Homogenization of fully overdamped Frenkel-Kontorova models

In this paper, we consider the fully overdamped Frenkel-Kontorova model. This is an infinite system of coupled first-order ODEs. Each ODE represents the microscopic evolution of one particle interacting with its neighbors and submitted to a fixed periodic potential. After a proper rescaling, a macroscopic model describing the evolution of densities of particles is obtained. We get this homogenization result for a general class of Frenkel-Kontorova models. The proof is based on the construction of suitable hull functions in the framework of viscosity solutions.     

Periodic solutions of a periodically forced and undamped beam resting on weakly elastic bearings

We study the problem of existence of periodic solutions to a partial differential equation modelling the behavior of an undamped beam subject to an external periodic force. We assume that the ordinary differential equation associated to the first two modes of vibration of the beam has a symmetric homoclinic solution. By using methods borrowed by dynamical systems theory we prove that, if the period is non resonant with the (infinitely many) internal periods of the PDE, the equation has a weak periodic solution of the same period as the external force. In particular we obtain continua of periodic solutions for the undamped beam in absence of external forces. This result may be considered as an infinite dimensional analogue of a result obtained in [16] concerning accumulation of periodic solutions to homoclinic orbits in finite dimensional reversible systems. Matteo Franca: Partially supported by G.N.A.M.P.A. – INdAM (Italy).     

Uniqueness and decay estimates for a class of parabolic partial differential equations with hysteresis and convection

We deal in detail with the question of existence, uniqueness and asymptotic behaviour of solutions to a parabolic equation with hysteresis and convection. This equation is part of a model system which describes the magnetohydrodynamic (MHD) flow of a conducting fluid between two ferromagnetic plates. The result of this paper complements the content of a previous paper of the first author, where existence of the solution has been proved under fairly general assumptions on the hysteresis operator and the uniqueness was only obtained for a restricted class of hysteresis operators.     

Weak convergence of approximate solutions of stochastic equations with applications to random differential and integral equations ∗

In this paper, we considerably extend our earlier result about convergence in distribution of approximate solutions: of random operator equations, where the stochastic inputs and the underlying deterministic equation are simultaneously approximated. As a by-product, we obtain convergence results for approximate solutions of equations between spaces of probability measures. We apply our results to random Fredholm integral equations of the second kind and to a random [nbar]onlinear elliptic boundary value problem.     

Little Hankel operators on the half plane

In this paper we consider a class of weighted integral operators onL ² (0, ) and show that they are unitarily equivalent to Hankel operators on weighted Bergman spaces of the right half plane. We discuss conditions for the Hankel integral operator to be finite rank, Hilbert-Schmidt, nuclear and compact, expressed in terms of the kernel of the integral operator. For a particular class of weights these operators are shown to be unitarily equivalent to little Hankel operators on weighted Bergman spaces of the disc, and the symbol correspondence is given. Finally the special case of the unweighted Bergman space is considered and for this case, motivated by approximation problems in systems theory, some asymptotic results on the singular values of Hankel integral operators are provided.     

Characterization of polynomial decay rate for the solution of linear evolution equation

In this paper, we study the decay rate of solutions to strongly stable, but not exponentially stable linear evolution equations. It is known that the resolvent operator of such an equation must be unbounded on the imaginary axis. Our main result is an estimate of the decay rate when the unboundedness is of polynomial order. We then apply our main theorem to three strongly stable but not exponentially stable systems to obtain the decay rate, which is not available in the literature.Received: July 21, 2003     

Asymptotic bounds of solutions for a periodic doubly degenerate parabolic equation

This paper is concerned with a doubly degenerate parabolic equation with logistic periodic sources. We are interested in the discussion of the asymptotic behavior of solutions of the initial-boundary value problem. In this paper, we first establish the existence of non-trivial nonnegative periodic solutions by a monotonicity method. Then by using the Moser iterative method, we obtain an a priori upper bound of the nonnegative periodic solutions, by means of which we show the existence of the maximum periodic solution and asymptotic bounds of the nonnegative solutions of the initial-boundary value problem. We also prove that the support of the non-trivial nonnegative periodic solution is independent of time.     

Paraproducts and Hankel operators of Schatten class via p-John-Nirenberg Theorem

We give an interpolation-free proof of the known fact that a dyadic paraproduct is of Schatten-von Neumann class S_p, if and only if its symbol is in the dyadic Besov space B_p^d. Our main tools are a product formula for paraproducts and a “p-John-Nirenberg-Theorem” due to Rochberg and Semmes.We use the same technique to prove a corresponding result for dyadic paraproducts with operator symbols.Using an averaging technique by Petermichl, we retrieve Peller's characterizations of scalar and vector Hankel operators of Schatten-von Neumann class S_p for 1<p<∞. We then employ vector techniques to characterise little Hankel operators of Schatten-von Neumann class, answering a question by Bonami and Peloso.Furthermore, using a bilinear version of our product formula, we obtain characterizations for boundedness, compactness and Schatten class membership of products of dyadic paraproducts.     

Zero measure Cantor spectra for continuum one-dimensional quasicrystals

We study Schrödinger operators on R$\mathbb{R}$ with measures as potentials. Choosing a suitable subset of measures we can work with a dynamical system consisting of measures. We then relate properties of this dynamical system with spectral properties of the associated operators. The constant spectrum in the strictly ergodic case coincides with the union of the zeros of the Lyapunov exponent and the set of non-uniformities of the transfer matrices. This result enables us to prove Cantor spectra of zero Lebesgue measure for a large class of operator families, including many operator families generated by aperiodic subshifts.     

