bluuebunny/crossref_metadata_2025 huggingface dataset:
A note on the penalty correction method
Authors: Hackbusch Wolfgang
Published: December 2016

The penalty correction method of J. Pasciak reduces the solution of an inhomogeneous Dirichlet boundary value problem to a sequence of finite element problems with natural boundary conditions. We show that the penalty correction method is a special case of the iterated defect correction. Error estimates are proved under weaker assumptions.
Mean Value Theorem for Functions Possessing First Order Convex Approximations. Applications in Optimization Theory
Authors: Studniarski Marcin
Published: December 2016

A mean value theorem for functions possessing first order convex approximations in the sense of Ioffe [6] is derived. It comprises two known results for convex and locally Lipschitzian functions as particular cases. This theorem is used in order to obtain a sufficient condition for a function defined on the Cartesian product of two topological vector spaces to possess a first order convex approximation. Some applications in optimization theory are also given.
A Note on Optimal Domains in a Reaction-Diffusion Problem
Authors: Bandle Catherine
Published: December 2016

A simple model for a chemical reaction is considered. The vessel where the reaction takes place is optimal if no dead core appears and if effectiveness is high. By means of the method of upper and lower solutions it is shown that, independent of the volume, sufficiently thin domains have this property. The proof is based on an estimate for the solution of the Dirichlet problem.
Das Poincarésche Randsteuerproblem bei elliptischen Differentialgleichungen zweiter Ordnung
Authors: Oestreich Dieter
Published: December 2016

There are considered the Poincaré boundary control problem for elliptic differential equations of second order and a mixed Riemann-Hilbert control problem with generalized analytic functions, where the control occurs non-linearly in the coefficients of the boundary condition or the differential equation too. Existence theorems are proved for these problems with integral cost functions, constraints and additional conditions.
Automatische Steuerung freier Ränder bei Stefan-Problemen durch Thermostatkontrollen am festen Rand
Authors: Hoffmann Karl-Heinz , Kornstaedt H.-J. , Sprekels Jürgen
Published: December 2016

The automatic control of the free boundary in one-phase Stefan problems via the input data at the fixed boundary is considered. In contrast to previous papers concerned with this subject the thermostat controlling the input data observes the temparature at the fixed boundary instead of the evolution of the free boundary. The corresponding parabolic initial-boundary value system is nonlinear and discontinuous. Unique solvability is shown and numerical results are presented.
Ritz-Regularization versus Least-Square-Regularization. Solution Methods for Integral Equations of the First Kind
Authors: Schock E.
Published: December 2016

Three principles of regularization methods are discussed (Tikhonov-, Ritz- and Galerkin-method), and it is shown that in the case of positive definite or of selfadjoint operators the Ritz- and the Galerkin-principles (the latter with non-real parameter) have better properties than the Tikhonov-principle.
On the Solvability of Transonic Potential Flow Problems
Authors: Feistauer Miloslav , Nečas Jindřich
Published: December 2016

The paper is devoted to the study of the solvability of transonic potential flow problems. The velocity potential equation governing irrotational non-viscous transonic flows is nonlinear, second order and of mixed type. There exist a series of numerical methods for the solution of the transonic flows. However, the existence of the solution has not yet been proved.

Here, with the use of the secant-modulus method and a convenient optimal control principle, we construct a functional

\psi

, whose minimization is equivalent to the solution of the problem. Since the so-called shocks, represented by jumps in the velocity, density and pressure, occur in the flow field, we consider weak solutions from the space

W^{1,2} (\Omega)

. From the physical point of view the entropy condition across the shock is very important. There exist various approaches how to embody this condition into the numerical method. Here we consider its simplified version by Glowinski, Pironneau etc. and besides, we propose its new natural, more complex formulation.

We show that these conditions introduce the missing compactness into the problem and allow to prove the existence of the solution in the following sense: If the minimizing sequence of the functional

\psi

satisfies (a posteriori) the entropy and bounded velocity conditions and converges weakly to a function

u

, then it converges strongly to

u

and

u

is a solution of the transonic flow problem.

The paper contains also some results concerning subsonic flows and the regularity of the minimizing sequence.
The Poisson Formula for Euclidean Space Groups and some of its Applications II. The Jacobi Transformation for Flat Manifolds
Authors: Günther Paul
Published: December 2016

Using the Poisson formula of part I of this paper we express the traces of the heat kernel and the wave equation kernel of a compact flat Riemannian manifold by data which are connected with the behaviour of its closed geodesics.
On Capillary Free Surfaces without Gravity
Authors: Miersemann Erich
Published: December 2016

The non-parametric capillary problem in the absence of gravity can be replaced by a Dirichlet problem for a quasilmnear equation of mixed type by introducing a stream function. We prove a maximum principle for an expression depending on the gradient of the stream function and, furthermore, regularity properties of

C^1(\bar {\Omega}

-solutions in domains

\Omega

with corners.
Über ebene Bimetallprobleme mit verschiedenen Kontaktbedingungen und Dirichletschen Randbedingungen
Authors: Jentsch Lothar
Published: December 2016

Existence and uniqueness theorems for boundary contact problems of plane elastostatics are proved. The special quality is that the line between two elastical homogeneous parts reaches to the external boundary

S

of the whole domain. Three different contact conditions and Dirichlet boundary conditions (displacement vector on

S

given) are considered. The problem is reduced with the aid of potentials, which have Green’s contact tensors in the kernel, to systems of singular integral equations about

