Properties of an integral operator preserved under pointwise multiplication of its kernel by a function

Let K be a kernel that determines an integral operator on some space of functions, and let H be a function. This paper investigates conditions under which certain properties of the integral operator determined by K (especially compactness properties) also hold for the integral operator determined by HK.     

On Some Integral Equations in Hilbert Space with an Application to the Theory of Elasticity

A Volterra type integral equation in a Hilbert space with an additional linear operator L and a spectral parameter depending on time is considered. If the parameter does not belong to the spectrum of L unconditional solvability of the considered problem is proved. In the case where the initial value of the parameter coincides with some isolated point of the spectrum of the operator L sufficient conditions for solvability are established. The obtained results are applied to the partial integral equations associated with a contact problem of the theory of elasticity.     

On the Spectrum of the Reflection Operator on Conical Surfaces

We study the mapping properties of the reflection operator on a conical surface. This allows us to derive regularity results for the solution of the radiosity equation on conical surfaces in a scale of weighted Sobolev spaces. To motivate the calculations we first study the operator on a cylinder. Here we estimate the asymptotic behavior of the spectrum of the reflection operator by partial integration. This method works also for the conical case, but first we have to find a simple representation for some hypergeometric functions.     

Eigenvalues of Positive Definite Integral Operators on Unbounded Intervals

Let k(x, y) be the positive definite kernel of an integral operator on an unbounded interval of ℝ. If k belongs to class defined below, the corresponding operator is compact and trace class. We establish two results relating smoothness of k and its decay rate at infinity along the diagonal with the decay rate of the eigenvalues. The first result deals with the Lipschitz case; the second deals with the uniformly C¹ case. The optimal results known for compact intervals are recovered as special cases, and the relevance of these results for Fourier transforms is pointed out.     

On the regularity of solutions to integral equations with nonsmooth kernels on a union of open intervals

The behaviour of a solution to a Fredholm integral equation of the second kind on a union of open intervals is examined. The kernel of the corresponding integral operator may have diagonal singularities, information about them is given through certain estimates. The weighted spaces of smooth functions with boundary singularities containing the solution of the integral equation are described.     

Boundary operator from matrix field formulation of boundary conditions for Friedrichs systems

Following the recent progress in understanding the abstract setting for Friedrichs symmetric positive systems by Ern, Guermond and Caplain (2007) [8], as well as Antoni? and Burazin (2010) [3], an attempt is made to relate these results to the classical Friedrichs theory.A comparison of two approaches, via the trace operator and the boundary operator, has been made, favouring the latter. Finally, a particular set of sufficient conditions for a boundary matrix field to define a boundary operator in that case is given, and the applicability of this procedure in realistic situations is shown by examples.     

Spectrum of the Kerzman-Stein Operator for the Ellipse

The skew-hermitian part of the Cauchy operator, defined with respect to arclength measure on the boundary, is known as the Kerzman-Stein operator. For an ellipse, the eigenvalues of this operator are shown to have multiplicity two. For an ellipse with small eccentricity, we compute the leading coefficient in the asymptotic expansion of the eigenvalues.     

Optimal order results for a class of regularization methods using unbounded operators

A class of regularization methods using unbounded regularizing operators is considered for obtaining stable approximate solutions for ill-posed operator equations. With an a posteriori as well as an a priori parameter choice strategy, it is shown that the method yields the optimal order. Error estimates have also been obtained under stronger assumptions on the generalized solution. The results of the paper unify and simplify many of the results available in the literature. For example, the optimal results of the paper include, as particular cases for Tikhonov regularization, the main result of Mair (1994) with an a priori parameter choice, and a result of Nair (1999) with an a posteriori parameter choice. Thus the observations of Mair (1994) on Tikhonov regularization of ill-posed problems involving finitely and infinitely smoothing operators is applicable to various other regularization procedures as well. Subsequent results on error estimates include, as special cases, an optimal result of Vainikko (1987) and also some recent results of Tautenhahn (1996) in the setting of Hilbert scales.     

The h-p version of the coupling of finite element and boundary element methods for transmission problems in polyhedral domains

Summary. This paper analyzes the rate of convergence of the h-p version of the coupling of the finite element and boundary element method for transmission problems with a linear differential operator with variable coefficients in a bounded polyhedral domain and with constant coefficients in the exterior domain . This procedure uses the variational formulation of the differential equation in and involves integral operators on the interface between and . The finite elements are used to obtain approximate solutions of the differential equation in and the boundary elements are used to obtain approximate solutions of the integral equations. For given piecewise analytic data we show that the Galerkin solution of this coupling procedure converges exponentially fast in the energy norm if the h-p version is used both for finite elements and boundary elements. Received February 10, 1996 / Revised version received April 4, 1997     

The wave equation on asymptotically de Sitter-like spaces

In this paper we obtain the asymptotic behavior of solutions of the Klein-Gordon equation on Lorentzian manifolds (X^○,g) which are de Sitter-like at infinity. Such manifolds are Lorentzian analogues of the so-called Riemannian conformally compact (or asymptotically hyperbolic) spaces. Under global assumptions on the (null)bicharacteristic flow, namely that the boundary of the compactification X is a union of two disjoint manifolds, Y_±, and each bicharacteristic converges to one of these two manifolds as the parameter along the bicharacteristic goes to +∞, and to the other manifold as the parameter goes to −∞, we also define the scattering operator, and show that it is a Fourier integral operator associated to the bicharacteristic flow from Y₊ to Y₋.     

