- journal articles metadata
European Mathematical Society Publishing House
2024-03-28 17:10:14
16
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=ZAA&vol=15&iss=4&update_since=2024-03-28
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)
15
1996
4
On the Existence of Connecting Orbits
Yu
Shu-Xiang
Academia Sinica, BEIJING, CHINA
Connecting orbits, parabolic sectors, exit points
Two existence criteria of orbits connecting a pair of critical points of planar differ-ential equations are given.
Ordinary differential equations
General
779
787
10.4171/ZAA/729
http://www.ems-ph.org/doi/10.4171/ZAA/729
Lifting Theorem as a Special Case of Abstract Interpolation Problem
S.
Kupin
Ukrainian Academy of Sciences, KHARKOV, UKRAINE
Non-unitary contractions, functional models, classical lifting theorem, abstract interpolation problem
Using properties of the de Branges-Rovnyak spaces we include the classical lifting problem into the general scheme of the abstract interpolation problem.
Operator theory
Functions of a complex variable
General
789
798
10.4171/ZAA/730
http://www.ems-ph.org/doi/10.4171/ZAA/730
Fourier Multipliers between Weighted Anisotropic Function Spaces. Part II: Besov-Triebel Spaces
P.
Dintelmann
Technische Hochschule Darmstadt, DARMSTADT, GERMANY
Fourier multipliers, weighted Besov and Triebel spaces, anisotropic spaces
We determine certain classes $M(X^{s_0}_{p_0, q_0} (w_0)), Y^{s_1}_{p_1, q_1} (w_1)$ of Fourier multipliers between weighted anisotropic Besov and Triebel spaces $X^{s_0}_{p_0, q_0} (w_0)$ and $Y^{s_1}_{p_1, q_1} (w_1)$ where $p_0 ≤ 1$ and $w_0, w_1$ are weight functions of polynomial growth. To this end we refine a method based on discrete characterizations of function spaces which was introduced in Part I of the paper. Thus widely generalized versions of known results of Bui, Johnson and others are obtained in a unified way.
Fourier analysis
Functional analysis
General
799
818
10.4171/ZAA/731
http://www.ems-ph.org/doi/10.4171/ZAA/731
Behavior of a Bounded Non-Parametric $H$-Surface Near a Reentrant Corner
K.E.
Lancaster
Wichita State University, WICHITA, UNITED STATES
D.
Siegel
University of Waterloo, WATERLOO, ONTARIO, CANADA
Minimal surfaces, $H$-surfaces, reentrant corners, dip-coating
We investigate the manner in which a non-parametric surface $z = f(x,y)$ of prescribed mean curvature approaches its radial limits at a reentrant corner. We find, for example, that the solution $f(x, y)$ approaches a fixed value (an extreme value of its radial limits at the corner) as a Hölder continuous function with exponent $\frac{2}{3}$ as $(x,y)$ approaches the reentrant corner non-tangentially from inside a distinguished half-space. We also mention an application of our results to a problem in the production of capacitors involving "dip-coating."
Partial differential equations
Differential geometry
General
819
850
10.4171/ZAA/732
http://www.ems-ph.org/doi/10.4171/ZAA/732
On the Green Function of the Landau Operator and its Properties Related to Point Interactions
V.A.
Geyler
Ogarev University Mordovia, SARANSK, RUSSIAN FEDERATION
V.V.
Demidov
Ogarev University Mordovia, SARANSK, RUSSIAN FEDERATION
Schrödinger operator, magnetic fields, Green functions, zero-range potentials
The Green function $G$ of the Schrödinger operator with a magnetic field (i.e. the Landau operator) $H$ is studied. Two representations of $G$ are used, namely in form of an integral and of a series. The space-variable asymptotics as well as the energetic ones are obtained. The analytical and asymptotical properties of $G$ we obtain are related to point perturbations of $H$.
Partial differential equations
General
851
863
10.4171/ZAA/733
http://www.ems-ph.org/doi/10.4171/ZAA/733
Hyperbolic Linear Skew-Product Semiflows
R.T.
Rau
Universität Tübingen, TÜBINGEN, GERMANY
$C_0$-semi groups, cocycles, exponential dichotomy, skew-product flows
A spectral theory for evolution operators on Banach spaces has been developed in [14, 15] considering associated $C_0$-semigroups on vector-valued function spaces. It is then quite natural to substitute the shift on $\mathbb R$ by an arbitrary flow $\sigma$ on a topological space $X$ and to substitute the evolution operator by a cocycle $\Phi$ over $\sigma$. This task was performed by Latushkin and Stepin (cf. [8, 9]) for hyperbolic linear skew-product flows assuming some norm continuity of this flow. In general only strong continuity can be obtained (cf. Sacker and Sell [18) and Example 2 below). Following a suggestion by Hale [7: p. 601 we consider strongly continuous linear skew-product flows in Banach spaces and characterize hyperbolicity through a spectral condition.
