Zeitschrift für Analysis und ihre Anwendungen
Journal for Analysis and its Applications

Volume 20 (2001)

Abstracts

R. Bader:A Topological Fixed-Point Index Theory for Evolution Inclusions, 20 (2001) 003--016

We construct a topological fixed-point theory for a class of set-valued maps which appears in natural way in boundary value problems for differential inclusions. Our construction is based upon the notion of (U, V)-approximation in the sense of Ben-El-Mechaiekh and Deguire. As applications we consider initial-value problems for nonlinear evolution inclusions of the type

x'(t) in -A(t, x(t)) + F(t, x(t)),    x(0) = x₀,

where the operator A satisfies various monotonicity assumptions and F is an upper semi-continuous set-valued perturbation.

P. Cerejeiras, K. Gürlebeck, U. Kähler, H. Malonek: A Quaternionic Beltrami-Type Equation and the Existence of Local Homeomorphic Solutions, 20 (2001) 017--034

The paper deals with a quaternionic Beltrami-type equation, which is a very natural generalization of the complex Beltrami equation to higher dimensions. Special attention is paid to the systematic use of the embedding of the set of quaternions H into C² and the corresponding application of matrix singular integral operators. The proof of the existence of local homeomorphic solutions is based on a necessary and sufficient criterion, which relates the Jacobian determinant of a mapping from R⁴ into R⁴ to the quaternionic derivative of a monogenic function.

S. N. Lal, S. Bhattacharya, C. Sreedhar: Complex 2-Normed Linear Spaces and Extension of Linear 2-Functionals, 20 (2001) 035--054

The known concept of 2-normed real linear spaces is extended to 2-normed complex linear spaces. This extension is not trivial. A Hahn-Banach type extension theorem for complex linear 2-functionals is established and it is shown that it is not possible to get this result from the known Hahn-Banach type extension theorem for real linear 2-functionals using the Bohnenblust-Sobczyk technique directly as is done in the case of linear functionals. As an application of our extension theorem, a 2-norm version of the Ascoli-Mazur theorem on tangent functionals is established. Several examples and counter examples illustrate the results obtained in the paper.

K. Oelschläger: A Sequence of Integro-Differential Equations Approximating a Viscous Porous Medium Equation, 20 (2001) 055--092

We consider a sequence of particular integro-differential equations, whose solutions r_n converge as n approaches infinity to the solution r of a viscous porous medium equation. First, it is demonstrated that under suitable regularity conditions the functions r_n are smooth uniformly in n of N. Furthermore, an asymptotic expansion for r_n as n approaches infinity is provided, which precisely describes the convergence to r. The results of this paper are needed in particular for the numerical simulation of a viscous porous medium equation by a particle method.

A. Rougirel: Blow-Up and Convergence Results for a One-Dimensional Non-Local Parabolic Problem, 20 (2001) 093--114

Considering a one-dimensional non-local semilinear parabolic problem, it is shown that blow-up in finite time occurs for suitable large initial conditions. The asymptotic behavior of global solutions corresponding to small initial conditions is also investigated. Their convergence in H¹-norm to a well determinated stationary solution is proved.

A. Bychowska, H. Leszczynski: Parabolic Equations with Functional Dependence, 20 (2001) 115--130

We consider the Cauchy problem for nonlinear parabolic equations with functional dependence and prove theorems on the existence of solutions to parabolic differential-functional equations.

G. Schmidt: Boundary Integral Operators for Plate Bending in Domains with Corners, 20 (2001) 131--154

The paper studies boundary integral operators of the bi-Laplacian on piecewise smooth curves with corners and describes their mapping properties in the trace spaces of variational solutions of the biharmonic equation. We formulate a direct integral equation method for solving interior and exterior mixed boundary value problems on non-smooth plane domains, analyze the solvability of the corresponding systems of integral equations and prove their strong ellipticity.

S. A. Nazarov, K. Pileckas: On the Fredholm Property of the Stokes Operator in a Layer-Like Domain, 20 (2001) 155--182

The Stokes problem is studied in the subdomain W of R³ coinciding with the layer P = {x = (y,z) : y = (y₁, y₂) in R², z in (0,1) } outside some ball. It is shown that the operator of such problem is of Fredholm type; this operator is defined on a certain weighted function space D^l_b (W) with norm determined by a stepwise anisotropic distribution of weight factors (the direction of z is distinguished). The smoothness exponent l is allowed to be a positive integer, and the weight exponent b is an arbitrary real number except for the integer set Z where the Fredholm property is lost. Dimensions of the kernel and cokernel of the operator are calculated in dependence of b. It turns out that, at any admissible b, the operator index does not vanish. Based on the generalized Green formula, asymptotic conditions at infinity are imposed to provide the problem with index zero.