S

containing Cauchy-type integrals and integrals with fixed singularities. From the theory of Duduchava it follows that the occuring integral operators are Fredholm operators in the space

L_2(S)

and have index zero. The null spaces of these operators contain only the zero.
Singulare Störung von Randwertproblemen durch ein kleines Loch im Gebiet
Authors: Göhde Dietrich
Published: December 2016

By some standard cases it is shown that perturbing a boundary value problem by a small hole in the domain leads to singular perturbation effects. The boundary layers will be constructed, in primary approximation, by means of the fundamental solutions.
Die Cartan-Algebra der äusseren Differentialformen als Supermannigfaltigkeit: Automorphismen, Jordan-Automorphismen und ihre Realisierung als Diffeomorphismen
Authors: Uhlmann Armin
Published: December 2016

In the Cartan algebra we constructively enumerate all automorphisms, all splitting and function subalgebras, and a class of Jordan automorphisms. We show representability of the automorphisms resp. Jordan automorphisms as diffeomorphisms on a fibred space over the given manifold

M

. Its dimensionality is 15 resp. 24 if dim

M

= 3, and 32 resp. 64 if dim

M

= 4.
Remarks on quadratic optimal control problems in Hilbert spaces
Authors: Benker Hans , Kossert Steffen
Published: December 2016

In this paper quadratic optimal control problems in Hilbert spaces are considered. For this problem properties of the optimal control and bounds are given.
On Duality and the Maximum Principle for Continuous Linear Programming Problems
Authors: Tröltzsch Fredi
Published: December 2016

In this paper, the Grinold maximum principle for continuous linear programming problems is extended to the case where the known duality theorems do not ensure the existence of an optimal solution of the dual problem in the usual reflexive

L_p

-spaces. The extended maximum principle is then applied to the investigation of a parabolic boundary control proble in with constraints on the control and the state.
A Theory of Quantum Measurement Based on the CCR Algebra L$^+(\mathcal W)$
Authors: Dubin Daniel A. , Sotelo-Campos J.
Published: December 2016

Starting from L

^+(\mathcal W)

as the algebra of observables,

\mathcal W

a space of type

\mathcal S

, a theory of quantum measurement is devised. It is based on the theory of Davies and Lewis, but adapted to unbounded symmetric operators defined on a common dense domain and no restriction on the spectra.
Approximation by Solutions of Elliptic Equations
Authors: Hamann Uwe , Wildenhain Günther
Published: December 2016

Let

\Omega \subset \mathbb R^n

be a bounded, smooth domain,

\Gamma

a closed, smooth,

(n-1)

-dimensional surface with boundary in the interior of

\Omega

and

V

an open subset of the boundary

\partial \Omega

. In

\Omega

we consider a properly elliptic differential operator

L

of order

2m

with smooth coefficients. Let

(B_1, \dots, B_m)

be normal system of boundary operators on

\partial \Omega

, which fulfils the classical root condition.

L_V(\Gamma)

denote the space of the restrictions on

\Gamma

of the functions from

L_V(\Omega) = \{u \in C^\infty \Omega \colon Lu = 0 \; \mathrm {in}\; \Omega, \; B_1u|_{\partial \Omega} = \cdots = B_mu|_{\partial \Omega} = 0 \; \mathrm {in} \; \partial \Omega \; \backslash \; V\}.

It is proved that

L_V(\Gamma)

is dense in the space

W_p^{2m-1/p} (\Gamma) (p > 1)

.
Deformations of Hyperbolic Structures without the Completeness Condition
Authors: Apanasov Boris N.
Published: December 2016

In the general case of deformations without the completeness condition we will show that the complex dimensionality of the space Def*

M^3

is equal to the nunber of isolated ends of the three-manifold

M^3

. For

n \geq 4

there exists a strong rigidity of deformations, similar to the rigidity of complete hyperbolic structures.
On the Majorization Method for Holomorphic Solutions of Linear Partial Differential Equations
Authors: Berg Lothar
Published: December 2016

The application of the majorization method to proving the existence of holomorphic solutions of initial value problems for two-dimensional partial differential equations requires the asymptotic estimation of inverse matrices. The article presents two examples concerning this subject.
Polysingular Operators and the Topology of Invertible Singular Operators
Authors: Khimshiashvili G.
Published: December 2016

The homotopy groups of the space of invertible singular integral operators with Cauchy kernel on the closed curve are computed. An index formula for a bisingular integral operator is presented.
Solution of a degenerated elliptic equation of second order in an unbounded domain
Authors: Berndt Werner
Published: December 2016

The paper deals with the equation div

(\varrho (x) (\bigtriangledown u + f(x))) = 0

in

\mathbb R^N

, where

\varrho \in L_1 (\mathbb R^N)

and

f \in L_2 (\mathbb R^N, \varrho)

are given smooth functions. The equation degenerates on the smooth surface

\Gamma = \{\varrho (x) = 0\}

where

\varrho (x)

behaves like a power of dist

(x, \Gamma)

. The following results are proved: 1. Existence and uniqueness (up to additive constants) of a solution with

\int | \bigtriangledown u|^2 \varrho \mathrm d x< \infty

; the proof uses a variational method in a weighted Sobolev space; 2. Regularity of the solution near

\Gamma

; 3. Convergence and correctness of a numerical (difference) method; 4. Convergence of an iteration method to solve the discrete problem.
On an Abstract Nonlinear Cauchy-Kowalewski Theorem – a Variant of L. Nirenberg’s and T. Nishida’s Proof
Authors: Tutschke Wolfgang
Published: December 2016

