- journal articles metadata
European Mathematical Society Publishing House
2024-03-29 03:12:08
47
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=ZAA&vol=3&update_since=2024-03-29
Zeitschrift für Analysis und ihre Anwendungen
Z. Anal. Anwend.
ZAA
0232-2064
1661-4534
Partial differential equations
Ordinary differential equations
Integral equations
Numerical analysis
10.4171/ZAA
http://www.ems-ph.org/doi/10.4171/ZAA
subscribers
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society (from 2006)
3
1984
1
Some estimates of the norms of derivatives of holomorphic functions, and their application to initial value problems (in Russian)
G.F.
Mandzhavidze
Tbilisi State University, TBILISI, GEORGIA
Wolfgang
Tutschke
Technische Universität Graz, GRAZ, AUSTRIA
The paper deals with the solution of initial value problems in scales of Banach spaces of generalized analytic functions. To this end a priori estimates of holomorphic functions are given, which allow to solve the initial value problems mentioned above in $L_p$-spaces.
General
1
5
10.4171/ZAA/86
http://www.ems-ph.org/doi/10.4171/ZAA/86
Riemannfunktionen and Differentialoperatoren
Karl Wilhelm
Bauer
Universität Graz, GRAZ, AUSTRIA
It is shown that the complex Riemann function of a formally hyperbolic differential equation can be represented by a differential operator if a representation of solutions is possible by at least one differential operator. The application of this method is illustrated by several examples.
General
7
17
10.4171/ZAA/87
http://www.ems-ph.org/doi/10.4171/ZAA/87
Flächensätze für quasikonform fortsetzbare Abbildungen
Erich
Hoy
, FRIEDBERG, GERMANY
In this paper an extension of the area principle to conformal mappings with a $Q_j$-quasiconformal continuation into the component $\mathcal B_j$ of the complement of a region $\mathcal G$ is given. A generalized area-theorem is proved for these mappings. The inequalities are sharp; the extrernal functions are connected with the solution of the equation $w_{\bar z} = \mu (z) \bar {w_z}$ with $\mu (z)$ being a piecewise constant function. These area theorems are applied to the estimations of the ranges of the coefficient for $z^{-1}$ of the Laurent expansion in the neighbourhood of infinity, the Schwarzian derivative and Golusin’s functional. Finally the possibility of an extension to conformal mappings with a quasiconformal continuation is shown. For Grunsky’s regions these inequalities are asymptotically sharp, if the restriction of the dilatation converges to a constant.
General
19
31
10.4171/ZAA/88
http://www.ems-ph.org/doi/10.4171/ZAA/88
Asymptotic conditions at the two first eigenvalues for the periodic solutions of Liénard differential equations and an inequality of E. Schmidt
C.P.
Gupta
Northern Illinois University, DEKALB, UNITED STATES
Jean
Mawhin
Université Catholique de Louvain, LOUVAIN-LA-NEUVE, BELGIUM
We study the periodic boundary problem $$x’’(t) + f(x(t)) x’(t) + g(t, x(t)) = e(t),$$ $$x(0) — x(2 \pi) = x’(0) —x’(2 \pi) = 0$$ under some non-resonance conditions on the asymptotic behavior of $x^{-1}g(t, x)$ for $|x| \to \infty$.
General
33
42
10.4171/ZAA/88a
http://www.ems-ph.org/doi/10.4171/ZAA/88a
Eine Realisierung der Theorie der abstrakten Besov-Räume $B_q^s(A) (s > 0, 1 \leq q \leq \infty$) und der Lebesgue-Räume $H^s_{p, \mu}$ auf der Grundlage Besselscher Differentialoperatoren
Gerd
Altenburg
Friedrich-Schiller-Universität Jena, JENA, GERMANY
The paper deals with the theory of abstract Besov-spaces, defined by H. Triebel, in the concrete case of weighted $L_p$-spaces on the base of Bessel-type differential operators. Also spaces of Lebesgue-type on the same base are defined and treated here. Furthermore interpolation theorems are given.
General
43
63
10.4171/ZAA/89
http://www.ems-ph.org/doi/10.4171/ZAA/89
Necessary optimality conditions for systems described by nonlinear elliptic equations. I (in Russian)
Uldis
Raitums
University of Latvia, RIGA, LATVIA
This paper considers optimal control problems for systems governed by second order elliptic differential equations in divergence form with nonlinear terms and control appear in all terms. The representation of the main part of increment for a solution of the state equation is given in the case of needle-like variations.
