- journal articles metadata
European Mathematical Society Publishing House
2024-03-29 09:42:32
13
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=ZAA&vol=17&iss=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)
17
1998
3
On the Dirac Operator with an Electromagnetic Potential
Vladislav
Kravchenko
Cinvestav del IPN - Querétaro, SANTIAGO DE QUERÉTARO, MEXICO
Dirac’s operator, biquaternions, eikonal equation, exact solutions
A new approach based on the construction of some special biquaternionic projection operators is proposed for analysis and solution of the Dirac equation with electromagnetic potential. There is given an example of the application of this technique which allows us to find the solutions for some class of potentials.
Quantum theory
General
549
556
10.4171/ZAA/837
http://www.ems-ph.org/doi/10.4171/ZAA/837
Orthogonal Projection and Restricted Subordination of Hubert-Schmidt Operator-Valued Stationary Processes
L.
Klotz
Universität Leipzig, LEIPZIG, GERMANY
Hilbert-Schmidt operator-valued stationary processes, subordination
The paper contains two results on subordination of stationarily correlated Hilbert-Schmidt operator-valued stationary processes. First an explicit form of the spectral measure of the orthogonal projection of one process onto another is stated. On the basis of this result B. Fritzsche’s and B. Kirstein’s solution of the restricted subordination problem for finite-dimensional processes is generalized to Hilbert-Schmidt operator-valued processes.
Probability theory and stochastic processes
Operator theory
General
557
563
10.4171/ZAA/838
http://www.ems-ph.org/doi/10.4171/ZAA/838
On Strong Closure of Sets of Feasible States Associated with Families of Elliptic Operators
O.
Zaytsev
University of Latvia, RIGA, LATVIA
Strong closure, feasible states, elliptic operators, systems of elliptic equations
The closure of sets of feasible states for systems of elliptic equations in the strong topology of the Cartesian product $[H^1_0 (\Omega)]^m$ of Sobolev spaces is considered. For $m = 2$ and $\Omega \subset \mathbb R^2$, it is shown that there is a family of linear elliptic operators of the type div $(\chi \mathcal A^1 + (1 - \chi)\mathcal A^2)\triangledown$, where $\chi$ belongs to the set of all characteristic functions of measurable subsets of $\Omega$, such that there does not exist a larger family of operators of the type div $\mathcal A \triangledown$ for which the sets of feasible states coincide with the closure of the original ones.
Calculus of variations and optimal control; optimization
Partial differential equations
General
565
575
10.4171/ZAA/839
http://www.ems-ph.org/doi/10.4171/ZAA/839
Regularity Properties and Generalized Inverses of Delta-Related Operators
L. P.
Castro
Universidade de Aveiro, AVEIRO, PORTUGAL
Frank-Olme
Speck
Instituto Superior Técnico, LISBOA, PORTUGAL
Delta-related operators, factorization theory, generalized inverses, singular integral operators with Carleman shift
As a central topic certain relations between operator matrices are investigated which are called delta relations. The main aim of these relations is to reduce questions about classes of operators without invertibility symbol to those which admit an invertibility symbol. Particular attention is devoted to the generalized inversion of such operators. Different kinds of relations are introduced in order to analyze the "information" contained in the symbols of the related operators. Several examples are considered and the theory is also applied to singular integral operators with Carleman shift. Asymptotic solutions of equations characterized by those operators are presented. The approach simplifies several known results, makes the theory more rigorous from the operator theoretic point of view, and allows further conclusions in a very compact form.
Operator theory
Linear and multilinear algebra; matrix theory
Integral equations
General
577
598
10.4171/ZAA/840
http://www.ems-ph.org/doi/10.4171/ZAA/840
An Extremal Problem Related to the Maximum Modulus Theorem for Stokes Functions
Werner
Kratz
Universität Ulm, ULM, GERMANY
Stokes system, maximum modulus theorem, Stokes-Poisson integral formula, norm of linear mappings
There are considered classical solutions $\nu$ of the Stokes system in the ball $B = \{Ix \in \mathbb R^n : |x| < 1\}$, which are continuous up to the boundary. We derive the-optimal constant $c = c_n$, such that, for all $x \in B$, $$| \nu (x) | ≤ c \ \mathrm {max}_{\xi \in \partial B} | \nu(\xi)| \ \ \ (*)$$ holds for all such functions. We show that $c_n = \mathrm {max}_{x \in \partial B} c_n (x)$ exists, where $c_n(x)$ is the minimal constant in (*) for any fixed $x \in B$. The constants $c_n(x)$ are determined explicitly via the Stokes-Poisson integral formula and via a general theorem on the norm of certain linear mappings given by some matrix kernel. Moreover, the asymptotic behaviour of the $c_n(x)$ as $x \to \partial B$ and as $n \to \infty$ is derived. In the concluding section the general result on the norm of linear mappings is used to prove two inequalities: one for linear combinations of Fourier coefficients and the other from matrix analysis.
