Wiener-Hopf factorization for a class of oscillatory symbols

Two classes of 2×2 matrix symbols involving oscillatory functions are considered, one of which consists of triangular matrices. An equivalence theorem is obtained, concerning the solution of Riemann-Hilbert problems associated with each of them. Conditions for existence of canonical generalized factorization are established, as well as boundedness conditions for the factors. Explicit formulas are derived for the factors, showing in particular that only one of the columns needs to be calculated. The results are applied to solving a corona problem.     

Weakly singular integral equations with operator-valued kernels and an application to radiation transfer problems

The standard problem of radiation transfer in a bounded regionG ⁿ can be reformulated as a weakly singular integral equation with an unknown functionu: GC(S ^n–1) and a kernelK: ((G × G }x=y}, which is continuously differentiable with respect to the operator strong convergence topology. We take these observations into the basis of an abstract treatment of weakly singular integral equations with (E)-valued kernels, whereE is a Banach space. Our purpose is to characterize the smoothness of the solution by proving that it belongs to special weighted spaces of smooth functions. On the way, realizing the proof techniques, we establish the compactness of the integral operator or its square inL _p (G,E),BC(G,E), and other spaces of interest in numerical analysis as well as in weighted spaces of smooth functions. The smoothness results are specified for the standard problem of radiation transfer as well as for the corresponding eigenvalue problem.     

Wiener-Hopf operators with oscillating symbols and convolution operators on a union of intervals

The present work was sponsored by JNICT (Portugal) under grant n. 87422/MATM and Programa Ciência.     

Galerkin method for Wiener-Hopf operators with piecewise continuous symbol

This paper is concerned with the applicability of maximum defect polynomial (Galerkin) spline approximation methods with graded meshes to Wiener-Hopf operators with matrix-valued piecewise continuous generating function defined on R. For this, an algebra of sequences is introduced, which contains the approximating sequences we are interested in. There is a direct relationship between the stability of the approximation method for a given operator and invertibility of the corresponding sequence in this algebra. Exploring this relationship, the methods of essentialization, localization and identification of the local algebras are used in order to derive stability criteria for the approximation sequences.Supported by grant Praxis XXI/BD/4501/94 from FCT.Partly supported by FCT/BMFT grant 423.     

Spline approximation methods for multidimensional periodic pseudodifferential equations

We investigate several numerical methods for solving the pseudodifferential equationAu=f on the n-dimensional torusT ⁿ . We examine collocation methods as well as Galerkin-Petrov methods using various periodical spline functions. The considered spline spaces are subordinated to a uniform rectangular or triangular grid. For given approximation method and invertible pseudodifferential operatorA we compute a numerical symbol ^C , resp. ^G , depending onA and on the approximation method. It turns out that the stability of the numerical method is equivalent to the ellipticity of the corresponding numerical symbol. The case of variable symbols is tackled by a local principle. Optimal error estimates are established.The second author has been supported by a grant of Deutsche Forschungsgemeinschaft under grant namber Ko 634/32-1.     

A shift-invariant algebra of singular integral operators with oscillating coefficients

LetB be the Banach algebra of all bounded linear operators on the weighted Lebesgue spaceL ^p (T, ) with an arbitrary Muckenhoupt weight on the unit circleT, and the Banach subalgebra ofB generated by the operators of multiplication by piecewise continuous coefficients and the operatorse _h,S _T e _h, ^–1 I (hR, T) whereS _T is the Cauchy singular integral operator ande _h,(t)=exp(h(t+)/(t–)),tT. The paper is devoted to a symbol calculus, Fredholm criteria and an index formula for the operators in the algebra and its matrix analogue . These shift-invariant algebras arise naturally in studying the algebras of singular integral operators with coefficients admitting semi-almost periodic discontinuities and shifts being diffeomorphisms ofT onto itself with second Taylor derivatives.Partially supported by CONACYT grant, Cátedra Patrimonial, No. 990017-EX and by CONACYT project 32726-E, México.     

Patterns and structure in systems governed by linear second-order differential equations

This article represents a survey of transmutation ideas and their interaction with typical physical problems. For linear second-order differential operatorsP andQ one deals with canonical connectionsB:PQ (transmutations) satisfyingQB=BP and the related transport of structure between the theories ofP andQ. One can study in an intrinsic manner, e.g., Parseval formulas, eigenfunction expansions, integral transform, special functions, inverse problems, integral equations, and related stochastic filtering and estimation problems, etc. There are applications in virtually any area where such operators arise.     