C. A. Stuart, G. Vuillaume: A Discretised Nonlinear Eigenvalue Problem with Many Spurious Branches of Solutions, 20 (2001) 183--192

We treat an example of a nonlinear eigenvalue problem in L²(0,1) which can be solved explicitly. It has a single branch of non-trivial solutions. Discretisation reduces the problem to a finite-dimensional one having many branches of non-trivial solutions. We investigate the convergence of these approximate solutions.

S. Saitoh, V. K. Tuan, M. Yamamoto: Conditional Stability of a Real Inverse Formula for the Laplace Transform, 20 (2001) 193--202

We establish a conditional stability estimate of a real inverse formula for the Laplace transform of functions under the assumption that the Bergman-Selberg norms of the Laplace transform of those functions are uniformly bounded. The rate of the stability estimate is shown to be of logarithmic order.

M. Miettinen, U. Raitums: On C¹-Regularity of Functions that Define G-Closure, 20 (2001) 203--214

we show that the functions which are used in the characterization of the G-closure or the G_q-closure of sets of matrices are continuously differentiable. These regularity results are based on the observation by Ball, Kirchheim and Kristensen ["Regularity of quasiconvex envelopes", Leipzig, Max-Planck-Institut für Mathematik, Preprint No. 72/1999] that separate convexity and upper semidifferentiability imply continuous differentiability.

A. Tiero: On Inequalities of Korn, Friedrichs, Magenes-Stampacchia-Necas and Babuska-Aziz, 20 (2001) 215--222

The equivalence between the inequalities of Korn, Friedrichs, Magenes-Stampacchia-Necas and Babuska-Aziz is derived using some elementary properties of the gradient, divergence and curl operators implied by these inequalities.

Yong Zhou, B. G. Zhang: Classification and Existence of Non-Oscillatory Solutions of Second-Order Neutral Delay Difference Equations, 20 (2001) 223--234

We give a classification of non-oscillatory solutions of a second-order neutral delay difference equation of the form

D²(x_n - c_nx_n-q) + f(n, x_g1(n), ... , x_gm(n)) = 0     (n >= n₀ in N).

Some existence results for each kind of non-oscillatory solutions are also established.

M. S. Stankovic, M. V. Vidanovic, S. B. Trickovic: Some Series over the Product of Two Trigonometric Functions and Series Involving Bessel Functions, 20 (2001) 235--246

The sum of the series

S_a = S _a(s, a, b, f(y), g(x)) = Sum_{n = 1}^∞ (s)^n-1 f((an - b)y) g((an - b)x) / (an - b)^a

involving the product of two trigonometric functions is obtained using the sum of the series

Sum_n=1^∞ (s)^(n-1) f((an - b)x) / (an - b)^a =

= [ cp / (2 G (a) f(pa/2) ] x^a-1 + Sum_i=0^∞ (-1)ⁱ [ F(a - 2i - d ) / (2i + d)! ] x^{2i + d}

whose terms involve one trigonometric function. The first series is represented as series in terms of the Riemann zeta and related functions, which has a closed form in certain cases. Some applications of these results to the summation of series containing Bessel functions are given. The obtained results also include as special cases formulas in some known books. We further show how to make use of these results to obtain closed form solutions of some boundary value problems in mathematical physics.

W. A. J. Luxemburg, M. Väth: The Existence of Non-Trivial Bounded Functionals Implies the Hahn-Banach Extension Theorem, 20 (2001) 267--279

We show that it is impossible to prove the existence of a linear (bounded or unbounded) functional on any L_infty / C₀ without an uncountable form of the axiom of choice. Moreover, we show that if on each Banach space there exists at least one non-trivial bounded linear functional, then the Hahn-Banach extension theorem must hold. We also discuss relations of non-measurable sets and the Hahn-Banach extension theorem.