General
65
79
10.4171/ZAA/90
http://www.ems-ph.org/doi/10.4171/ZAA/90
On the existence of the solution of an abstract optimization problem related to a quasi-variational inequality
Gottfried
Bruckner
Karl-Weierstraß-Institut für Mathematik, BERLIN, GERMANY
A general concept is given to get existence and regularity results for an optimization problem that is closely connected to a quasi-variational inequality.
General
81
86
10.4171/ZAA/91
http://www.ems-ph.org/doi/10.4171/ZAA/91
Estimates by Lozinsky’s functional improved in the linear autonomous case
Barnabas
Garay
Technical University of Budapest, BUDAPEST, HUNGARY
Viktor
Kertész
Technical University of Budapest, BUDAPEST, HUNGARY
Using a real-valued functional (denoted by $\mu$) introduced by Lozinsky and defined on matrices, lower and upper bounds can be given for the norm of solutions of linear differential equations. This functional also depends on the norm applied. It is pointed out that the best possible bounds can be obtained applying an appropriate linear transformation of the differential equation. In the real and autonomous case this appropriate linear transformation is real and does not depend on the time if $\mu$ is induced by the Euclidean norm. Moreover, close correspondence between $\mu$ and quadratic Liapunov functions is shown. It is proved that in the general case (not Euclidean norms) the best possible bounds can, generally, not be obtained if the linear transformation does not depend on the time.
General
87
95
10.4171/ZAA/92
http://www.ems-ph.org/doi/10.4171/ZAA/92
2
Fredholmness and finite section method for Toeplitz operators in $l^p (Z_+ \times Z_+)$ with piecewise continuous symbols I
Albrecht
Böttcher
Technische Universität Chemnitz, CHEMNITZ, GERMANY
We consider discrete Toeplitz operators on the space $l^p$ over the quarter-plane for a class of piecewise continuous symbols. This class of symbols is usually denoted by $PC_p(\mathbb T^2)$ and it contains, in particular, all finite sums of the form $\Sigma a_i (\xi) b_i (\eta), (\xi, \eta) \in \mathbb T^2$, where $a_i$ and $b_i$ are of bounded variation. Necessary and sufficient conditions for Fredholmness of such operators and for the applicability of the finite section method to them are obtained. The present part I contains the necessary definitions, the formulation of the main results, and the proofs of the necessity of the given conditions. Their sufficiency will be proved in part II of this work.
General
97
110
10.4171/ZAA/93
http://www.ems-ph.org/doi/10.4171/ZAA/93
Eine Verallgemeinerung der Lambert-Transformation I
Hans-Jürgen
Glaeske
Friedrich-Schiller-Universität Jena, JENA, GERMANY
K.
Stallknecht
Friedrich-Schiller-Universität Jena, JENA, GERMANY
In this paper a generalized Lambert-transformation is investigated, which includes all the well known Lambert-series and Lambert-transformations as special cases. We prove a convergence theorem and by suitable choice of the kernel of the transformation the results are specialized to the well known cases.
General
111
118
10.4171/ZAA/94
http://www.ems-ph.org/doi/10.4171/ZAA/94
Lageabschätzung für einen Kondensator minimaler Kapazität
Siegfried
Kirsch
Universität Halle-Wittenberg, HALLE, GERMANY
With the help of the method of interior variation we get informations about the geometrical form and locus from that continuum, which contains $n ≥ 2$ various points and has minimal generalized transfinite diameter connected with a fundamental solution for generalized Cauchy-Riemann equations.
General
119
131
10.4171/ZAA/95
http://www.ems-ph.org/doi/10.4171/ZAA/95
Necessary optimality conditions for systems described by nonlinear elliptic equations. II (in Russian)
Uldis
Raitums
University of Latvia, RIGA, LATVIA
This paper considers necessary conditions of optimality for systems governed by second elliptic differential equations in divergence form with nonlinear terms. The set of admissible controls is not assumed to be convex, control appears in all coefficients of operator and the case with integral constraints is considered.