Partial differential equations
Linear and multilinear algebra; matrix theory
Operator theory
Fluid mechanics
599
613
10.4171/ZAA/841
http://www.ems-ph.org/doi/10.4171/ZAA/841
The Stokes System in Domains with Outlets of Bounded and Connected Cross-Sections
A.
Passerini
Università di Ferrara, FERRARA, ITALY
G.
Thäter
Universität Bonn, BONN, GERMANY
Stokes systems, non-compact boundaries, weighted spaces, local spaces
The Stokes system with prescribed fluxes is investigated. By smoothness assumptions on the boundary and by the boundedness of the diameters of the outlets it is ensured that the divergence equation in each bounded subdomain is solvable, the Poincaré inequality is valid and the constants in all the corresponding estimates are bounded $independently of the location$. We derive existence, uniqueness and regularity results in two different frameworks: On one hand we use weighted function spaces generated by $L^q$-norms, $1 < q < \infty$, where the weight is of exponential type and apply a technique of Maz’ya and Plamenevskii. On the other hand we use local spaces, since in order to solve the problem with non-zero flux it seems to us that to formulate results in local spaces is more adequate and physical senseful.
Partial differential equations
General
615
639
10.4171/ZAA/842
http://www.ems-ph.org/doi/10.4171/ZAA/842
$L_p$-Theory of Boundary Integral Equations on a Contour with Inward Peak
Vladimir
Maz'ya
Linköping University, LINKÖPING, SWEDEN
A.
Soloviev
Chelyabinsk State University, CHELYABINSK, RUSSIAN FEDERATION
Boundary integral equations, logarithmic potential, asymptotics of solution
Boundary integral equations of the second kind in the logarithmic potential theory - are studied under the assumption that the contour has an inward peak. For each equation we find a pair of function spaces such that the corresponding operator bijectively maps one of them onto another.
Potential theory
Integral equations
General
641
673
10.4171/ZAA/843
http://www.ems-ph.org/doi/10.4171/ZAA/843
On the Stabilization of the Inversion of Some Kontorovich-Lebedev Like Integral Transforms
Hans-Jürgen
Glaeske
Friedrich-Schiller-Universität Jena, JENA, GERMANY
Semyon
Yakubovich
Faculdade de Ciências do Porto, PORTO, PORTUGAL
Kontorovich-Lebedev transform, Lebedev-Skalskaya transforms, index transforms, Macdonald function
In this paper we construct a special type of regularization operators for the Kontorovich-Lebedev type integral transforms to stabilize their inversion in weighted $L_{\nu, p} spaces. Some estimates of norms in these spaces are obtained.
Integral transforms, operational calculus
General
675
690
10.4171/ZAA/844
http://www.ems-ph.org/doi/10.4171/ZAA/844
On Two-Point Right Focal Eigenvalue Problems
Patricia
Wong
Nanyang Technological University, SINGAPORE, SINGAPORE
R.P.
Agarwal
National University of Singapore, SINGAPORE, SINGAPORE
Eigenvalues, positive solutions, boundary value problems
We consider the boundary value problem $$(–1)^{n–p}y^{(n} = \lambda F(t, y, y', \dots, y^{(p)} \ \ \ (n ≥ 2, t \in (0,1))$$ $$y^{(i)} (0) = 0 \ \ \ (0 ≤ i ≤ p–1)$$ $$y^{(i)} (1) = 0 \ \ \ (p ≤ i ≤ n–1)$$ where $\lambda > 0$ and $1 ≤ p ≤ n–1$ are fixed. The values of $\lambda$ are characterized so that the boundary value problem has a positive solution. We also establish explicit intervals of $\lambda$. Examples are included to dwell upon the importance of the results obtained.