G. Loaiza, J. A. Lopez Molina, M. J. Rivera: Characterization of the Maximal Ideal of Operators Associated to the Tensor Norm Defined by an Orlicz Function, 20 (2001) 281--293

Given an Orlicz function H satisfying the D₂ property at zero, one can use the Orlicz sequence space l_H to define a tensor norm g_H^c and the minimal (H^c-nuclear) and maximal (H^c-integral) operator ideals associated to g_H^c in the sense of Defant and Floret. The aim of this paper is to characterize H^c-integral operators by a factorization theorem.

F. Bagarello, A. Inoue, C. Trapani: Unbounded C*-Seminorms and *-Representations of Partial *-Algebras, 20 (2001) 295--314

The main purpose of this paper is to construct *-representations from unbounded C*-seminorms on partial *-algebras and to investigate their *-representations.

J. Wolf: Partial Regularity of Weak Solutions to Nonlinear Elliptic Systems Satisfying a Dini Condition, 20 (2001) 315--330

This paper is concerned with systems of nonlinear partial differential equations $$ - D_\alpha a_i^\alpha(x,u,\nabla u) = b_i(x,u,\nabla u) \qquad(i = 1,\ldots,N) $$ where the coefficients $a_i^\alpha$ are assumed to satisfy the condition $$ \big|a_i^\alpha(x,u,\xi) - a_i^\alpha(y,v,\xi)\big| \le \omega\big(|x - y| + |u - v|\big)(1 + |\xi|) $$ for all $\{x,u\}, \{y,v\} \in \Omega \times \R^N$ and all $\xi \in \R^{nN}$, and where $\int_0^1 {\omega(t) \over t}\,dt < +\infty$ while the functions ${\p a_i^\alpha \over \p \xi_\beta^j}$ satisfy the standard boundedness and ellipticity conditions and the function $\xi \mapsto b_i (x,u,\xi)$ may have quadratic growth. With these assumptions we prove partial H\"older continuity of bounded weak solutions $u$ to the above system provided the usual smallness condition on $\|u\|_{L^\infty(\Omega)}$ is fulfilled.

Jiaxin Hu: Nonlinear Diffusion Equations on Bounded Fractal Domains, 20 (2001) 331--345

We investigate nonlinear diffusion equations du / dt = Du + f (u)  with initial data and zero boundary conditions on bounded fractal domains. We show that these equations possess global solutions for suitable f and small initial data by employing the iteration scheme and the maximum principle that we establish on the bounded fractals under consideration. The Sobolev-type inequality is the starting point of this work, which holds true on a large class of bounded fractal domains and gives rise to an eigenfunction expansion of the heat kernel.

M. Kirane, N.-e. Tatar: Convergence Rates for a Reaction-Diffusion System, 20 (2001) 347--357

A class of reaction-diffusion systems is investigated. This class is motivated by some diffusive epidemic models, which serve to modelise the spread of Feline Immunodeficiency Virus (FIV) in the cat population, and sexually transmitted diseases. We obtain exponential convergence rates for a system with unbounded time dependent coefficients.

V. Pata: Hyperbolic Limit of Parabolic Semilinear Heat Equations with Fading Memory, 20 (2001) 359--377

This paper is devoted to the comparison of two models describing heat conduction with memory, arising in the frameworks of Coleman-Gurtin and Gurtin-Pipkin. In particular, the second model entails an equation of hyperbolic type, where the dissipation is carried out by the memory term solely, and can be viewed as the limit of the first model as the coefficient W of the laplacian of the temperature tends to zero. Results concerning the asymptotic behavior, with emphasis on the existence of a uniform attractor, are provided, uniformly in W. The attractor of the hyperbolic model is shown to be upper semicontinuous with respect to the family of attractors of the parabolic models, as W tends to zero.

L. Heinrich: On the Asymptotic Behaviour of the Integral  int₀^infty e^itx ( (1 / x^a ) - ( 1 / (x^a + 1) ) ) dx    (t --> 0)  and Rates of Convergence to a-Stable Limit Laws, 20 (2001) 379--394

Let X₁, X₂, ... be  independent identically distributed, positive, integer-valued random variables which take the value n with probablity 1/(n(n+1)). It is easy to check that the distribution of the power  X₁^1/a for 0 < a < 2 belongs to the normal domain of attraction of an a-stable law. It turns out that the rates of convergence of the power sums X₁^1/a + ... + X_n^1/a to this stable law are determined by the asymptotic behaviour of the Fourier transform of the function x^-a (1+[x^a])^-1 near the origin.  This problem is closely related to estimates of remainder terms in Tauberian theorems and exponential sums in analytic number as well as lattice point theory.