General
133
152
10.4171/ZAA/96
http://www.ems-ph.org/doi/10.4171/ZAA/96
Maximal inequalities and Fourier multipliers for spaces with mixed quasinorms. Applications
Hans-Jürgen
Schmeisser
Friedrich-Schiller-University, JENA, GERMANY
The paper concerns with maximal inequalities and Fourier multipliers for systems of entire analytic functions of exponential type belonging to mixed quasinormed spaces. As an application 4 types of spaces of functions with dominating mixed smoothness properties are introduced. Relations to classical spaces of that type (spaces of S.M. Nikol’skij, T.I. Amanov, P.I. Lizorkin) are treated. As a special case the Sobolev spaces with a dominating mixed derivative are contained.
General
153
166
10.4171/ZAA/97
http://www.ems-ph.org/doi/10.4171/ZAA/97
Some properties of a new kind of modulus of smoothness
Vilmos
Totik
University of Szeged, SZEGED, HUNGARY
The modulus of smoothness $$\omega (f, \delta)_{\varphi, p} = \mathrm {sup}_{0 < h ≤ \delta} \| \Delta^2_{h, \varphi} \|_{L^p}$$ has arisen during the investigation of positive operators of the Kantorovich type. Here we show that $\omega_{\varphi, p}$ resembles the ordinary case $\varphi = 1$ and we give the characterization of those functions $f$ for which $\omega (f, \delta)_{\varphi, p} = O (\delta^2)$. The results obtained have applications to positive operators.
General
167
178
10.4171/ZAA/98
http://www.ems-ph.org/doi/10.4171/ZAA/98
Asymptotische Abschätzung der Inversen Toeplitzscher Bandmatrizen im Grenzfall
Lothar
Berg
Universität Rostock, ROSTOCK, GERMANY
For the elements of the inverse of Toeplitz band matrices there are constructed $O$-estimations in the limit case of multiple zeros of the stability polynomial, if the order of the matrices tends to infinity.
General
179
191
10.4171/ZAA/99
http://www.ems-ph.org/doi/10.4171/ZAA/99
3
Fredholmness and finite section method for Toeplitz operators in $l^p (Z_+ \times Z_+)$ with piecewise continuous symbols II
Albrecht
Böttcher
Technische Universität Chemnitz, CHEMNITZ, GERMANY
In this paper we prove sufficient conditions for Fredholmness of discrete Toeplitz operators with piecewise continuous symbols on the space $l^p$ over the quarter-plane and for the applicability of the finite section method to such operators. The methods used here are based on a bilocalization technique and the local principle of Douglas and Krupnik. Part I of this work contained the proofs of the necessity of the corresponding conditions, the necessary definitions, and the formulation of the main results.
General
193
202
10.4171/ZAA/100
http://www.ems-ph.org/doi/10.4171/ZAA/100
Eine Verallgemeinerung der Lambert-Transformation (II)
Hans-Jürgen
Glaeske
Friedrich-Schiller-Universität Jena, JENA, GERMANY
K.
Stallknecht
Friedrich-Schiller-Universität Jena, JENA, GERMANY
In this paper an inversion formula is proved for the generalized Lambert-transformation which was introduced by the authors in part (I). The proof is possible by means of the representation of this transformation as a Laplace-Stieltjes-transformation, the representation of the Laplace-Stieltjes-transformation as a sum of generalized Lambert transformations and reverse.
General
203
211
10.4171/ZAA/101
http://www.ems-ph.org/doi/10.4171/ZAA/101
Asymptotische Entwicklungen der hypergeometrischen Funktionen $F(a, b, c; z)$ für $|a| \to \infty$ und konstante $b, c, z$
Eberhard
Wagner
Martin-Luther-Universität Halle-Wittenberg, HALLE, GERMANY
Asymptotic expansions of the hypergeometric function $F(a, b, c; z)$ for $|a| \to \infty$ are derived where $b, c (c \neq 0, -1, -2, \dots)$ and $z (z \neq 0, |Arg (1— z)| < \pi)$ are fixed complex numbers.