A Wiener-Hopf approach for the scattering by a disk

Dedicated to Professor Erhard Meister on the occasion of his 65^th birthday.     

Localized minimizers of flat rotating gravitational systems

We study a two-dimensional system in solid rotation at constant angular velocity driven by a self-consistent three-dimensional gravitational field. We prove the existence of stationary solutions of such a flat system in the rotating frame as long as the angular velocity does not exceed some critical value which depends on the mass. The solutions can be seen as stationary solutions of a kinetic equation with a relaxation-time collision kernel forcing the convergence to the polytropic gas solutions, or as stationary solutions of an extremely simplified drift-diffusion model, which is derived from the kinetic equation by formally taking a diffusion limit. In both cases, the solutions are critical points of a free energy functional, and can be seen as localized minimizers in an appropriate sense.     

Global existence and maximal regularity of solutions of gradient systems

In this article, we use a Galerkin method to prove a maximal regularity result for the following abstract gradient system

     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-maximal regularity for second order Cauchy problems

We introduce the concept of L^p-maximal regularity for second order Cauchy problems. We prove L^p-maximal regularity for an abstract model problem and we apply the abstract results to prove existence, uniqueness and regularity of solutions for nonlinear wave equations. The author acknowledges with thanks the support provided by the Department ofApplied Analysis, University of Ulm, and the travel grants provided by NBMH India and MSF Delhi, India.     

The forward problem for the electromagnetic Helmholtz equation with critical singularities

We study the forward problem of the magnetic Schrödinger operator with potentials that have a strong singularity at the origin. We obtain new resolvent estimates and give some applications on the spectral measure and on the solutions of the associated evolution problem.     

Attractors for nonautonomous 2D Navier-Stokes equations with less regular symbols

We introduce a new class of functions satisfying normal Condition (C*), denoted by , which are translation bounded but not translation compact — in particular, which are more general than normal functions (see [S.S. Lu, H.Q. Wu, C.K. Zhong, Attractors for nonautonomous 2D Navier-Stokes equations with normal external forces, Discrete Contin. Dyn. Syst., 13 (2005) 701-719] for the definition), denoted by . Furthermore, we prove the existence of uniform attractors for 2D Navier-Stokes equations with external forces belonging to in .     

Pseudodifferential operators with heavy spectrum

We establish Fredholm criteria and index formulas for one-dimensional zero-order pseudodifferential operators with piecewise continuous generating functions onL ^p spaces with Muckenhoupt weights. The Fredholm symbol of such operators is shown to be a matrix function defined on a set which, roughly speaking, is a cylinder with a certain collection of horn shaped handles. The presence of these horns implies that, unlike the case ofL ^p spaces without weight or with so-called power weights, the spectrum may contain heavy parts, i. e. the set of the interior points of the spectrum need not be empty. Our proof makes essential use of recent results by Finck, Roch, Silbermann, Gohberg, and Krupnik on the inverse closedness of certain Banach algebras.Research supported by the Alfried Krupp Förderpreis für junge Hochschullehrer of the Krupp Foundation.     

On the approximation of functions satisfying defective renewal equations

Functions satisfying a defective renewal equation arise commonly in applied probability models. Usually these functions do not admit an explicit expression. In this work, we consider their approximation by means of a gamma-type operator given in terms of the Laplace transform of the initial function. We investigate which conditions on the initial parameters of the renewal equation give the optimal order of uniform convergence of the approximation. We apply our results to ruin probabilities in the classical risk model, paying special attention to mixtures of gamma claim amounts.     

Inversion formulas for infinite generalized Toeplitz matrices

Inversion formulas are obtained for a certain class of infinite matrices that possess displacement structure similar to that of finite block Toeplitz matrices. Consequences are symmetric inversion formulas for matrix-valued singular integral operators and infinite Toeplitz plus Hankel matrices.     

Convergence in Variation and Rate of Approximation for Nonlinear Integral Operators of Convolution Type

In this paper we obtain estimates, convergence results and rate of approximation for functions belonging to BV–spaces (spaces of functions with bounded variation) by means of nonlinear convolution integral operators. We treat both the periodic and the non-periodic case using, respectively, the classical Jordan variation and the multidimensional variation in the sense of Tonelli.     

Regularity and mass conservation for discrete coagulation–fragmentation equations with diffusion

We present a new a priori estimate for discrete coagulation–fragmentation systems with size-dependent diffusion within a bounded, regular domain confined by homogeneous Neumann boundary conditions. Following from a duality argument, this a priori estimate provides a global L²$L^2$ bound on the mass density and was previously used, for instance, in the context of reaction–diffusion equations.