P. Kogut, G. Leugering: On S-Homogenization of an Optimal Control Problem with Control and State Constraints, 20 (2001) 395--429

We study the limiting behavior of an optimal control problem for a linear elliptic equation subject to control and state constraints. Each constituent of the mathematical description of such an optimal control problem may depend on a small parameter e. We study the limit of this problem when e approaches 0 in the framework of variational S-convergence which generalizes the concept of G -convergence. We also introduce the notion of G*-convergence generalizing the concept of G-convergence to operators with constraints. We show convergence of the sequence of optimal control problems and identify its limit. We then apply the theory to an elliptic problem on a perforated domain.

M. Petzoldt: Regularity Results for Laplace Interface Problems in Two Dimensions, 20 (2001) 431--455

We investigate the regularity of solutions of interface problems for the Laplacian in two dimensions. Our objective are regularity results which are independent of global bounds of the data (the diffusion). Therefore we use a restriction on the data, the quasi-monotonicity condition, which we show to be sufficient and necessary to provide H(1 + (1/4)) -regularity. In the proof we use estimates of eigenvalues of a related Sturm-Liouville eigenvalue problem. Additionally we state regularity results depending on the data.

L. Berg, M. Krüppel: Eigenfunctions of Two-Scale Difference Equations and Appell Polynomials, 20 (2001) 457--488

Both classical and distributional solutions of two-scale difference equations are interpreted as eigenfunctions, which are closely connected with Appell polynomials. Different generating functions are analyzed and the relations between them. Equivalent eigenfunctions as well as equivalent and minimal characteristic polynomials are defined and investigated in detail via the rational solution of a basic functional equation. Finally, reversed eigenfunctions are introduced and characterized.

L. Berezansky, E. Braverman: On Oscillation of Equations with Distributed Delay, 20 (2001) 489--504

For the scalar delay differential equation with a distributed delay dx/dt + integral_{-infty}^t x(s) ds R(t, s) = f(t), where t > t₀, a connection between the properties "non-oscillation" "positiveness of the fundamental function" and "existence of a non-negative solution for a certain nonlinear integral inequality" is established. This enables to obtain comparison theorems and explicit non-oscillation and oscillation conditions being generalizations of some known results for delay equations and integro-differential equations and leads to oscillation results for equations with infinite number of delays.

A. S. Mshimba: The Generalized Riemann Problem of Linear Conjugation for Polyanalytic Functions of Order n in W_{n, p}(D), 20 (2001) 505--512

We consider a homogeneous polyanalytic differential equation of order n in a simply-connected domain D with a smooth boundary dD in the complex plane C. We pose and then prove solvability of a generalized Riemann problem of linear conjugation to the differential equation. This is done by reducing the problem into n classical Riemann problems of linear conjugation for holomorphic functions, the solution of which is available in the literature.

A. S. Mshimba: The Generalized Riemann Problem of Linear Conjugation for Non-Homogeneous Polyanalytic Equations of Order n in W_{n, p}(D), 20 (2001) 513--524

We consider a non-homogeneous polyanalytic partial differential equation of order n in a simply-connected domain D with smooth boundary dD in the complex plane C. Initially we transform the given equation into an equivalent system of integro-differential equations and then find the general solution of the former in W_{n, p}(D). Next we pose and prove the solvability of a generalized Riemann problem of linear conjugation to the differential equation. This is effected by first reducing the Riemann problem to a corresponding one for a polyanalytic function. The latter is solved by first transforming it into n classical Riemann problems of linear conjugation for n holomorphic functions expressed in terms of the analytic functions which define the polyanalytic function. The solution of the classical Riemann problem is available in the literature.

L. Klotz: An Interpolation Problem for Hilbert-Schmidt Operator-Valued Stationary Processes, 20 (2001) 525--535

The paper contains a solution of the following interpolation problem for Hilbert-Schmidt operator-valued stationary processes on the real line: Assume that the values of the process on the integers are known. Determine the best linear approximation of an unknown value on the basis of the known values and compute the approximation error. Our results generalize previous results of Yaglom and Salehi for univariate and q-variate processes, respectively.

