34 research outputs found
Periodic Riemannian manifold with preassigned gaps in spectrum of Laplace-Beltrami operator
It is known (E.L. Green (1997), O. Post (2003)) that for an arbitrary
one can construct a periodic non-compact Riemannian manifold
with at least gaps in the spectrum of the corresponding
Laplace-Beltrami operator . In this work we want not only to produce
a new type of periodic manifolds with spectral gaps but also to control the
edges of these gaps. The main result of the paper is as follows: for arbitrary
pairwise disjoint intervals (\a_j,\b_j)\subset[0,\infty),
(), for an arbitrarily small and for an arbitrarily
large we construct a periodic non-compact Riemannian manifold with at
least gaps in the spectrum of the operator , moreover the
edges of the first gaps belong to -neighbourhoods of the edges of
the intervals (\a_j,\b_j), while the remaining gaps (if any) are located
outside the interval .Comment: 30 pages, 2 Figures; Journal of Differential Equations (2012
Operator estimates for the crushed ice problem
Let be the Dirichlet Laplacian in the domain
.
Here and is a family of
tiny identical holes ("ice pieces") distributed periodically in
with period . We denote by the
capacity of a single hole. It was known for a long time that
converges to the operator
in strong resolvent sense provided the limit exists and is finite. In the
current contribution we improve this result deriving estimates for the rate of
convergence in terms of operator norms. As an application, we establish the
uniform convergence of the corresponding semi-groups and (for bounded )
an estimate for the difference of the -th eigenvalue of
and . Our proofs
relies on an abstract scheme for studying the convergence of operators in
varying Hilbert spaces developed previously by the second author.Comment: now 24 pages, 3 figures; some typos fixed and references adde
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
Operator estimates for Neumann sieve problem
Let be a domain in , be a hyperplane
intersecting , be a small parameter, and
, be a family of small "holes" in
; when , the number of holes tends to
infinity, while their diameters tends to zero. Let be
the Neumann Laplacian in the perforated domain
, where
("sieve"). It
is well-known that if the sizes of holes are carefully chosen,
converges in the strong resolvent sense to the
Laplacian on subject to the so-called
-conditions on . In the current work we improve this result:
under rather general assumptions on the shapes and locations of the holes we
derive estimates on the rate of convergence in terms of and
operator norms; in the latter case a special corrector is
required.Comment: 33 pages, 3 figure