Operator theory
Ordinary differential equations
General
865
880
10.4171/ZAA/734
http://www.ems-ph.org/doi/10.4171/ZAA/734
Heat Semi-Group and Function Spaces on Symmetric Spaces of Non-Compact Type
Leszek
Skrzypczak
Adam Mickiewicz University, POZNAN, POLAND
Function spaces, symmetric spaces, heat semigroup, atoms
Besov-Triebel scales of function spaces defined on symmetric spaces of non-compact type are investigated. We prove an atomic decomposition theorem for the function spaces and give their characterization in terms of heat semigroup. In consequence we can describe the spectrum of the Laplace-Beltrami operator in these spaces and improve the generalized Riemann-Lebesgue lemma for the spherical Fourier transform.
Functional analysis
Abstract harmonic analysis
General
881
899
10.4171/ZAA/735
http://www.ems-ph.org/doi/10.4171/ZAA/735
Approximation of Solutions of Stochastic Differential Equations by Discontinuous Galerkin Methods
Wilfried
Grecksch
Universität Halle-Wittenberg, HALLE, GERMANY
A.
Wadewitz
Universität Halle-Wittenberg, HALLE, GERMANY
Approximation of solution of a stochastic differential equation, Stratonovzch integral, Galerkin method
The generalized solution of a system of Stratonovich equations is approximated by a discontinuous Galerkin method. A piecewise polynomial approximation is introduced. The convergence and error estimates are proved. The solution of Galerkin equations can be approximated by the solution of a system of equations with an inhomogeneous random part and the simulation of a stochastic integral.
Probability theory and stochastic processes
General
901
916
10.4171/ZAA/736
http://www.ems-ph.org/doi/10.4171/ZAA/736
On a Class of Nonlinear Neumann Problems of Parabolic Type: Blow-Up of Solutions
M.A.
Pozio
Università di Roma La Sapienza, ROMA, ITALY
Alberto
Tesei
Università di Roma La Sapienza, ROMA, ITALY
Nonlinear Neumann parabolic equations, reaction terms of indefinite type, blow-up of solutions, existence and non-existence of stationary solutions
We investigate large time behaviour of solutions for a class of nonlinear Neumann parabolic problems of indefinite type, possibly degenerate. Depending on the features of the problem, several parameters play a role to establish global boundedness or finite time blow-up of solutions. The occurrence of either situation is related with the existence of stationary solutions. Proofs make extensive use of monotonicity methods.
Partial differential equations
General
917
933
10.4171/ZAA/737
http://www.ems-ph.org/doi/10.4171/ZAA/737
On a Class of Multilinear Operator Equations
Jaan
Janno
Tallin Technical University, TALLINN, ESTONIA
Lothar
von Wolfersdorf
Technische Universität, FREIBERG, GERMANY
Nonlinear operator equations, scale of norms, existence, stability
By means of contraction principle in a Banach space $E$ with a scale of norms $\| \cdot \|_{\sigma} (\sigma ≥ 0)$ existence, uniqueness and stability of solutions are proved for a general class of operator equations $u + G_0u + G_1u = g$ including multilinear ones where $G_0, G_1 \in (E \to E)$ are some operators. The theorems are applicable to equations with operators of generalized convolution type.
Operator theory
Integral equations
General
935
948
10.4171/ZAA/738
http://www.ems-ph.org/doi/10.4171/ZAA/738
Stability of Linear Evolution Equations in Lattice Normed Spaces
M.I.
Gil'
Ben Gurion University of the Negev, BEER-SHEBA, ISRAEL
Linear evolution equations, stability
Linear evolution equations in a space with a generalized norm are considered. Stability conditions are obtained. In particular, the "freezing" method for ordinary differential equations is extended to equations in Banach spaces.