B. Sciunzi: On the Three "Essential" Critical Values Theorem, 20 (2001) 553--563

Global methods of the calculus of variations and the infinite dimensional critical point theory of Ljusternik and Schnirelmann are applied to give results on the existence of so-called critical values and essential critical values. The case of continuous, not necessarily differentiable functionals is considered and studied introducing a suitable variant of the Palais-Smale condition.

J. Appell, E. De Pascale, A. Vignoli: A Semilinear Furi-Martelli-Vignoli Spectrum, 20 (2001) 565--577

We extend a spectrum which was introduced by M. Furi, M. Martelli and A. Vignoli ["Contributions to the Spectral Theory for Nonlinear Operators in Banach Spaces", Ann. Mat. Pura Appl. 118 (1978) 229-294] for continuous nonlinear maps F to a certain new spectrum for a "semilinear pair" (L, F), with L being a linear Fredholm operator of index zero, and F being nonlinear and continuous.

K. Ahmad, R. Kumar, L. Debnath: On Fourier Transforms of Wavelet Packets, 20 (2001) 579--588

This paper deals with generalizations of some results on Fourier transforms of wavelet packets. This is followed by a result on the quadratic mirror filter based on the Fourier transform of wavelet packets.

M. Bildhauer: A Note on Degenerate Variational Problems with Linear Growth, 20 (2001) 589--598

Given a class of strictly convex and smooth integrands f with linear growth, we consider the minimization problem
Integral_W f(nabla u) dx -->  min
and the dual problem with maximizer s. Although degenerate problems are studied, the validity of the classical duality relation is proved in the following sense: there exists a generalized minimizer u* in BV(W; R^N) of the original problem such that s(x) = nabla f (nabla^a u*) holds almost everywhere, where nabla^a u* denotes the absolutely continuous part of nabla u* with respect to the Lebesgue measure. In particular, this relation is also true in regions of degeneracy, i.e. at points x such that D² f (nabla^a u*(x)) = 0. As an application, we can improve the known regularity results for the dual solution.

S. Bernard: Entropy Solution for a Hyperbolic Equation, 20 (2001) 599--615

Nonlinear hyperbolic systems of conservation laws play a central role in Science and Engineering, and their mathematical theory as well as their numerical approximation have made recent significative progress. This paper deals with the existence and uniqueness of an entropy solution of the Cauchy problem for the quasi-linear equation u_t + a(f(u))_x = 0 in one space dimension, where a is a non-smooth coefficient.

S. Marchi: C^{1, a} Local Regularity for the Solutions of the p-Laplacian on the Heisenberg Group for 2 ≤ p < 1 + Squareroot(5), 20 (2001) 617--636

We prove local Hölder continuity of the homogeneous gradient for weak solutions u in W^1,p_loc of the p-Laplacian on the Heisenberg group Hⁿ for 2 ≤ p < 1 + Squareroot(5).

I. Kmit, G. Hörmann: Semilinear Hyperbolic Systems with Singular Non-Local Boundary Conditions: Reflection of Singularities and Delta Waves, 20 (2001) 637--659

We study initial-boundary value problems for first-order semilinear hyperbolic systems where the boundary conditions are non-local. We focus on situations involving strong singularities, of the Dirac delta type, in the initial data as well as in the boundary conditions. In such cases we prove an existence and uniqueness result in an algebra of generalized functions. Furthermore, we investigate the existence and structure of delta waves, i.e., distributional limits of solutions to the regularized systems. Due to the additional singularities in the boundary data the search for delta waves requires a delicate splitting of the solution into a linearly evolving singular part and a regular part satisfying a nonlinear equation. A new feature in the splitting procedure used here, compared to delta waves in pure initial value problems, is the dependence of the singular part also on part of the regular part due to singularities enetering from the boundary. Finally, we include simple examples where the existence of delta waves breaks down.

S. Heinze: Wave Solutions to Reaction-Diffusion Systems in Perforated Domains, 20 (2001) 661--676

Traveling waves in periodically perforated domains are shown to exist for large classes of reaction-diffusion systems, provided the homogenized equation admits a non-degenerate traveling wave. This can be applied e.g. to a single equation with bistable non-linearity and to bistable monotone systems. The proof uses the implicit function theorem of a suitably transformed problem in the space H¹. Furthermore, corrector-type estimates are given for the wave profile and the wave velocity.