The paper deals with a variant of the proof of an abstract nonlinear Cauchy-Kowalewski theorem. The original proof as well as its modification regarded in this paper make use of the method of successive approximations where in both cases the interval of convergence is characterized by an infinite product. In the present paper the infinite product of the original proof is replaced by one with arbitrary factors. In this way in some cases the convergence is proved in a larger interval.
The Poisson formula for Euclidean space groups and some of its applications I
Authors: Günther Paul
Published: December 2016

The classical Poisson summation formula refers to a translation group with

n

linearly independent generators in the

n

-dimensional affine or euclidean space. In this paper a generalization of the Poisson formula is given which belongs to a general properly discontinuous group of affine transformations with compact fundamental domain.
The maximum principle in optimal control problems for an elliptic equation (in Russian)
Authors: Raitums Uldis
Published: December 2016

Problems of optimal control are considered under state equations in the form of quasilinear second-order elliptic differential equations of divergence type, with equations and inequalities as integral constraints. The main stress is on those cases with non-convex sets of admissible controls where all coefficients of the state equation depend on the controls. A maximum principle in integrated form, embodying necessary conditions for optimality, is derived under assumptions analogous to those under which the linearized maximum principle holds in the ease of convex sets of admissible controls.
Some Applications of Degree Theory to Bifurcation Problems
Authors: Tan Nguyen Xuan
Published: December 2016

Some results on the vanishing of a generalized degree of multivalued mappings are proved and applied to study bifurcation solutions of the equation

F(\lambda, u) = 0

in a Banach space; as a special case,

F(\lambda, u) = u—M(u)—N(\lambda, u)—H(\lambda, u)

, where the parameter

\lambda

is from a normed space. For concrete application there are considered bifurcation solutions of a nonlinear elliptic differential equation of second order.
Approximation einer parabolischen Itogleichung
Authors: Grecksch Wilfried
Published: December 2016

A stochastic equation of evolution on Hilbert spaces

V \subset H \subset V*

is approximated by an Ito equation with values in a Hilbert space.
On the existence of dense ideals in LMC*-algebras
Authors: Fritzsche Marlen
Published: December 2016

In this paper we prove the following proposition: The existence of an unbounded element in an LMC*-algebra (with unity) implies the existence of a dense ideal in this algebra.
Remarks on the dual least action principle
Authors: Willem Michel
Published: December 2016

Let

L

be a self-adjoint operator with a closed range in a Hilbert space

H

and let

\psi

be a convex function on

H

. Under a non resonance assumption the surjectivity of

L + \partial \psi

is studied.
Invarianzeigenschaften bei Variationsproblemen mit nichtdifferenzierbaren Integranden
Authors: Thämelt Wolfgang
Published: December 2016

A new proof of Rvacev’s invariance theorem for variational problems with nonsmooth but convex Lagrangians is given. This proof is valid even for the multidimensional case. The results are extended to the physically interesting case where the invariance of the Lagrangian can only be obtained with the help of some divergence terms and to the case where the Lagrangian is merely convex for some part of the variables.
Über funktionentheoretische Eigenschaften der Lösungen gewisser elliptischer Differentialgleichungen
Authors: Berglez Peter
Published: December 2016

Some function theoretic properties of the solutions of the elliptic differential equation

w_{z, \bar {z}} + a(z, \bar {z}) w_{\bar {z}} + b(z, \bar {z}) w = 0

are discussed which can be represented by differential operators of K. W. Bauer. After a discussion of general properties, formal powers are defined. With this it is possible to prove some theorems essential in the classical theory of functions for certain subclasses of the solutions of this equation which are generalizations of holomorphic functions.
Ein abstraktes nichtlineares Cauchy–Kowalewskaja-Theorem mit singulären Koeffizienten I
Authors: Reissig Michael
Published: December 2016

An abstract nonlinear Cauchy–Kowalewskaja problem with singular coefficients is regarded. We want to prove the existence of continuous solutions on the singular face

t = 0

. By applying the method of the Banach space scales we show which role the order of the singularity plays for the solutions of the abstract problem.
Eine gemischte Randwertaufgabe für stationäre Mehrschichtenströmungen
Authors: Socolowsky Jürgen , Bergmann Joachim
Published: December 2016

A mixed boundary-value problem for the multi-layer flow of Newtonian liquids is discussed, which corresponds to free boundary problems. Basing on an appropriate generalized solution, theorems of existence and uniqueness are proved. For the linear problem local regularity properties of the generalized solution can be shown.
On Solutions of First-Order Partial Differential-Functional Equations in an Unbounded Domain
Authors: Kamont Zdzisław , Prządka Katarzyna
Published: December 2016

Under the assumptions of continuity and the existence of first-order partial derivatives of the solutions it is proved that the Cauchy problem

z_x^{(i)} (x, y) = f^{(i)} (x, y, z (x, y), z, z_y^{(i)} (x, y))

z^{(i)} (x, y) = \varphi_i (x, y) for (x, y) \in [–\tau_0, 0] \times \mathbb R^n \;\; (i= 1, \dots, m)

admits at most one solution if the function

f = (f^{(i)}, \dots, f^{(m)}

of the variables

(x, y, p, z, q)

satisfies a Lipschitz condition with respect to

(p, z, q)

, or a Lipschitz condition with respect to

(p, z)

and a Hölder condition with respect to

q

.
On a Coupled System of Partial Differential Equations
Authors: Bruckner Gottfried , Kluge Reinhard
Published: December 2016

