2,128 research outputs found
The localization effect for eigenfunctions of the mixed boundary value problem in a thin cylinder with distorted ends
A simple sufficient condition on curved end of a straight cylinder is found
that provides a localization of the principal eigenfunction of the mixed
boundary value for the Laplace operator with the Dirichlet conditions on the
lateral side. Namely, the eigenfunction concentrates in the vicinity of the
ends and decays exponentially in the interior. Similar effects are observed in
the Dirichlet and Neumann problems, too.Comment: 25 pages, 10 figure
Water-waves modes trapped in a canal by a body with the rough surface
The problem about a body in a three dimensional infinite channel is
considered in the framework of the theory of linear water-waves. The body has a
rough surface characterized by a small parameter while the
distance of the body to the water surface is also of order . Under a
certain symmetry assumption, the accumulation effect for trapped mode
frequencies is established, namely, it is proved that, for any given and
integer , there exists such that the problem has at
least eigenvalues in the interval of the continuous spectrum in the
case . The corresponding eigenfunctions decay
exponentially at infinity, have finite energy, and imply trapped modes.Comment: 25 pages, 8 figure
Scalar boundary value problems on junctions of thin rods and plates. I. Asymptotic analysis and error estimates
We derive asymptotic formulas for the solutions of the mixed boundary value
problem for the Poisson equation on the union of a thin cylindrical plate and
several thin cylindrical rods. One of the ends of each rod is set into a hole
in the plate and the other one is supplied with the Dirichlet condition. The
Neumann conditions are imposed on the whole remaining part of the boundary.
Elements of the junction are assumed to have contrasting properties so that the
small parameter, i.e. the relative thickness, appears in the differential
equation, too, while the asymptotic structures crucially depend on the
contrastness ratio. Asymptotic error estimates are derived in anisotropic
weighted Sobolev norms.Comment: 34 pages, 4 figure
Exact dynamical exchange-correlation kernel of a weakly inhomogeneous electron gas
The dynamical exchange-correlation kernel of a non-uniform electron
gas is an essential input for the time-dependent density functional theory of
electronic systems. The long-wavelength behavior of this kernel is known to be
of the form where is the wave vector and is a
frequency-dependent coefficient. We show that in the limit of weak
non-uniformity the coefficient has a simple and exact expression in
terms of the ground-state density and the frequency-dependent kernel of a {\it
uniform} electron gas at the average density. We present an approximate
evaluation of this expression for Si and discuss its implications for the
theory of excitonic effects.Comment: 5 pages, 2 figure
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