M. Winkler: On the Cauchy Problem for a Degenerate Parabolic Equation, 20 (2001) 677--690

Existence and uniqueness of global positive solutions to the degenerate parabolic problem
u_t = f (u) D u in Rⁿ ´ (0, infinity), u|_t=0 = u₀
with f taken from the intersection of C⁰([0, infinity)) and C¹((0, infinity)) satisfying f(0) = 0 and f(s) > 0 for s > 0 are investigated. It is proved that, without any further conditions on f, decay of u₀ in space implies uniform zero convergence of u(t) as t approaches infinity. Furthermore, for a certain class of functions f explicit decay rates are established.

S. Ding: Parametric Weighted Integral Inequalities for A-Harmonic Tensors, 20 (2001) 691--708

We prove the A_r(W)-weighted Hardy-Littlewood inequality, the A_r(W)-weighted weak reverse Hölder inequality and the A_r(W)-weighted Caccioppoli-type estimate for A-harmonic tensors all being generalizations of classical results.

J. Andres, L. Malaguti, V. Taddei: Floquet Boundary Value Problems for Differential Inclusions: a Bound Sets Approach, 20 (2001) 709--725

A technique is developed for the solvability of the Floquet boundary value problem associated to a differential inclusion. It is based on the usage of a not necessarily C¹-class of Liapunov-like bounding functions. Certain viability arguments are applied for this aim. Some illustrating examples are supplied.

R. P. Agarwal, D. O'Regan, V. Lakshmikantham: Quadratic Forms and Nonlinear Non-Resonant Singular Second Order Boundary Value Problems of Limit Circle Type, 20 (2001) 727--737

New existence results are presented for non-resonant second order singular boundary value problems
(1 / p(t)) (p(t) y'(t))' + t(t) y(t) = l f (t, y(t))   a. e. on [0,1]
lim_{t --> 0+} p(t) y'(t) = y(1) = 0
where one of the endpoints is regular and the other may be singular or of limit circle type.

A. Ja. Lepin, F. Zh. Sadyrbaev: The Upper and Lower Functions Method for Second Order Systems, 20 (2001) 739--753

Two-point boundary value problems for $m$-dimensional second order systems are considered. The method of upper and lower functions is applied to problems of the Dirichlet type and problems with nonlinear boundary conditions. The conditions on upper and lower functions are substantially relaxed comparing with the classical C²-class and properties of them are studied for systems with monotone in x right sides. Consequences for even order differential equations with mixed monotonicities are given.

D. D. Trong: Crack Detection in Plane Semilinear Elasticity, 20 (2001) 755--760

Let W be a two-dimensional semilinear elastic body limited by a known outer boundary G represented by a Jordan curve and an unknown inner boundary g represented by a finite disjoint union of piecewise C¹ Jordan curves. Plane stress is considered. We assume that the Lame coefficient l depends on the spatial variables x, y and the displacements u, v. Our main result asserts that g is uniquely determined by the displacements and stresses prescribed on an open portion G₀ of G.

M. E. H. Ismail: Orthogonality and Completeness of q-Fourier Type Systems, 20 (2001) 761--775

We establish orthogonality and completeness of the system of q-exponential functions {E_q(. ; iw_n)} using orthogonality and dual orthogonality of a q-analogue of Lommel polynomials. We also set up a very general procedure by which one can produce similar orthogonal systems using bilinear generating functions formed by products of two complete orthogonal function systems.

B. Damyanov: Some Distributional Products of Mikusinski Type in the Colombeau Algebra G(R^m), 20 (2001) 777--785

Particular products of Schwartz distributions on the Euclidean space R^m are derived when the latter have coinciding point singularities and the products are 'balanced' so that their sum to give an ordinary distribution. These products follow the pattern of a known distributional product published by J. Mikusinski ["On the Square of the Dirac Delta-Distribution", Bull. Acad. Polon. Ser. Sci. Math. Astron. Phys. 43 (1966) 511--513]. The results are obtained in the Colombeau algebra G(R^m) of generalized functions. G(R^m) is a relevant algebraic construction, with the distribution space linearly embedded, which by the notion of 'association' allows the results to be evaluated on the level of distributions.