Boundary value problems are considered for a system of stationary partial differential equations that stems from the models describing the implantation of impurities into semiconductor devices and is similar to the equations of carrier transport. The existence of solutions by functional analytic means (energetic method) is proved. Besides, an upper estimate for the set of solutions is obtained.
Dualität bei Steuerungsproblemen mehrfacher Integrale
Authors: Pickenhain Sabine
Published: December 2016

A duality is presented for problems of optimal control with multiple integrals. In the centre of interest the question on strong duality stands. The mean idea is to study the bi-dual problem to the given optimal control problem. It succeeds to get relations between flow problems in, the sense of Young and Klötzler and to find a new possible approach to assertions on strong duality for control problems with multiple integrals.
An Application of B. N. Sadovskij’s Fixed Point Principle to Nonlinear Singular Equations
Authors: Appell Jürgen , De Pascale Espedito , Zabrejko Petr P.
Published: December 2016

This paper is concerned with the applicability of Sadovskij’s fixed point principle to the solvability of nonlinear singular integral equations of the form

x = \lambda SFx

, with

F

being a non-linear superposition operator, and

S

a singular linear integral operator. If the operator

F

acts in the "little" Hölder space

H^0_\alpha

, or in some space

J_{\alpha, \beta}

of functions which satisfy an integral-type Hölder condition,

F

turns out to be

k

-condensing under fairly general hypotheses such that the above equation has it solution at least for small

\lambda

. On the other hand, under these general hypotheses the classical fixed point principles of Schauder does not apply immediately, and that of Banach–Caccioppoli not at all; in fact, it is shown that the Bunach-Caccioppoli principle applies only if the above equation is linear. Moreover, in this paper it is shown that the above equation may be studied as well by means of the topological degree theory for vector fields involving condensing operators.
Zur Modellierung stochastischer Randwertprobleme partieller Differentialgleichungen
Authors: Girlich Hans-Joachim
Published: December 2016

Measuring errors, random noise, and hydrodynamic instabilities in connection with boundary value problems will be considered as examples which lead to stochastic analysis. In the main part of this paper the concept of the weakly correlated random field is applied on Dirichlet’s problem under uncertainties using simplifying dependence assumptions.
Some problems in the theory of approximately differentiable functions (in Russian)
Authors: Isakov Nikolay M.
Published: December 2016

Global and local properties of approximatively differentiable functions are studied.
Zur Lösung des linearisierten Knickstab-Problems mit beschränkter Ausbiegung
Authors: Phú Hoàng Xuân
Published: December 2016

The problem of rod capable of creasing under a load on its upper end and situated in a narrow strip-shaped channel is replaced by a linearized problem, which is then solved completely.
Über zufällige Störungen einer Basis aus Exponentialfunktionen im Raum $L_2 [-\pi, \pi]$
Authors: Kürsten Klaus-Detlef
Published: December 2016

We investigate the limit case

|s_n| = 1/4

for the theorem of Paley and Wiener and some later authors on the basis properties of the perturbed system {exp

(i(n + s_n) t): n \in \mathbb Z

} in the space

L_2[-\pi, \pi]

. We prove that the perturbed system almost sure is no Riesz basis, if {

s_n

} is taken from

[- 1/4, 1/4]^{\mathbb Z}

equipped with natural product measures.
Studies On Transonic Flow Problems by Nonlinear Variational Inequalities
Authors: Gittel H.-P.
Published: December 2016

The transonic flow problem is handled as a variational inequality in a convex set which is given by a suitable entropy condition, by a bound for the gas velocity and by boundary conditions for the velocity potential. Using Katchanov’s method and the compactness of the convex set this variational inequality is solved. Furthermore, a result on flows in the subsonic region is given.
Stability Properties of Space Periodic Standing Waves
Authors: Scarpellini B.
Published: December 2016

Equilibrium solutions of parabolic systems of the form

u = \Delta u + F(\alpha, u)

are considered, where

D

designates as

2 \times 2

diagonal matrix,

\alpha

a bifurcation parameter,

u = (u_1, u_2)

a state vector and

F

a polynomial nonlinearity. A trivial solution branch

u(\delta) \in \mathbb R^2, \delta \in I = (– \epsilon, \epsilon)

, is supposed to be given, i.e.

F(\alpha_0 + \delta, u(\delta)) = 0

for some

\alpha_0

and every

\delta \in I

. Then a period

L

is fixed and under suitable assumptions space-

L

-periodic bifurcating standing waves are constructed. It is shown that these bifurcating branches become generically unstable as

L \uparrow \infty

. Under the condition of

d_{uu}F(\alpha_0, u(0)) = 0

however, they will remain stable against

nL

-periodic perturbations

(1 < n \in \mathbb N)

, provided that the trivial solution-branch

u(\delta)

behaves alike for small

\delta < 0

. The so-called Landau-Ginzburg equations arising in Landau’s theory of phase transitions constitute a special example in physics.
Factorization of operators mapping (F)-spaces into (DF)-spaces
Authors: Junek Heinz
Published: December 2016

This paper deals with the factorization of linear operators mapping (F)-spaces into (DF)-spaces through Banach spaces and through operators of given operator ideals. Roughly speaking, we answer the question of, to what extend global properties of such operators are determined by their behaviour on the hounded subsets. The results are used to characterize the geometric structure of the neighbourhoods of zero in (F)-spaces by the geometry of their bounded sub-sets. Moreover, they allow further insights into the theory of nuclear (F)-spaces.
The Convergence of Rothe’s Method for Parabolic Differential Equations
Authors: Schumann Rainer
Published: December 2016