Conditions for blow-up of solutions of some nonlinear Volterra integral equations

In this paper we give necessary and sufficient conditions for blow-up of solutions for a particular class of nonlinear Volterra equations. We also give some examples.     

Diffraction problems with impedance conditions

The two- or three-dimensional electromagnetic diffraction problem for a half-plane impedance or reactance sheet belongs to a class of elliptic transmission problems of mixed type. Sobolev spaces of order 1 and ±1/2 are naturally involved according to the energy norm and the trace theorem, respectively. This operator theoretic approach presents the equivalence to systems of Wiener-Hopf equations and their solution in the sense of a well-posed problem with respect to the spaces under consideration. Slightly different impedance numbers for the two banks of the screen lead to a perturbation problem. All results yield direct a priori estimates for the solutions.     

Mean Ergodicity and Mean Stability of Regularized Solution Families

We prove general theorems on mean ergodicity and mean stability of regularized solution families with respect to fairly general summability methods. They can be applied to integrated solution families, integrated semigroups and cosine functions. In particular, through applications with modified Cesàro, Abel, Gauss, and Gamma like summability methods we deduce particular results on mean ergodicity and mean stability of polynomially bounded C₀-semigroups and cosine operator functions.     

Finite Interval Convolution Operators with Transmission Property

We study convolution operators in Bessel potential spaces and (fractional) Sobolev spaces over a finite interval. The main purpose of the investigation is to find conditions on the convolution kernel or on a Fourier symbol of these operators under which the solutions inherit higher regularity from the data. We provide conditions which ensure the transmission property for the finite interval convolution operators between Bessel potential spaces and Sobolev spaces. These conditions lead to smoothness preserving properties of operators defined in the above-mentioned spaces where the kernel, cokernel and, therefore, indices do not depend on the order of differentiability. In the case of invertibility of the finite interval convolution operator, a representation of its inverse is presented in terms of the canonical factorization of a related Fourier symbol matrix function.     

On approximation and representation of K-regularized resolvent families

We study the problem of approximation and representation for a family of strongly continuous operators defined in a Banach space. It allows us to extend, and in some cases to improve results from the theory ofC ₀-semigroups of operators to, among others, the theories of cosine families, n-times integrated semigroups, resolvent families and k-generalized solutions by means of an unified method.The author was supported by FONDECYT grants 1980812; 1970722 and DICYT (USACH).     

An adaptive wavelet shrinkage approach to the Spektor-Lord-Willis problem

The stereological problem of unfolding the sphere size distribution from linear sections is considered. A minimax estimator of the intensity function of a Poisson process that describes the problem is introduced and an adaptive estimator is constructed that achieves the optimal rate of convergence over Besov balls to within logarithmic factors. The construction of these estimators uses Wavelet-Vaguelette Decomposition (WVD) of the operator that defines our inverse problem.     

A numerical method for solving Fredholm-Volterra integral equations in two-dimensional spaces using block pulse functions and an operational matrix

In this paper, the block pulse functions (BPFs) and their operational matrix are used to solve two-dimensional Fredholm-Volterra integral equations (F-VIE). This method converts F-VIE to systems of linear equations whose solutions are the coefficients of block pulse expansions of the solutions of F-VIE.Finally some numerical examples are presented to show the efficiency and accuracy of the method.     

Maximal regularity of type L p for abstract parabolic Volterra equations

We prove maximal regularity results of type L_p for abstract parabolic Volterra equations including problems with inhomogeneous boundary data. Our approach is purely operator theoretic. It uses the inversion of the convolution, the Dore-Venni theorem, the Mikhlin theorem in the operator-valued version, and real interpolation. Known results on L_p-regularity of abstract Cauchy problems and abstract parabolic pdes with inhomogeneous boundary conditions are recovered. As an application we consider the heat equation of memory type with inhomogeneous boundary condition.     

On some nonlocal evolution equations in Banach spaces

This note is concerned with the initial value problem for the abstract nonlocal equation where A is a maximal monotone operator from a reflexive Banach space E to its dual E^*, while B is a nonlocal maximal monotone operator from . Under proper boundedness and coercivity assumptions on the operators, a solution is achieved by means of a discretization argument. Uniqueness and continuous dependence are also discussed and we prove some estimates for the discretization error. Finally, we deal with the approximation of linear Volterra integrodifferential operators.