Ordinary differential equations
General
691
713
10.4171/ZAA/845
http://www.ems-ph.org/doi/10.4171/ZAA/845
Approximation of Stochastic Differential Equations with Modified Fractional Brownian Motion
Wilfried
Grecksch
Universität Halle-Wittenberg, HALLE, GERMANY
V.V.
Anh
Queensland University of Technology, BRISBANE, AUSTRALIA
Modified fractional Brownian motion, splitting method, stochastic integral, $\epsilon$-optimal control
The modified fractional Brownian motion is a special semimartingale. This stochastic process is suitable for studying the phenomenon of long-range dependence in a wide range of fields. This paper introduces stochastic differential equations with respect to modified fractional Brownian motion. The solution of these equations is approximated by a splitting method whose convergence in probability is proved. An application of this method to determine $\epsilon$-optimal controls for a stochastic control problem is also given.
Probability theory and stochastic processes
Systems theory; control
General
715
727
10.4171/ZAA/846
http://www.ems-ph.org/doi/10.4171/ZAA/846
Semi-Infinite Transportation Problems
H.
Voigt
Universität Leipzig, LEIPZIG, GERMANY
Transportation problem, Kantorovic-Monge problem, transportation flow problem, set partitioning, market area problem
An old set partitioning problem is treated as a special case of the Kantorovic-Monge transportation problem. This problem is then related to Klötzler’s transportation flow problems which allow the consideration of a local cost rate, instead of the constant cost rate in the Kantorovic-Monge problem. Three possibilities for the; numerical solution of the problem are discussed and briefly compared.
Operations research, mathematical programming
Calculus of variations and optimal control; optimization
General
729
741
10.4171/ZAA/847
http://www.ems-ph.org/doi/10.4171/ZAA/847
The Riemann Problem for a Two-Dimensional Hyperbolic System of Nonlinear Conservation Laws: Multiplication of Distribution Solutions
Jiaxin
Hu
Tsinghua University, BEIJING, CHINA
Two-dimensional conservation laws, Ricmann problems, non-classical waves, multiplication of distributions
In the papers [6-8], the author has constructed the Riemann solutions to a two-dimensional hyperbolic system of nonlinear conservation laws for any piecewise constant initial data having two discontinuity rays with origin as vertex. It has been found that, for some initial data, the Riemann solutions no longer lie in $L^{\infty}_{loc} (\mathbb R^2 \times \mathbb R_+)$, and the non-classical waves (labelled as Dirac-contact waves) have arisen. But it remains open in [6-8] to verify that the non-classical solutions constructed satisfy the system considered. In the present paper, we borrow the new mathematical theory of generalized functions, chiefly initiated by J. F. Colombeau and Rosinger, to deal with the diffculty of the multiplication of distribution solutions. The non-classical Riemann solutions we constructed in [6-8] satisfy the system in the sense of association. The present paper provides a good example of applications for this new mathematical theory in powerfully handling the product of generalized functions.
Partial differential equations
General
743
757
10.4171/ZAA/848
http://www.ems-ph.org/doi/10.4171/ZAA/848
Some Operator Ideals in Non-Commutative Functional Analysis
F.
Fidaleo
Università di Roma Tor Vergata, ROMA, ITALY
Linear spaces of operators; Operator algebras and ideals on Hubert spaces; Classifications, factors
We study classes of linear maps between operator spaces $E$ and $F$ which factorize through maps arising in a natural manner by the Pisier vector-valued non-commutative $L^p$-spaces $S_p[E]$ based on the Schatten classes on the separable Hilbert space $\ell ^2$. These classes of maps, firstly introduced in [28] and called p-nuclear maps, can be viewed as Banach operator ideals in the category of operator spaces, that is in non-commutative (quantized) functional analysis. We also discuss some applications to the split property for inclusions of $W*$-algebras such as those describing the physical observables in Quantum Field Theory.
Operator theory
Functional analysis
General
759
776
10.4171/ZAA/849
http://www.ems-ph.org/doi/10.4171/ZAA/849