General
213
226
10.4171/ZAA/102
http://www.ems-ph.org/doi/10.4171/ZAA/102
Über Grundlösungen von Differenzenoperatoren
Eckehard
Pfeifer
Technische Universität Dresden, DRESDEN, GERMANY
Angela
Rauhöft
Technische Universität Dresden, DRESDEN, GERMANY
The method to solve problems of the mathematical physics using the so-called fundamental solutions and its properties is well-known. There is the question how does it work in the case of finite difference equations approximating the mentioned problems. An answer can be given treating difference equations as functional equations in certain spaces of distributions. By a suitable construction it is shown that every linear forward difference operator with constant coefficients possesses a tempered fundamental solution in a neighbourhood (with respect to the discretisation parameter) of the fundamental solution of the corresponding differential operator. An initial-value problem for the one-dimensional equation of heat conduction as an example for a useful application of discrete fundamental solutions closes the paper.
General
227
236
10.4171/ZAA/103
http://www.ems-ph.org/doi/10.4171/ZAA/103
The Laguerre Transform of some Elementary Functions
Hans-Jürgen
Glaeske
Friedrich-Schiller-Universität Jena, JENA, GERMANY
Oleg
Maričev
Belarusian State University, MINSK, BELARUS
This paper deals with the calculation of the generalized Laguerre transformation of a class of functions, which often appear in applications. The result is used for the derivation of the solution of an ordinary differential equation.
General
237
244
10.4171/ZAA/104
http://www.ems-ph.org/doi/10.4171/ZAA/104
Lokale Reflexivität und lokale Dualität von Ultraprodukten für halbgeordnete Banachräume
Klaus-Detlef
Kürsten
Universität Leipzig, LEIPZIG, GERMANY
Let $(E_i)_U$ be the ultraproduct of ordered Banach spaces and let $L$ be a finite dimensional sub-space of the dual $(E_i)_u*$. Then there exists a positive iscmetric imbedding of $L$ into $(E_i*)_U$ also satisfying some additional conditions In the case of Banach lattices the isometric imbedding may be chosen as a lattice homomorphism. Similar but somewhat weaker positivity properties can be satisfied for the almost isometric imbeddings of finite dimensional subspaces of $E**$ into $E$ which exist by the principle of local reflexivity.
General
245
262
10.4171/ZAA/105
http://www.ems-ph.org/doi/10.4171/ZAA/105
Zur Vollständigkeit des Systems der Eigenfunktionen irregulärer Eigenwertprobleme mit $\lambda$-abhängigen Randbedingungen
Gerhard
Freiling
Universität Duisburg-Essen, DUISBURG, GERMANY
We show that the system of eigen- and associated functions of the eigenvalue problem $$y^{(n)} + \sum \limits^{n}_{v=2} p_v (x) y^{(n-v)}) = \lambda y \quad 0 \leq x \leq 1,$$ $$U_v (y, \lambda) = 0, \quad 1 \leq v \leq n,$$ with irregular two-point boundary conditions depending on $\lambda$ is complete in $L_2[0,1]$.
General
263
269
10.4171/ZAA/106
http://www.ems-ph.org/doi/10.4171/ZAA/106
Parabolische Regularisierung einer hyperbolischen Itogleichung
Wilfried
Grecksch
Universität Halle-Wittenberg, HALLE, GERMANY
The solution of the first boundary value problem of a random hyperbolic equation is approximated by a method of the parabolic regularization. Considered an application of this method to the development of a maximum principle and to the existence of the $\epsilon$-optimal controls of an optimal control problem of a random hyperbolic equation.
General
271
284
10.4171/ZAA/107
http://www.ems-ph.org/doi/10.4171/ZAA/107
4
Abschätzungen für das Spektrum von $\Delta_p$ auf Räumen konstanter Krümmung
Jürgen
Eichhorn
Universität Greifswald, GREIFSWALD, GERMANY
We present estimations for the first eigenvalue of $\Delta_p$ on spaces $M$ of constant curvature in terms of vol ($M$), $d_M$ and the radius of the greatest geodesic ball contained in $M$. The applied method in an essential way uses the zeros of spherical functions.
General
289
302
10.4171/ZAA/108
http://www.ems-ph.org/doi/10.4171/ZAA/108
Spektrale Geometrie und Huygenssches Prinzip für Tensorfelder und Differentialformen II
Rainer
Schimming
Ernst-Moritz-Arndt-Universität Greifswald, GREIFSWALD, GERMANY
Gunter
Teumer
Ernst-Moritz-Arndt-Universität Greifswald, GREIFSWALD, GERMANY
Geometrical properties (especially local flatness) of a Riemannian manifold are recognized from analytical properties (spectrum or Huygens’ principle) of a Laplace operator. Especially, the "definiteness problem" of the spectral geometry of closed manifolds is solved for the canonical Laplace operator which acts on diffrential forms.
