46 research outputs found
Integral equations PS-3 and moduli of pants
More than a hundred years ago H.Poincare and V.A.Steklov considered a problem
for the Laplace equation with spectral parameter in the boundary conditions.
Today similar problems for two adjacent domains with the spectral parameter in
the conditions on the common boundary of the domains arises in a variety of
situations: in justification and optimization of domain decomposition method,
simple 2D models of oil extraction, (thermo)conductivity of composite
materials. Singular 1D integral Poincare-Steklov equation with spectral
parameter naturally emerges after reducing this 2D problem to the common
boundary of the domains. We present a constructive representation for the
eigenvalues and eigenfunctions of this integral equation in terms of moduli of
explicitly constructed pants, one of the simplest Riemann surfaces with
boundary. Essentially the solution of integral equation is reduced to the
solution of three transcendent equations with three unknown numbers, moduli of
pants. The discreet spectrum of the equation is related to certain surgery
procedure ('grafting') invented by B.Maskit (1969), D.Hejhal (1975) and
D.Sullivan- W.Thurston (1983).Comment: 27 pages, 13 figure
Localized boundary-domain singular integral equations based on harmonic parametrix for divergence-form elliptic PDEs with variable matrix coefficients
This is the post-print version of the Article. The official publised version can be accessed from the links below. Copyright @ 2013 Springer BaselEmploying the localized integral potentials associated with the Laplace operator, the Dirichlet, Neumann and Robin boundary value problems for general variable-coefficient divergence-form second-order elliptic partial differential equations are reduced to some systems of localized boundary-domain singular integral equations. Equivalence of the integral equations systems to the original boundary value problems is proved. It is established that the corresponding localized boundary-domain integral operators belong to the Boutet de Monvel algebra of pseudo-differential operators. Applying the Vishik-Eskin theory based on the factorization method, the Fredholm properties and invertibility of the operators are proved in appropriate Sobolev spaces.This research was supported by the grant EP/H020497/1: "Mathematical Analysis of Localized Boundary-Domain Integral Equations for Variable-Coefficient Boundary Value Problems" from the EPSRC, UK
Radiative corrections to the excitonic molecule state in GaAs microcavities
The optical properties of excitonic molecules (XXs) in GaAs-based quantum
well microcavities (MCs) are studied, both theoretically and experimentally. We
show that the radiative corrections to the XX state, the Lamb shift
and radiative width , are
large, about of the molecule binding energy , and
definitely cannot be neglected. The optics of excitonic molecules is dominated
by the in-plane resonant dissociation of the molecules into outgoing
1-mode and 0-mode cavity polaritons. The later decay channel,
``excitonic molecule 0-mode polariton + 0-mode
polariton'', deals with the short-wavelength MC polaritons invisible in
standard optical experiments, i.e., refers to ``hidden'' optics of
microcavities. By using transient four-wave mixing and pump-probe
spectroscopies, we infer that the radiative width, associated with excitonic
molecules of the binding energy meV, is
meV in the microcavities and
meV in a reference GaAs single quantum
well (QW). We show that for our high-quality quasi-two-dimensional
nanostructures the limit, relevant to the XX states, holds at
temperatures below 10 K, and that the bipolariton model of excitonic molecules
explains quantitatively and self-consistently the measured XX radiative widths.
We also find and characterize two critical points in the dependence of the
radiative corrections against the microcavity detuning, and propose to use the
critical points for high-precision measurements of the molecule bindingenergy
and microcavity Rabi splitting.Comment: 16 pages, 11 figures, accepted for publication in Phys. Rev.
A Survey on the Krein-von Neumann Extension, the corresponding Abstract Buckling Problem, and Weyl-Type Spectral Asymptotics for Perturbed Krein Laplacians in Nonsmooth Domains
In the first (and abstract) part of this survey we prove the unitary
equivalence of the inverse of the Krein--von Neumann extension (on the
orthogonal complement of its kernel) of a densely defined, closed, strictly
positive operator, for some in a Hilbert space to an abstract buckling problem operator.
This establishes the Krein extension as a natural object in elasticity theory
(in analogy to the Friedrichs extension, which found natural applications in
quantum mechanics, elasticity, etc.).
In the second, and principal part of this survey, we study spectral
properties for , the Krein--von Neumann extension of the
perturbed Laplacian (in short, the perturbed Krein Laplacian)
defined on , where is measurable, bounded and
nonnegative, in a bounded open set belonging to a
class of nonsmooth domains which contains all convex domains, along with all
domains of class , .Comment: 68 pages. arXiv admin note: extreme text overlap with arXiv:0907.144
Integral potential method for a transmission problem with Lipschitz interface in R^3 for the Stokes and Darcy–Forchheimer–Brinkman PDE systems
The purpose of this paper is to obtain existence and uniqueness results in weighted Sobolev spaces for transmission problems for the non-linear Darcy-Forchheimer-Brinkman system and the linear Stokes system in two complementary Lipschitz domains in R3, one of them is a bounded Lipschitz domain with connected boundary, and the other one is the exterior Lipschitz domain R3 n. We exploit a layer potential method for the Stokes and Brinkman systems combined with a fixed point theorem in order to show the desired existence and uniqueness results, whenever the given data are suitably small in some weighted Sobolev spaces and boundary Sobolev spaces
Transmission Problems for the Navier–Stokes and Darcy–Forchheimer–Brinkman Systems in Lipschitz Domains on Compact Riemannian Manifolds
M. Kohr acknowledges the support of the Grant PN-II-ID-PCE-2011-3-0994 of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI. The research has been also partially supported by the Grant EP/M013545/1: “Mathematical Analysis of Boundary-Domain Integral Equations for Nonlinear PDEs” from the EPSRC, UK
Analysis of segregated boundary-domain integral equations for BVPs with non-smooth coefficients on Lipschitz domains
Segregated direct boundary-domain integral equations (BDIEs) based on a parametrix and associated with the Dirichlet and Neumann boundary value problems for the linear stationary diffusion partial differential equation with a variable Hölder-continuous coefficients on Lipschitz domains are formulated. The PDE right-hand sides belong to the Sobolev (Bessel potential) space Hs−2(Ω ) or H˜s−2(Ω ) , 12<s<32, when neither strong classical nor weak canonical co-normal derivatives are well defined. Equivalence of the BDIEs to the original BVP, BDIE solvability, solution uniqueness/non-uniqueness, and the Fredholm property and invertibility of the BDIE operators are analysed in appropriate Sobolev spaces. It is shown that the BDIE operators for the Neumann BVP are not invertible; however, some finite-dimensional perturbations are constructed leading to invertibility of the perturbed (stabilised) operators.EPSR
Encyclopaedia of mathematical sciences. vol.LXXIX, Partial differential equations IX : Elliptic boundary value problems
280 p. ; 25 cm