The convergence of Rothe’s method for quasilinear parabolic differential equations is proved under typical assumptions which guarantee existence. The investigations are based on appropriate approximation schemes for evolution tripels and on the observation that Rothe’s method satisfies the requirements of stability and consistency.
Gieichgewichtsfiguren magnetischer Flüssigkeiten
Authors: Beyer Klaus
Published: December 2016

Equilibrium configurations of magnetic liquid drops subject to an applied magnetic field and to surface tension are investigated. Existence and uniqueness of such configurations in the neighbourhood of a sphere in sufficiently weak fields are proved.
On the Elliptic Sturmian Theory for General Domains
Authors: Müller-Pfeiffer Erich
Published: December 2016

The well-known comparison theorem by Sturm and Picone for ordinary, self-adjoint, second order differential equations is extended to self-adjoint elliptic differential equations. The basic domain

G

and the coefficients of the equation are not necessarily bounded, and no regularity hypotheses on the boundary

\partial G

are required.
A Variational Principle for General Polymer Systems
Authors: Warstat Volker
Published: December 2016

A new model for classical statistical mechanics of interacting extended particles (polymers) is introduced. First, the construction of the configuration space for polymer systems is given. Then the existence of the thermodynamic limit of the local pressure is derived. For this limit which is called the pressure of the system a variational principle is proved. Finally, for given interaction, the class of equilibrium states determined by this variational principle is proved to coincide with the class of translation invariant Gibbs states.
Nodal Domains for One- and Two-Dimensional Elliptic Differential Equations
Authors: Müller-Pfeiffer Erich
Published: December 2016

Concerning the Schrödinger equation

-\Delta u + q(x) u = 0, x = (x_1, x_2) \in \mathbb R^2

, a condition for

q

is formulated to ensure that there exists a nodal domain being a circle. Similarly, conditions for the Sturm–Liouville equation are obtained for the existence of a pair of conjugate points.
Inequalities between Entropy and Approximation Numbers of Compact Maps
Authors: Edmunds R.M.
Published: December 2016

It is shown that if

T

is a compact linear map, with infinite-dimensional range, from one Hilbert space to another, then given any

N \in \mathbb N, e_n(T) ≤ 2a_{N+1}(T)

for any

n

satisfying a certain inequality. Applications of this are given.
The Galerkin approximation for quasilinear elliptic equations with rapidly (or slowly) increasing coefficients
Authors: Schumann Rainer
Published: December 2016

In this paper the convergence of Galerkin’s method is proved for quasilinear elliptic partial differential equations with rapidly (or slowly) increasing coefficients. The investigations are based on Gossez’s theory of pseudomonotone operators in complementary systems of Sobolev-Orlicz spaces.
On the Analysis of a Particular Volterra-Stieltjes Convolution Integral Equation
Authors: Hofmann Bernd
Published: December 2016

A particular Volterra–Stieltje convolution integral equation arising in the mathematical modelling of aquifers is investigated. One direct and two inverse problems are formulated and analysed with respect to the uniqueness, existence and stability of associated solutions.
Spaces of Continuous Sesquilinear Forms Associated with Unbounded Operator Algebras
Authors: Schmüdgen Konrad
Published: December 2016

Let

\mathcal A

be a closed *-algebra of unbounded operators on a dense invariant domain

\mathcal D

of a Hilbert space, and let

\mathcal {L_A} (\mathcal {D, D'})

be the vector space of all continuous sequilinear forms on

\mathcal D \times \mathcal D

relative to the graph topology of

\mathcal A

. We generalize some basic results of the von Neumann algebra theory (von Neumann bicommutant theorem, Kaplansky density theorem) to certain linear subspaces of

\mathcal {L_A} (\mathcal {D, D'})

.
Products of Distributions: Nonstandard Methods
Authors: Oberguggenberger Michael
Published: December 2016

Nonstandard tools are developed which are suitable for studying products of distributions defined by regularization and passage to the limit. We obtain nonstandard criteria for the existence of the products (which are demonstrated to be useful for calculating standard examples) as well as new standard results clarifying the relationship between different types of such products. As an offspring we are able to construct algebras of distributions – as quotients of external spaces of smooth functions – which have properties similar to the Colombeau algebras.
Steuerungstheoretische Behandlung einer Schachtelaufgabe
Authors: Klötzler Rolf
Published: December 2016

An optimization problem is studied that was already considered by A. Fricke. It consists in finding among all pairs of congruent axialsymmetric plane surface patches with given length of the major axis that are bent and fastened to cylindric surfaces that which includes maximal volume. By a methodically new approach, this problem ist solved completely from the analytical and numerical point of view, including hitherto unsettled questions (existence, unity and convergence of approximate methods).
Die Methode der Grenzschichtverbesserung für eine Masse entarteter gewöhnlicher Differentialgleichungen
Authors: Meyer Sybille
Published: December 2016

In this paper we generalize Visik’s and Ljusternik’s method for a symptotic solution of boundary value problems with a small parameter in the highest derivatives of the differential equation for a class of boundary value problems for non-self-adjoint ordinary differential equations of the order