General
303
313
10.4171/ZAA/109
http://www.ems-ph.org/doi/10.4171/ZAA/109
Interior Estimates for Singularly Perturbed Problems
Dietrich
Göhde
, LEIPZIG, GERMANY
The solution of the Dirichlet problem for a singularly perturbed elliptic differential equation $\epsilon L_1u + L_0u = h$ of order $2m$ converges, for $\epsilon \to 0$, outside of the boundary layer uniformly to a solution of the degenerate elliptic equation $L_0w = h$ of lower order. It is shown in the case of order zero of $L_0$ this assertion may be proved immediately, i.e., without the usual construction of boundary layer terms, but rather elementary and on weak smoothness conditions with respect to the boundary of the domain.
General
315
328
10.4171/ZAA/110
http://www.ems-ph.org/doi/10.4171/ZAA/110
Nonlocal Nonlinear Problems for One-Dimensional Parabolic System
Eugeniusz
Chrzanowski
Warsaw Technical University, WARSAW, POLAND
In the paper two nonlocal, nonlinear problems for a system of parabolic equations are considered: to find a solution of the system $$\vec u_t (x,t) = D\vec u_{xx}(x, t) + \vec f (x, t, \vec u (x, t))$$ subject to the conditions $$\vec u (0, t) = \vec \varphi (t), \quad t \in (0, T),$$ $$\vec u (x, 0) = \vec \psi (x), \quad x \in (0, 1),$$ $$\vec u (1, t) - \vec u (x_0, t) = \vec h (x_0, t, \vec u (x_0, t))$$ or $$\int ^1_0 \vec u (x, t) dx = \vec g (t).$$ For this an operator $L: C(\bar \Omega) \to C(\bar \Omega)$ being a sum of four potentials is constructed. It is shown that the operator $L$ has only one fixed point. Moreover it is proved that the fixed point is the only solution of the considered problem.
General
329
336
10.4171/ZAA/111
http://www.ems-ph.org/doi/10.4171/ZAA/111
Zur numerischen Bestimmung des Abbildungsgrades im $\mathbb R^n$ I
Wolfgang
Kliesch
Universität Leipzig, LEIPZIG, GERMANY
Two formulas computing the topological degree of a continuous function $\Phi$ relative to an $n$-dimensional polyhedron $P^n$ are presented. Both formulas are based on the same idea of construction. Let T be a triangulation of the boundary of $P^n$ and let f$(c)$ = sgn $\Phi (e), e \in$ E(T), be a simplicial mapping from T into a boundary triangulation of the $n$-dimensional unit cube, then the topological degree of the function $\Phi$ relative to $P^n$ is given by $$\mathrm {deg} (\Phi, \mathrm {int} \; P^n) = k^{-1} \sum_{\sigma \in \mathrm T_{n-1}} \mathrm {sgn \; d}(\sigma, \Phi),$$ $$\mathrm d(\sigma, \Phi := \mathrm {det} (\mathrm f (a^1) \dots \mathrm f (a^n)), \quad \sigma = [a^1 \dots a^n],$$ if T is oriented in a suitable manner. The second computation formula is based on a simplicial mapping from T into the natural boundary triangulation of the $n$-dimensional unit octahedron.
General
337
355
10.4171/ZAA/112
http://www.ems-ph.org/doi/10.4171/ZAA/112
Infinite Representability of Schrödinger Operators with Ergodic Potential
Harald
Englisch
Universität Leipzig, LEIPZIG, GERMANY
Klaus-Detlef
Kürsten
Universität Leipzig, LEIPZIG, GERMANY
Analogous to the notion of finite representability in the theory of Banach spaces, the notions of the representability and the infinite representability of self-adjoint operators are introduced. It is proved that the infinite representability of the operator $A$ in $B$ yields that the essential spectrum of $B$ contains the spectrum of $A$. This result applied to ergodic Schrödinger operators yields a new proof for the nonrandomness of the spectrum and for the connection between the spectrum and the density of states. A formula for the spectrum of the Hamiltonian of a substitutional alloy is presented, which clarifies the bowing effect. Similar results were found independcntly by Kirsch and Martinelli.