O. Christensen: Linear Combinations of Frames and Frame Packets, 20 (2001) 805--815

We find coefficients c_mn (m, n in Z) such that for an arbitrary frame {f_n}_{n in Z} the set of vectors {f _m}_{m in Z} = {c_m1 f₁ + c_m2 f₂ + ...}_{m in Z} will again be a frame. Appropriate coefficients can always be chosen as function values c_mn = g(n/b - ma), where g belongs to a broad class of functions generating a Gabor frame {E_bm T_an g}_{m,n in Z} for L²(R). We also prove a version of the splitting trick, which allows to construct a large family of frames based on a single (wavelet or Gabor) frame.

P. Krejci, A. Vladimirov: Lipschitz Continuity of Polyhedral Skorokhod Maps, 20 (2001) 817--844

We show that a special stability condition of the associated system of oblique projections (the so-called l-paracontractivity) guarantees that the corresponding polyhedral Skorokhod problem in a Hilbert space X is solvable in the space of absolutely continuous functions with values in X. If moreover the oblique projections are transversal, the solution exists and is unique for each continuous input and the Skorokhod map is Lipschitz continuous in both spaces C([0,T]; X) and W^1,1(0,T;X). An explicit upper bound for the Lipschitz constant is derived too.

M. Girardi, L. Mastroeni, M. Matzeu: Existence and Regularity Results for Non-Negative Solutions of some Semilinear Elliptic Variational Inequalities via Mountain Pass Techniques, 20 (2001) 845--857

The main result stated in the present paper is the existence of a non-negative solution for a semilinear variational inequality through the use of some estimates for the Mountain-Pass critical points obtained for the penalized equations associated with the variational inequality. The positivity of the solution is achieved through a regularity result and the strong maximum principle.

C. Kuttler: Free Boundary Problem for a One-Dimensional Transport Equation, 20 (2001) 859--881

For a linear transport equation in one space dimension with speeds in a compact interval and a general symmetric kernel for the change of velocity a problem with free boundary (Stefan problem) is stated. The case of constant speed corresponds to a Stefan problem for the damped wave equation (telegraph equation). Existence and uniqueness of the free boundary is shown, and the connection to the classical Stefan problem (parabolic limit) is exhibited.

S. Gatti: Automatic Control of the Temperature in Phase Change Problems with Memory, 20 (2001) 883--914

We study a parabolic two-phase system with memory occupying a bounded and smooth domain.The heat exchange at part of the boundary is controlled by a thermostat. Assuming on the phase variable either a relaxation dynamics or a Stefan condition, we prove existence and uniqueness results for feedback control problems corresponding to two different types of thermostat: the relay switch and the Preisach operator. These results are strictly related to the continuous dependence of the solution on the boundary datum, which is investigated in advance.

M. Grillot, P. Grillot: Asymptotical Behavior of Solutions of Nonlinear Elliptic Equations in R^N, 20 (2001) 915--928

We study the behavior near infinity of non-negative solutions u from C²(R^N) of the semi-linear elliptic equation
-D u + u^q - u^p = 0
where 0 < q < 1,  p > q and N >= 2. Especially, for a non-negative radial solution of this equation we prove the following alternative : either u has a compact support or u tends to one at infinity. Moreover, we prove that if a general solution is sufficiently small in some sense, then it is compactly supported. To prove this result we use some inequalities between the solution and its spherical average at a shift point and consider a differential inequality. Finally, we prove the existence of non-trivial solutions which converge to one at infinity. 

Y. Amirat, O. Bodart: Boundary Layer Correctors for the Solution of Laplace Equation in a Domain with Oscillating Boundary, 20 (2001) 929--940

We study the asymptotic behaviour of the solution of Laplace equation in a domain with very rapidly oscillating boundary. The motivation comes from the study of a longitudinal flow in an infinite horizontal domain bounded at the bottom by a plane wall and at the top by a rugose wall. The rugose wall is a plane covered with periodic asperities which size depends on a small parameter e > 0. The assumption of sharp asperities is made, that is the height of the asperities does not vanish as e approaches 0. We prove that, up to an exponentially decreasing error, the solution of Laplace equation can be approximated, outside a layer of width 2e, by a non-oscillating explicit function.

T. A. Mel'nyk: Hausdorff Convergence and Asymptotic Estimates of the Spectrum of a Perturbed Operator, 20 (2001) 941--957

A family of self-adjoint compact operators A_e  (e > 0)  acting in Hilbert spaces H_e is considered. The asymptotic behaviour as e approaches 0 of eigenvalues and eigenvectors of the operators A_e is studied; the limiting operator A₀: H₀ --> H₀ is non-compact. Asymptotic estimates of the differences between eigenvalues of A_e and points of the spectrum s(A₀) (both of the discrete spectrum and the essential one) are obtained. Asymptotic estimates for eigenvectors of A_e are also proved.