2m

, in which the coefficients have a zero of integer order in a boundary point. A part of the arising boundary layer equations having singularities. In Chapter 3 of this paper we calculate the index of the differential operators of this boundary layer equations in a certain pair of spaces of the positive half-axis. In Chapter 4 we construct the first term of an asymptotic solition for small parameter values.
How to Measure Smoothness of Distributions on Riemannian Symmetric Manifolds and Lie Groups?
Authors: Triebel Hans
Published: December 2016

We discuss several types of means on Riemannian (globally symmetric) manifolds, hyperbolic manifolds and Lie groups, their mutual interrelations and related function spaces.
Inverse Probleme für die Wärmeleitungsgleichung
Authors: Handrock-Meyer Sybille
Published: December 2016

A summary is given of the results obtained in the past years for the unique determination of the heat conductivity in the heat equation where the cases of a constant, a space dependent, a time dependent heat conductivity and the quasilinear case, i.e. the heat conductivity is a function of the temperature are considered. Moreover a simple proof is given for the validity of the maximum principle in the quasilinear case.
Phasenrekonstruktion für Stapel homogener dielektrischer Schichten gleicher optischer Dicke - eine Ambiguitätsstudie
Authors: Schachtzabel H.
Published: December 2016

A detailed analysis of the possible ambiguities in recognizing of homogeneous multilayer systems is given. Also the physical realizibility of all theoretical possible solutions of the inverse problem is shown.
Regularized Parameter Identification in Elliptic Boundary Value Problems
Authors: Friedrich Volkmar , Tautenhahn Ulrich
Published: December 2016

In this paper the regularization method applied to the numerical identification of the diffusion coefficient in second order elliptic equations is investigated. For solving the corresponding regularized discrete and continuous nonlinear minimization problems the Gauss-Newton method is analyzed. We describe an effective way for performing one iteration step which requires to solve one problem of 2nd order and one problem of 4th order.
Behandlung Fredholmscher Integralgleichungen mit schwachen Singularitäten und Unstetigkeiten erster Art in den Kernfunktionen
Authors: Wisotzki Christel
Published: December 2016

Twice interpolatory type quadrature formulas with adapted product integration have been used for solving Fredholm integral equations of the second kind, where the kernel function has discontinuities of the first kind along any continuous curve and weak singularities. The series of approximation operators developed by means of such quadrature formulas with the help of two recommended adaption principles are collectively compact, i.e. they give a convergent approximation method.
Comparison Theorems for Nonself-Adjoint Differential Equations of Second Order
Authors: Müller-Pfeiffer Erich
Published: December 2016

Concerning the zeros of the derivatives of solutions for nonself-adjoint second order differential equations Sturm–Picone type comparison theorems are proved.
On the Number of Stable Local Minima of Some Functionals
Authors: Benkert Frank
Published: December 2016

Using the theories of locally monotone operators and of proper Fredholm operators, results on the number of stable local minima of some functionals are obtained and related examples are considered.
Ein Gitterpunktproblem in der hyperbolischen Ebene
Authors: Martin Tobias
Published: December 2016

A lattice point problem in hyperbolic plane on which acts a Fuchsian group of the first kind is considered. In this connection the asymptotic behaviour at

t \to \infty

of the function

A (t, z, z’)

giving the number of

z’

-equivalent points in the circle around

z

with radius

t

is investigated. Besides the main terms there is given an estimation of the remaining part improving previous results. As a most important instrument there serves the mean value method based on a functional equation between

A (t, z, z’)

and the iterated lattice point function.
The Spectrum and Quadrature Formulas of Spherical Space Forms
Authors: Prüfer Friedbert
Published: December 2016

The spectrum of compact Riemannian manifolds of constant sectional curvature 1 with cyclic fundamental group is computed. From this corollaries on the first eigenvalue of 3-dimensional spherical space forms and quadrature formulas are obtained.
Asymptotics of the Green function and Poisson kernels of a mixed parabolic problem in a cone. I.
Authors: Kozlov Vladimir
Published: December 2016

General boundary value problems for

2b

-parabolic equations in a multidimensional cone are considered. Theorems on solvability and smoothness of solutions are proved. Infinite asymptotic expansions of solutions are obtained in a neighbourhood of the conic vertex. The representations for the coefficients of these expansions are also obtained. At last formal asymptotics of the Green function and the Poisson kernels are obtained.
Continuous Dependence of the Solution of a System of Differential Equations with Impulses on the Initial Condition
Authors: Dishliev A. B. , Bainov D. D.
Published: December 2016

Initial value problems for systems of differential equations with impulses are considered. The impulses are realized in the moments when the integral curves meet some of previously fixed hypersurfaces in the extended phase space. Sufficient conditions for the continuous dependence of the solutions of these systems on the initial conditions are found.
Quadrature and Collocation Methods for Singular Integral Equations on Curves with Corners
Authors: Prössdorf Siegfried , Rathsfeld A.
Published: December 2016

This paper is concerned with approximation methods for singular integral equations on curves with corners. Necessary and sufficient conditions for the stability of the piecewise constant

\epsilon

-collocation and of certain quadrature methods are given.
On the Noethericity of linear singular integral equations in weighted Lebesgue spaces
Authors: Paatashvili V.A.
Published: December 2016

Under general assumptions with respect to the integration contour

\Gamma

and a weight function

\omega

, the question of noetherianness of a linear singular integral equation with bounded measurable coefficients in a weighted Lebesgue space

L_p(\Gamma ; \omega)

is reduced to the investigation of noetherianness of a similar equation in a space without weight

L_p(\Gamma)