General
357
366
10.4171/ZAA/113
http://www.ems-ph.org/doi/10.4171/ZAA/113
A Remark on the Qualitative Spectral Theory or Sturm-Liouville Operators
Erich
Müller-Pfeiffer
Pädagogische Hochschule, ERFURT, GERMANY
If $N (\Lambda)$ denotes the maximal number of zeros of the non-trivial solutions of the Sturm-Liouville equation $$—(p(x) u’)’ + q(x) u = \Lambda u, \quad -\infty \leq a < x < b \leq \infty,$$ then under some hypothesis the number of eigenvalues of a special selfadjoint operator (Friedrichs extension) is equal to $N\Lambda - 1$ below $\Lambda$.
General
367
369
10.4171/ZAA/114
http://www.ems-ph.org/doi/10.4171/ZAA/114
The Convergence of Galerkin and Collocation Methods with Splines for Pseudodifferential Equations on Closed Curves
Gunther
Schmidt
Angewandte Analysis und Stochastik, BERLIN, GERMANY
The present paper studies the approximate solution of pseudodifferential equations on closed curves using Galerkin and nodal collocation methods with polynomial splines. We give sufficient and in general necessary conditions for the convergence of these methods in Sobolev spaces.
General
371
384
10.4171/ZAA/115
http://www.ems-ph.org/doi/10.4171/ZAA/115
5
On Strongly Nonlinear Poincaré Boundary Value Problems for Harmonic Functions
Lothar
von Wolfersdorf
Technische Universität, FREIBERG, GERMANY
A class of strongly nonlinear Poincaré problems for harmonic functions in the unit disk is studied by reducing them to a new integral equation system to which Schauder’s fixed point theorem is applied. Specific existence results are given for several special cases, in particular the quasilinear case is dealt with in detail.
General
385
399
10.4171/ZAA/116
http://www.ems-ph.org/doi/10.4171/ZAA/116
Konforminvarianten vom Gewicht –1 elnes Zusammenhanges oder Eichfeldes
Rainer
Schimming
Ernst-Moritz-Arndt-Universität Greifswald, GREIFSWALD, GERMANY
A connection or a gauge field in a vector bundle over a riemannian manifold is introduced by means of some given Laplace-like operator. Applying new results of P. Günther and V. Wünsch we construct a sequence of conformal invariants of weight –1 for a gauge field. These are relevant e.g. in the theory of Huygens’ principle. We calculate the relative conformal invariants for some examples; especially they vanish for any instanton gauge field.
General
401
412
10.4171/ZAA/117
http://www.ems-ph.org/doi/10.4171/ZAA/117
Ein Randwertproblem für eine nichtlineare Gleichung gemischtenTyps im $\mathbb R^3$
Andreas
Müller-Rettkowski
Karlsruhe Institute of Technology (KIT), KARLSRUHE, GERMANY
A boundary value problem for the equation $Tu = Lu — u |u|^p = f(x, u), p > 0$, is studied in a simply connected bounded domain $G$ of $\mathbb R^3$. Here $L$ denotes a linear second order differential operator which is elliptic, parabolic or hyperbolic if $x_3 > 0, x_3 = 0$ or $x_3 < 0$, respectively. The boundary of 0 is formed by a non-characteristic and by two characteristic surfaces. The boundary value problem to be solved is to find a solution of the equation in 0 which assumes zero data on the non-characteristic and on one of the characteristic boundary surfaces. It is proved that this problem has a generalized solution belonging to $L^{p+2}$ and to a Sobolew space with weight. Using apriori estimates the solubility of a sequence of approximate problems is shown whose solutions turn out to converge towards a solution of the boundary value problem in question.