Circles in the spectrum and the geometry of orbits: A numerical ranges approach

We prove that a bounded linear Hilbert space operator has the unit circle in its essential approximate point spectrum if and only if it admits an orbit satisfying certain orthogonality and almost-orthogonality relations. This result is obtained via the study of numerical ranges of operator tuples where several new results are also obtained. As consequences of our numerical ranges approach, we derive in particular wide generalizations of Arveson's theorem as well as show that the weak convergence of operator powers implies the uniform convergence of their compressions on an infinite-dimensional subspace.     

Compact linear operators on functional spaces with two norms

Some principles of the operator theory in a linear space with two norms are established in this paper. The well-known Hilbert-Schmidt theorem on the eigenfunction expansion of sourcewise represented functions, Mercer's theorem and other results can be consider as special cases of the statements presented. The general approach proposed is used to construct the theory of symmetrizable operators and to investigate the asymptotic behaviour of eigenvalues of compact operators.This paper was translated by M. Gorbuchuk and V. GorbachukThis paper was translated by M. Gorbuchuk and V. Gorbachuk.     

Asymptotic almost-equivalence of Lipschitz evolution systems in Banach spaces

In this paper we introduce a notion of asymptotic almost-equivalence of two evolution systems and provide simple tests that guarantee that two evolution systems have the same qualitative asymptotic properties. In this way we are able to unify and extend many previously known results and also to understand what is behind equally behaved systems. In particular, we establish convergence, ergodic convergence and almost-convergence of almost-orbits both for the weak and the strong topologies based on the behavior of the orbits.     

Exact periodic wave solutions for the hKdV equation

In this paper, by using the Hirota bilinear method and the Jacobian theta functions for the higher order KdV equation, the existence of periodic wave solutions with one and two period are obtained. The asymptotic properties of the periodic wave solutions are analyzed in detail. It is shown that the well-known soliton solutions can be reduced from the periodic wave solutions.     

Evolution equations of second order with nonconvex potential and linear damping: existence via convergence of a full discretization

Global existence of solutions for a class of second-order evolution equations with damping is shown by proving convergence of a full discretization. The discretization combines a fully implicit time stepping with a Galerkin scheme. The operator acting on the zero-order term is assumed to be a potential operator where the potential may be nonconvex. A linear, symmetric operator is assumed to be acting on the first-order term. Applications arise in nonlinear viscoelasticity and elastodynamics.     

Deterministic homogenization of non-linear non-monotone degenerate elliptic operators

Deterministic homogenization has been till now applied to the study of monotone operators, the determination of the limiting problem being systematically based on the monotonicity of the operator under consideration. Here we mean to show that deterministic homogenization also tackle non-monotone operators. More precisely, under an abstract general hypothesis, we study the homogenization of non-linear non-monotone degenerate elliptic operators. We obtain some general homogenization result, which result is applied to the resolution of several concrete homogenization problems such as the periodic homogenization and the almost periodic homogenization problems. Our main tool is the theory of homogenization structures.     

On statistical convergence of difference sequences of fractional order and related Korovkin type approximation theorems

Abstract

Prior to investigating on sequence spaces and their convergence, we study the notion of statistical convergence of difference sequences of fractional order α ∈ ?. As generalizations of previous works, this study includes several special cases under different limiting conditions of α, such as the notion of statistical convergence of difference sequences of zeroth and mth (integer) order. In fact, we study certain new results on statistical convergence via the difference operator Δ^α and interpret them to those of previous works. Also, by using the convergence of Δ^α-summable sequences which is stronger than statistical convergence of difference sequences, we apply classical Bernstein operator and a generalized form of Meyer-Konig and Zeller operator to construct an example in support of our result. Also, we study the rates of Δ^α-statistical convergence of positive linear operators.     

Asymptotic behavior in time periodic parabolic problems with unbounded coefficients

We study asymptotic behavior in a class of nonautonomous second order parabolic equations with time periodic unbounded coefficients in R×Rd. Our results generalize and improve asymptotic behavior results for Markov semigroups having an invariant measure. We also study spectral properties of the realization of the parabolic operator u?A(t)u−ut in suitable Lp spaces.     

The regularized Benjamin-Ono and BBM equations: Well-posedness and nonlinear stability

Nonlinear stability of nonlinear periodic solutions of the regularized Benjamin-Ono equation and the Benjamin-Bona-Mahony equation with respect to perturbations of the same wavelength is analytically studied. These perturbations are shown to be stable. We also develop a global well-posedness theory for the regularized Benjamin-Ono equation in the periodic and in the line setting. In particular, we show that the Cauchy problem (in both periodic and nonperiodic case) cannot be solved by an iteration scheme based on the Duhamel formula for negative Sobolev indices.     

New results on the invertibility of the finite interval convolution operator

Conditions for the invertibility and explicit formulas for the inverse of the convolution operator on a finite interval are obtained making use of solutions of corona problems. Using these results, a family of classes of functions is defined for which the study of invertibility can be carried through. An example of one class of this family is presented and a smaller class, for which the calculations are simpler, is more thoroughly studied.Work sponsored by F.C.T. (Portugal) under Project Praxis XXi/2/2.1/MAT/441/94