.
On Spectral Properties of Elliptic Pseudo-Differential Operators Far from Self-Adjoint Ones
Authors: Agranovich M.S. , Markus A.S.
Published: December 2016

We establish a rough asymptotics for eigenvalues of elliptic operators under some assumptions; one of them is that the values of the principal symbol do not cover the whole complex plane. We consider also a collection of some examples of non-self-adjoint elliptic operators: in particular, with unusual formulas for asymptotics of eigenvalues; with a regular asymptotics of eigenvalues but without the completeness of root functions; with a complete system of eigenfunctions which is not a basis.
The Explicit Solution of Elastodynamical Diffraction Problems by Symbol Factorization
Authors: Meister E. , Speck Frank-Olme
Published: December 2016

A class of linear boundary value problems is considered due to time-harmonic elastic wave propagation in the exterior domain of a half-plane in

\mathbb R^3

. The explicit solution is obtained from a Wiener-Hopf factorization of specific types of non-rational 3 x 3 matrix functions, which occur as Fourier symbols in the corresponding boundary integral equations.
Solvability of a noncoercive initial-boundary value problem for the Stokes system in Hölder classes of functions (the half-space case) (in Russian)
Authors: Mogilevskii Ilya S. , Solonnikov Vsevolod A.
Published: December 2016

The initial-boundary value problem for the Stokes system with the surface tension in boundary conditions is considered in the half-space

\mathbb R_+^3 (x_3 > 0)

. This unique solubility of this problem in Hölder spaces is proved. The proof is based on estimates of the solution in $\mathbb R_{\infty} = \{ (x, t): x \in \mathbb R_+^3, t > 0\}. These estimates are obtained by methods of the Fourier multipliers theory.
Über die Halbierung der Ordnung bei reziproken Polynomen und Anwendungen
Authors: Filippi Siegfried , Schöne F.
Published: December 2016

A theorem for halving the order of reciprocal polynomials is derived and numerical applications are given.
Multipoint Problem for an Extraordinary Differential Equation
Authors: Michalski Marek W.
Published: December 2016

A certain multipoint problem for an extraordinary differential equation is considered. This problem is equivalent to a certain nonlinear integral equation and in this way the existence of its global solution is shown.
Removable Singularities of the Functional Wave Equation
Authors: Chlebík Miroslav , Král Josef
Published: December 2016

Removable singularities for solutions of the functional wave equation satisfying a strengthened Lipschitz condition are characterized with help of one-dimensional Lebesgue measure.
Asymptotics of the densities of harmonic potentials near the apex of a cone (in Russian)
Authors: Levin A.V. , Maz'ya Vladimir G.
Published: December 2016

Boundary integral equations of the method of potentials for the Laplace differential equation in a domain containing a conic point are considered. For the density of a double-layer (single-layer) potential in the case of Dirichlet’s (Neumann’s) problem, the asymptotic expansion in a neighbourhood of the conic point is derived and the corresponding coefficients are calculated. In the case of a circular cone, the summands of the asymptotic expansion and the dependence of the power exponent of its principal term on the apex angle of the cone are given.
A problem of the flow of a viscous fluid (in Russian)
Authors: Kopachevskii Nikolay D. , Krein Selim G.
Published: December 2016

The motion of a viscous incompressible fluid in a bounded area is considered in a linear form under certain rules for the influx and outflow through the surface.
Overdetermined Systems of Nonlinear Partial Differential Equations
Authors: Berg Lothar
Published: December 2016

For the systems of partial differential equations appearing in the theory of Lie series we deduce the general structure of the solutions without assuming the data to be holomorphic ones.
On Function Spaces Related to Finite Element Approximation Theory
Authors: Oswald Peter
Published: December 2016

Certain function spaces of Besov type based on triangular finite elements in a two-dimensional domain are introduced and investigated. These spaces allow a systematic approach to the main estimates of finite element approximation theory in Besov-Sobolev norms, to the problem of improving the rate of approximation by choosing appropriate triangulations, and to asymptotic error estimates for elliptic boundary value problems.
Error Estimates for Finite-Element Solutions of Elliptic Boundary Value Problems in Non-Smooth Domains
Authors: Sändig Anna-Margarete
Published: December 2016

Error estimates in different norms (namely in

W^{m,2}(\Omega)

and

L^p(\Omega), 2 ≤ p ≤ \infty

) of standard finite element solutions of elliptic boundary value problems in bounded domains in

\mathbb R^N

with conical points or non-intersecting edges are considered.
Investigation of extremals in a generalized isoperimetric problem by means of higher-order conditions
Authors: Osmolovskii N.P.
Published: December 2016

The problem of a material point going round maximum area on a plane in a fixed period of time, with its velocity vector varying in a compact set

U

, is considered. Under some assumptions on

U

and with the help of higher-order conditions it is proved that multiply-traversing extremals do not even give a so-called 0-weak maximum (which would at the same time be a weak maximum in the case of continuous velocity). For simply traversing extremals providing an absolute maximum; a non-trivial result on the kind of the maximum is obtained. Furthermore, finite-dimensional and infinite-dimensional quadratic forms connected with the pointed-out generalizationof the isoperimetric problem are studied; their non-negativeness on the corresponding subspaces and some other properties are established.
Convergence Statements for Projection Type Linear Iterative Methods with Relaxations
Authors: Schott Dieter
Published: December 2016