General
413
423
10.4171/ZAA/118
http://www.ems-ph.org/doi/10.4171/ZAA/118
On an Evolution Equation for a Non-Hypoelliptic Linear Partial Differential Operator from Stochastics
Karl
Doppel
Freie Universität Berlin, BERLIN, GERMANY
Niels
Jacob
University of Wales Swansea, SWANSEA, UNITED KINGDOM
Recently E. B. Dynkin [2] introduced and studied a non-hypoelliptic linear partial differential operator of even order (with constant coefficients) which originates from the theory of multi-parametric stochastic processes. Motivated by the consideiations of Dynkin the authors have solved a generalized Dirichlet problem for this differential operator in their work [1]. Our aim in the present paper is to investigate the Cauchy problem for the corresponding evolution equation (in the time variable of first order); such a Cauchy problem could have applications to some questions from the stochastics.
General
425
433
10.4171/ZAA/119
http://www.ems-ph.org/doi/10.4171/ZAA/119
Einige Bemerkungen über die Fortsetzung positiv definiter Funktionen
Zoltán
Sasvári
Technische Universität Dresden, DRESDEN, GERMANY
In this note we give some extensions of continuous, positive definite functions on $(–a, a), 0 < a < \infty$. Theorem 3 yields as a special case a result of P. Lévy on periodic extensions.
General
435
440
10.4171/ZAA/120
http://www.ems-ph.org/doi/10.4171/ZAA/120
Three-dimensional dynamic problems of the nonclassical theory of thermoelasticity (in Russian)
Tengiz
Burchuladze
Georgian Acadademy of Sciences, TBILISI, GEORGIA
The paper deals with the dynamical differential equations of nonclassical thermoelasticity initiated by Green and Lindsay, which are characterized by using of two different relaxation times for describing the finite velocity of heat flux. Uniqueness theorems are proved by the aid of integral formulas. Some remarks are made on the applicability of potential methods to the study of the related initial boundary value problems.
General
441
455
10.4171/ZAA/121
http://www.ems-ph.org/doi/10.4171/ZAA/121
Einige Bemerkungen zur Anfangs-Randwertaufgabe $\frac{\partial}{\partial t} u - h(x, t) \Delta u =f$ mit meßbarem Koeffizienten
Jörg
Heinrich
Technische Universität Dresden, DRESDEN, GERMANY
In the present paper the mixed problem $\frac{\partial}{\partial t} u - h(x, t) \Delta u =f$ with measurable coefficient is considered at first for special domains (cubes) by a modification of Banach’s fixed point theorem. Starting from this we get an existence and uniqueness theorem for domains of the class $O^2$. Several properties of the solution are discussed.
General
457
479
10.4171/ZAA/122
http://www.ems-ph.org/doi/10.4171/ZAA/122
6
On the Limit of some Diffusion-Reaction System with Small Parameter
H.
Gajewski
Karl-Weierstraß-Institut für Mathematik, BERLIN, GERMANY
H.-D.
Sparing
Karl-Weierstraß-Institut für Mathematik, BERLIN, GERMANY
A diffusion-reaction system with small parameter $\epsilon$ is considered describing some process of polycondensation in which the chemical reactions are faster than the mass transport. For $\epsilon \to 0$ results a nonlinear evolution equation like $v_t = \Delta f(v)$.
General
481
487
10.4171/ZAA/123
http://www.ems-ph.org/doi/10.4171/ZAA/123
Zur numerischen Bestimmung des Abbildungsgrades im $\mathbb R^n$ II
Wolfgang
Kliesch
Universität Leipzig, LEIPZIG, GERMANY
In this part of the paper, special formulas are deduced from the representations of the topological degree introduced in the first part. Partially, these formulas are generalizations of well-known formulas. Thus, possible means are given for unified approach to the computational formulas for the topological degree used in numerical analysis. Further it is shown how, using Kuhnsimplices, the topological degree can be used for existence proofs and error estimations for zeros of nonlinear equations. Results of some computational tests are given in the last section.
