordering of Dirichlet and
Neumann eigenvalues at gap edges. We show that the ordering can be determined
in the first two gaps via variational analysis for potentials satisfying
certain monotonicity conditions. Numerical computations of interface
eigenvalues are presented to corroborate the analysis.Comment: 1. finiteness of the number of additive interface eigenvalues proved
in a remark below Corollary 3.6.; 2. small modifications and typo correction
Beyond local effective material properties for metamaterials
To discuss the properties of metamaterials on physical grounds and to
consider them in applications, effective material parameters are usually
introduced and assigned to a given metamaterial. In most cases, only weak
spatial dispersion is considered. It allows to assign local material
properties, i.e. a permittivity and a permeability. However, this turned out to
be insufficient. To solve this problem, we study here the effective properties
of metamaterials with constitutive relations beyond a local response and take
strong spatial dispersion into account. The isofrequency surfaces of the
dispersion relation are investigated and compared to those of an actual
metamaterial. The significant improvement provides evidence for the necessity
to use nonlocal material laws in the effective description of metamaterials.
The general formulation we choose here renders our approach applicable to a
wide class of metamaterials
Interface conditions for a metamaterial with strong spatial dispersion
Local constitutive relations, i.e. a weak spatial dispersion, are usually
considered in the effective description of metamaterials. However, they are
often insufficient and effects due to a nonlocality, i.e. a strong spatial
dispersion, are encountered. Recently (K.~Mnasri et al., arXiv:1705.10969), a
generic form for a nonlocal constitutive relation has been introduced that
could accurately describe the bulk properties of a metamaterial in terms of a
dispersion relation. However, the description of functional devices made from
such nonlocal metamaterials also requires the identification of suitable
interface conditions. In this contribution, we derive the interface conditions
for such nonlocal metamaterials
Enclosure for the Biharmonic Equation
In this paper we give an enclosure for the solution of the biharmonic problem and also for its gradient and Laplacian in the -norm, respectively
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