6 research outputs found
Homogenization of Ferromagnetic Energies on Poisson Random Sets in the Plane
We prove that by scaling nearest-neighbour ferromagnetic energies
de ned on Poisson random sets in the plane we obtain an isotropic perimeter
energy with a surface tension characterised by an asymptotic formula. The
result relies on proving that cells with `very long' or `very short' edges of the
corresponding Voronoi tessellation can be neglected. In this way we may apply
Geometry Measure Theory tools to de ne a compact convergence, and a characterisation
of metric properties of clusters of Voronoi cells using limit theorems
for subadditive processes
Large deviations for Markov jump processes in periodic and locally periodic environments
The paper deals with a family of jump Markov process defined in a medium with a periodic or locally periodic microstructure. We assume that the generator of the process is a zero order convolution type operator with rapidly oscillating locally periodic coefficient and, under natural ellipticity and localization conditions, show that the family satisfies the large deviation principle in the path space equipped with Skorokhod topology. The corresponding rate function is defined in terms of a family of auxiliary periodic spectral problems. It is shown that the corresponding Lagrangian is a convex function of velocity that has a superlinear growth at infinity. However, neither the Lagrangian nor the corresponding Hamiltonian need not be strictly convex, we only claim their strict convexity in some neighbourhood of infinity. It then depends on the profile of the generator kernel whether the Lagrangian is strictly convex everywhere or not
On the spectrum of convolution operator with a potential
This paper focuses on the spectral properties of a bounded self-adjoint operator in L2(Rd) being the sum of a convolution operator with an integrable convolution kernel and an operator of multiplication by a continuous potential converging to zero at infinity. We study both the essential and the discrete spectra of this operator. It is shown that the essential spectrum of the sum is the union of the essential spectrum of the convolution operator and the image of the potential. We then provide a number of sufficient conditions for the existence of discrete spectrum and obtain lower and upper bounds for the number of discrete eigenvalues. Special attention is paid to the case of operators possessing countably many points of the discrete spectrum. We also compare the spectral properties of the operators considered in this work with those of classical Schrödinger operators
Mathematical multi-scale model of water purification
In this work, we consider a mathematical model of the water treatment process and determine the effective characteristics of this model. At the microscopic
length scale, we describe our model in terms of a lattice random walk in a
high-contrast periodic medium with absorption. Applying then the upscaling
procedure, we obtain the macroscopic model for total mass evolution. We discuss both the dynamic and the stationary regimes and show how the efficiency
of the purification process depends on the characteristics of the macroscopic
model
On operator estimates in homogenization of nonlocal operators of convolution type
The paper studies a bounded symmetric operator in
with here is a small
positive parameter. It is assumed that is a non-negative
function such that and the moments , , are finite. It is also assumed
that is -periodic both in and function such
that and . Our
goal is to study the limit behaviour of the resolvent
, as . We show that, as
, the operator
converges in the operator norm in to the resolvent
of the effective operator being a
second order elliptic differential operator with constant coefficients of the
form . We then obtain sharp in
order estimates of the rate of convergence