General
489
502
10.4171/ZAA/124
http://www.ems-ph.org/doi/10.4171/ZAA/124
Orthonormalreihenentwicklungen für gewisse quasikonforme Normalabbildungen
Erich
Hoy
, FRIEDBERG, GERMANY
The paper deals with the construction of solutions for the equation $f_{\bar z} (z) = v(z) \overline{f_z(z)}$ with $v(z) \equiv 0$ in a finitely connected region $\mathfrak g$ and $v(z) \equiv q_j =$ const in the complementary continua $\mathfrak B_j$ of $\mathfrak G (0 < q_j < 1, j = 1, 2, \dots, n)$. The construction starts with well-known and in a simple way explicitly computable analytic functions in $\mathfrak G$, and series for the solutions are received only by the use of orthogonalization processes. These series converge in the well-known norm produced by the integral over $\mathfrak G$ of the square of derivative’s absolute value. If the boundary of $\mathfrak G$ consists of analytic Jordan curves only, then there is even an upper bound of the form $M \ast \varrho \ast ^m$ with $M \ast > 0$ and $0 < \varrho \ast < 1$ for the supremum of deviation of the $m$-th partial sum of these series from the sought solutions over $\mathfrak G$. Simple methods are given for the computation of $\varrho \ast$.The results are generalized for the case, that in $\mathfrak B_j$ analytic functions take the place of the constants $q_j$. At the conclusion a possible extension of the procedure to more generalized functions $v(z)$ is discussed.
General
503
521
10.4171/ZAA/125
http://www.ems-ph.org/doi/10.4171/ZAA/125
Zur Unität der Lösung der Theodorsenschen Integralgleichung der konformen Abbildung
Lothar
von Wolfersdorf
Technische Universität, FREIBERG, GERMANY
In the paper a new proof of the uniqueness of the continuous solution to the Theodorsen integral equation of conformal mapping is given.
General
523
526
10.4171/ZAA/126
http://www.ems-ph.org/doi/10.4171/ZAA/126
Zur Stetigkeit der Lösung der adjungierten Gleichung bei Aufgaben der optimalen Steuerung mit Zustandsbeschränkungen
Hoàng Xuân
Phú
Institute of Mathematics, HANOI, VIETNAM
The continuity of the solution of the adjoint equation in optimal control problems with state restrictions has already been shown by other authors under certain conditions that are often not fulfilled. Here the continuity of this solution is shown under different conditions. Furthermore, some other interesting results and applications are given.
General
527
539
10.4171/ZAA/127
http://www.ems-ph.org/doi/10.4171/ZAA/127
On the Existence of Solutions for a General Form of Variational and Quasi-Variational Inequalities
Le Van
Chong
Institute of Mathematics, HANOI, VIETNAM
The paper studies the existence of solutions for a general form of variational and quasi-variational inequalities. For this there is used a modification of the classical coerciveness condition and a monotonicity of inequalities.
General
541
548
10.4171/ZAA/128
http://www.ems-ph.org/doi/10.4171/ZAA/128
Redundancy Conditions for the Functional Equation $f(x + h(x)) = f(x) + f(h(x))$
Gian Luigi
Forti
Università di Milano, MILANO, ITALY
Consider the functional equation $f(x+ h(x)) = f(x) + f(h(x))$, where $h: \mathbb R \to \mathbb R$ is a given continuous function, $h(0) = 0$. It is proved if the set of all zeros of $h$ and of all points where $h(x) = -x$ is not "too much dense", then the continuous and at $x = 0$ differentiable solution $f: \mathbb R \to \mathbb R$ of the functional equation under consideration is $f(x) = xf’(0)$ for all real $x$.
General
549
554
10.4171/ZAA/129
http://www.ems-ph.org/doi/10.4171/ZAA/129
Extremalprinzipien zur Charakterisierung von quasikonformen Normalabbildungen und ihre Anwendung zur Abschätzung von Gebietsfunktionalen
Siegfried
Kirsch
Universität Halle-Wittenberg, HALLE, GERMANY
For a certain class of quasiconformal canonical mappings of a multiple-connected domain the extremal principle is derived with the help of the energy-principle of the modern potential theory, and several distorsion theorems and estimations of domain functionals are proved.
General
555
568
10.4171/ZAA/130
http://www.ems-ph.org/doi/10.4171/ZAA/130
Remarks on Duality Mapping and the Lax-Milgram Property
Josef
Kolomý
Charles University Prague, PRAGUE, CZECH REPUBLIC
Some conditions under which the duality map is a homeomorphism from a Banach space $X$ onto $X \ast$ and upper-semicontinuous at some dense subset of $X$ are derived. Some further pro-perties of the duality mapping are established in connection with the structure of the Banach space $X$. The so-called Lax-Milgram property of the bilinear forms is also investigated.
General
569
576
10.4171/ZAA/131
http://www.ems-ph.org/doi/10.4171/ZAA/131