M. Bildhauer, M. Fuchs, G. Mingione: A Priori Gradient Bounds and Local C^{1, a}-Estimates for (Double) Obstacle Problems under Non-Standard Growth Conditions, 20 (2001) 959--985

We prove local gradient bounds and interior Hölder estimates for the first derivatives of functions u from W¹_{1, loc} (W) which locally minimize the variational integral I(u) = Integral over W of f (nabla u) dx subject to the side condition y₁ <= u <= y₂. We establish these results for various classes of integrands f with non-standard growth. For example, in the case of smooth f the (s, m, q)-condition is sufficient. A second class consists of all convex functions f with (p, q)-growth.

G. Grün: On Bernis' Interpolation Inequalities in Multiple Space Dimensions, 20 (2001) 987--998

We prove a multi-dimensional substitute for the well-known Bernis' interpolation inequalities. Applied to fourth order degenerate parabolic equations like the thin film equation, these new estimates are a key ingredient to control third order derivatives or to prove certain qualitative results like finite speed of propagation or occurrence of a waiting time phenomenon.

M. Z. Garaev, Ka-Lam Kueh: L₁-Norms of Exponential Sums and the Corresponding Additive Problem, 20 (2001) 999--1006

A new estimate of L₁-norm of certain exponential sum is obtained. At the same time, we establish a sharp lower bound for the cardinality of corresponding sumsets. In some cases this lower bound gives the true order of the cardinality.

R. S. Kraußhar: On a New Type of Eisenstein Series in Clifford Analysis, 20 (2001) 1007--1029

We deduce a recursion formula for the partial derivatives of the fundamental solution of the generalized Cauchy-Riemann operator in R^k+1 in terms of permutational products. These functions generalize the classical negative power functions to Clifford analysis. We exploit them to introduce a new generalization of the classical complex analytic Eisenstein series on the half-plane to higher dimensions satisfying the generalized Cauchy-Riemann differential equation. Under function-theoretical and number-theoretical aspects we investigate their Fourier series expansion in which multiple divisor sums and certain generalizations of the Riemann zeta function play a crucial role.

D. Rachinskii: Iteration Procedures of Shuttle Iteration Type in Continuous Non-Monotone Problems, 20 (2001) 1031--1054

We suggest and study iteration procedures converging from below and above to robust stable solutions and to robust stable continuous branches of solutions for quasilinear boundary-value problems with continuous non-monotone non-linearities. The iterations are constructed by modifications of the shuttle iteration method, which is used in problems with monotone operators leaving invariant a cone interval.

G. Herzog: Differential-Functional Inequalities for Bounded Vector-Valued Functions, 20 (2001) 1055--1063

For the space Rⁿ ordered by a cone and some functions f : R^n+mn --> Rⁿ and h₁, ... , h_m: R --> R we consider differential-functional inequalities of the type
v''+ c v'+ f (v, v (h₁), ... , v (h_m)) <= u'' + c u' + f (u, u (h₁), ... , u(h_m))
and conclude u <=  v under suitable conditions on u, v, h_k and f. The result can be applied to obtain existence and uniqueness results for differential-functional boundary value problems of the form
u'' + c u' + f (u, u( h₁), ... , u( h_m)) = q
with u from C²(R, Rⁿ) bounded.

Yong Zhou, Y. Q. Huang: Existence of Non-Oscillatory Solutions of Second-Order Neutral Delay Difference Equations, 20 (2001) 1065--1074

We consider the second-order neutral delay difference equation with positive and negative coefficients
D (r_n D (x_n + cx_n-k)) + p_n+1 x_n+1-m - q_n+1 x_n+1-j = 0
where c is a real, k >= 1 and m, j >= 0 are integers, {r_n} _n=n0^infinity, {p_n} _n=n0^infinity, {q_n} _n=n0^infinity, are sequences of non-negative real numbers. We obtain global results (with respect to c) which are some sufficient conditions for the existences of non-oscillatory solutions.

D. Bugajewska: On Topological Structure of Solution Sets for Delay and Functional-Differential Equations, 20 (2001) 1075--1080

We characterize the topological structure of global solution sets for classical delay and functional-differential equations in terms of R_d sets.