To determine solutions or generalized solutions of the linear operator equation

Ax = b

linear iterative methods of the form

x_{n+1} = (I – D_n) x_n + D_n b

are considered for which the operators

T_n = I – D_nA

are norm-reducing outside their fixed point set. Conditions are given which guarantee the convergence of the methods and certain properties of the limit elements.
Asymptotische Entwicklungen der Gaußschen hypergeometrischen Funktion für unbeschränkte Parameter
Authors: Wagner Eberhard
Published: December 2016

Asymptotic expansions of the hypergeometric function

F(a, b; d; z)

are derived in the case

a^2 = o(c), b^2 = o(c) (|c| \to \infty)

.
Zur Lösung räumlicher Probleme der Elastizitätstheorie für torusberandete Gebiete mit Potentialen vom Typ der einfachen Schicht
Authors: Maul Johannes
Published: December 2016

The problem of the elastic homogeneous and isotropic whole space, with a simple connected inclusion of other material in clastotatics and for stationary elastic vibrations is considered. On the contact surface

L

the jumps of displacements and tractions are given.

L

is assumed to be homeomorphic to the boundary surface of a regular torus, and to belong to the class

C^{1,\beta}

. Using the elastic single layer potential, a system of integral equations is obtained, which is transformed to a twodimensional singular integral equation system by the aid of a Beltrami differential operator

\Omega

. The singular system is normal solvable and has the index 0. Studying the kernel and the range of the operator

\Omega

, the existence of solutions of the original problems can be proved. Some generalisations of the approach on other problems are discussed.
Maximal Op*-Algebras on DF-Domains
Authors: Junek Heinz
Published: December 2016

Let

D

be any dense domain in a Hilbert space and let

\mathcal L^+ (D)

be the maximaximal Op*-algebra of (possibly unbounded) linear operators. In this paper the uniform topology

\tau_D

on

\mathcal L^+ (D)

is investigated for the case where

D

is a DF-space with respect to the graph topology. As a main result, a characterization of the bounded subsets of

D

and of the topology

\tau_D

by strongly bounded selfadjoint operators is given. Especially, each bounded subset of

D

is contained in some bounded ellipsoid. This is applied to approximate the operators in

\mathcal L^+ (D)

by bounded ones.
Potential Flow Past a Circular Cylinder with Permeable Surface and Nonlinear Filtration Law
Authors: Wegert Elias , von Wolfersdorf Lothar
Published: December 2016

The plane potential flow past a circular cylinder with porous surface for two classes of nonlinear filtration laws is dealt with by reducing the problem to a nonlinear singular integral equation and to a linear Riemann-Hilbert problem for a quasilinear equation of Beltrami type, respectively. A general topological existence theorem for the problem will be proven.
Zur Regularität einer Kontakt-Randwertaufgabe
Authors: Schumann Rainer
Published: December 2016

It is proved that the solution of a contact boundary value problem in linear

n

-dimensional elasticity belongs to

C^{1+\alpha}

if the normal component of the displacement vector does.
Double Walsh Series with Coefficients of Bounded Variation
Authors: Móricz Ferenc
Published: December 2016

Convergence properties of double Walsh series are studied whose coefficients form a null sequence of bounded variation. These series converge regularly at all points of (0, 1) x (0, 1) and converge in the pseudometric of

L^r

for all

r \in (0, 1)

. Sufficient conditions for convergence are also proved which involve the second-order differences of the coefficients.
Duality for Nonlinear Abstract Evolution Differential Equations
Authors: Nowakowski Andrzej , Rogowski Andrzej
Published: December 2016

The existence of periodic solutions and duality results for an abstract differential equation in HiIbert space are established. The dual variational method and a modification of the duality theory of Toland are used.
On Imbeddings between Weighted Orlicz Spaces
Authors: Krbec Miroslav , Pick Luboš
Published: December 2016

Necessary and sufficient conditions are given for weight functions

\varrho

and

\sigma

, which guarantee validity of the imbedding between the Orlicz spaces

L_{Q,\varrho} \hookrightarrow L_{P, \sigma}

. We make use of two methods: the first of them is a limit procedure leading to the imbedding of weighted Zygnwnd classes into weighted

L_1

, the second method employs the concept of duality in weighted Orlicz spaces, index of a Young function and results in a general imbedding theorem.
Über ein eindimensionales Modell der finiten Elastostatik
Authors: Langenbach Arno
Published: December 2016

The deformation of a rod

S = [0,1]

is given by a diffeomorphism

w: S \to S_w = [0, w(1)]

from the set

\{w \in W^2_2 (0, 1): w(0) = 0, w' (s) >0 (s \in S) \}

. The function ln

w’

appears as measure of deformation. The function

w

is solution to a second order ordinary differential equation with the second boundary condition

w'(1) = p \in \mathbb R^+

. By an open - and - closed argument we show that the set of those

p

for which the boundary problem is uniquely solvable, is all

\mathbb R^+

.
On a Certain Differential Equation of Non-Integer Order
Authors: Michalski Marek W.
Published: December 2016

Some differential equation of non-integer order is considered. It plays an important role in polarography.
Eine Beziehung zwischen den Algebren verallgemeinerter Distributionen von Colombeau und Ivanov
Authors: Föllinger Olaf
Published: December 2016

We show a relation between an algebra of generalized distributions, which is generated by homogeneous distributions and defined by lvainov, and the algebra of generalized distributions defined by Colombeau.
A Note on Benford’s Law for Second Order Linear Recurrences with Perodical Coefficients
Authors: Schatte Peter , Nagasaka Kenji
Published: December 2016

The solutions of second order linear recurrences with periodical coefficients are shown to obey Benford’s law.