Ordinary differential equations
Integral equations
General
949
959
10.4171/ZAA/739
http://www.ems-ph.org/doi/10.4171/ZAA/739
Optimal Stable Solution of Cauchy Problems for Elliptic Equations
Ulrich
Tautenhahn
University of Applied Sciences, ZITTAU, GERMANY
Ill-posed problems, elliptic partial differential equations, Cauchy problems, optimal regularization methods, optimal error bounds
We consider ill-posed Cauchy problems for elliptic partial differential equations $u_{tt} - Lu = 0 (0 < t ≤ T, x \in \Omega \subset \mathbb R^n)$ with linear densely defined self-adjoint and positive definite operators$ L : D(L) \subset H \to H$ where $H$ denotes a Hilbert space with norm $\| \cdot \|$ and inner product $(\cdot, \cdot)$. We assume that instead of exact data $y = u(x,0)$ or $y = u_t(x,0)$ noisy data $y^{\delta} = u^{\delta} (x,0)$ or $y^{\delta} = u(x,0)$ are available, respectively, with $\|y – y^{\delta} \| ≤ \delta$. Furthermore we assume certain smoothness conditions $u(x,t) \in M$ with appropriate sets $M$ and answer the question concerning the best possible accuracy for identifying $u(x,t)$ from the noisy data. For special sets $M$ the best possible accuracy depends either in a Hölder continuous way or in a logarithmic way on the noise level $\delta$. Furthermore, we discuss special regularization methods which realize this best possible accuracy.
Numerical analysis
Partial differential equations
General
961
984
10.4171/ZAA/740
http://www.ems-ph.org/doi/10.4171/ZAA/740
A Variant of the Mountain Pass Theorem and its Application to Hammerstein Integral Equations
V.
Moroz
The Academy of Sciences of Belarus, MINSK, BELARUS
P. P.
Zabrejko
The Academy of Sciences of Belarus, MINSK, BELARUS
Critical point theory, mountain pass lemma, Hammerstein integral equations, non-trivial solutions
A new variant of the mountain pass theorem based on a ’strong’ deformation lemma is presented. Some applications to the existence of non-trivial solutions of nonlinear Hammerstein integral equations are given too.
Global analysis, analysis on manifolds
Integral equations
Operator theory
General
985
997
10.4171/ZAA/741
http://www.ems-ph.org/doi/10.4171/ZAA/741
$\epsilon k^0$-Subdifferentia1s of Convex Functions
E.-Ch.
Henkel
Universität Halle-Wittenberg, HALLE, GERMANY
Subdifferentials, $\epsilon-subdifferentials, order complete vector lattices, scalarization, properly efficient elements
The paper as a contribution to convex analysis in ordered linear topological spaces. For any convex function $f$ from a Banach space $X$ into a partially ordered one $Y$ endowed with a convex cone $K$ some properties of the $\epsilon k^0$-subdifferential $\partial ^≥_{\epsilon k^0}f(x)$ of $f$ are examined. The non-emptyness of $\partial ^≥_{\epsilon k^0}f(x)$ is proved, whenever $Y$ is a normal order complete vector lattice and $f$ belongs to the class of functions which are continuous and convex with respect to the cone $K$. For the real-valued case Bronsted and Rockafellar have proved that the set of subgradients of a lower semicontinuous function f on a Banach space $X$ is dense in the set of $\epsilon$-subgradients [21]. We deduce a similar result for a class of $\epsilon k^0$-subdifferentials of functions which takes values in an ordered linear topological space $Y$.
Operations research, mathematical programming
General
999
1013
10.4171/ZAA/742
http://www.ems-ph.org/doi/10.4171/ZAA/742
On the Uncertainty Principle for Positive Definite Densities
I.
Dreier
Technische Universität Dresden, DRESDEN, GERMANY
Uncertainty principle, positive definite probability density
The products of variances of adjoint positive definite densities have a greatest lower bound $\Lambda$. We improve the known estimates of $\Lambda$ showing $0.527 < \Lambda < 0.8609....
Probability theory and stochastic processes
General
1015
1023
10.4171/ZAA/743
http://www.ems-ph.org/doi/10.4171/ZAA/743
Some Discrete Inequalities
S.
Varošanec
University of Zagreb, ZAGREB, CROATIA
J.
Pečarić
University of Zagreb, ZAGREB, CROATIA
J.
Šunde
The University of Adelaide, ADELAIDE SA, AUSTRALIA
Bellman inequality, Cebysev inequality, Hölder inequality, Jensen inequality, Minkowski inequality, Popoviciu inequality, weighted mean, quasiarithmetic mean, logarithmic mean
A number of inequalities with finite differences which are connected with weighted, quasiarithmetic and logarithmic means and some well-known general inequalities are considered.
Real functions
General
1025
1032
10.4171/ZAA/744
http://www.ems-ph.org/doi/10.4171/ZAA/744