- journal articles metadata
European Mathematical Society Publishing House
2024-03-29 08:08:11
7
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=ZAA&vol=4&iss=4&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)
4
1985
4
Approximation durch Lösungen partieller Differentialgleichungen
Alfred
Göpfert
Universität Halle-Wittenberg, HALLE, GERMANY
Gert
Wanka
Technische Universität Chemnitz, CHEMNITZ, GERMANY
Johanna
Wanka
Technische Hochschule Leuna Merseburg, MERSEBURG, GERMANY
Remembering earlier $L_2$-approximation theorems of H. Beckert and the authors such approximations are studied in the uniform sense along a surface $\Gamma$, which is lying in the interior of the domain of a boundary value problem of a linear elliptic differential equation of second order, having eigen-solutions in the domain bounded by $\Gamma$.
General
291
303
10.4171/ZAA/154
http://www.ems-ph.org/doi/10.4171/ZAA/154
On the Solvability of Transonic Potential Flow Problems
Miloslav
Feistauer
Charles University, PRAGUE 8, CZECH REPUBLIC
Jindřich
Nečas
Charles University, PRAHA 8, CZECH REPUBLIC
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.
General
305
329
10.4171/ZAA/155
http://www.ems-ph.org/doi/10.4171/ZAA/155
On the Pendant Liquid Drop
Robert
Finn
Stanford University, STANFORD, UNITED STATES
The volume, diameter and other geometrical quantities associated with a pendent liquid drop are shown to be equibounded among all symmetric drops in equilibrium configuration. Explicit bounds are given, and they are shown to be valid also for configurations that are known to be statically unstable.
General
331
339
10.4171/ZAA/156
http://www.ems-ph.org/doi/10.4171/ZAA/156
The Poisson Formula for Euclidean Space Groups and some of its Applications II. The Jacobi Transformation for Flat Manifolds
Paul
Günther
Universität Leipzig, LEIPZIG, GERMANY
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.
General
341
352
10.4171/ZAA/157
http://www.ems-ph.org/doi/10.4171/ZAA/157
Bernstein-Sätze nullter Ordnung und Liouville-Sätze für eine Klasse elliptischer Gleichungen
Jens
Frehse
Universität Bonn, BONN, GERMANY
Subject of § 1 are certain second order nonlinear partial differential equations $Lu = 0$ which allow a so called zero order Bernstein theorem: If $u$ is a solution which is defined in all of $\mathbb R^n$ then $u$ is constant. In § 2 Liouville theorems for powers of certain linear elliptic operators $L$ of second order are presented, this means that solutions of $L^mu = 0$ which are defined and bounded in all of $\mathbb R^n$ must be constant. A connection to the hyperbolic equation $\varphi_{tt} + L\varphi = 0$ is shown.
General
353
362
10.4171/ZAA/158
http://www.ems-ph.org/doi/10.4171/ZAA/158
Twistprodukt und Quasi-*-AIgebren
Gerd
Lassner
Universität Leipzig, LEIPZIG, GERMANY
Gisela
Lassner
Universität Leipzig, LEIPZIG, GERMANY
The Schwartz distribution space $\mathcal S’$ becomes a topological quasi-*-algebra with the distinguished subspace $\mathcal S$, if one defines the so-called twisted product in it. In the paper it is pointed out that the Weyl quantization $f \to W(f)$ IV(/) is an isomorphism of this topological quasi-*-algebra onto the topological quasi-*-algebra $\mathcal L (\mathcal S, \mathcal S’)$ of all linear continuous operators of $\mathcal S$ in $\mathcal S’$. Furthermore, the problem of the extensions of the multiplications in these quasi-*-algebras is discussed.
General
363
372
10.4171/ZAA/159
http://www.ems-ph.org/doi/10.4171/ZAA/159
Konvexe Bereiche kleinster Oberfläche bei gegebener Dicke
Rolf
Klötzler
, BORSDORF, GERMANY
A till now unconfirmed conjecture is proved analytically: The balls are in $\mathbb E^n (n \geq 3)$ the single convex domains of given thickness with smallest surface area. The proof is effected by means of a generalized duality theory of restricted multidimensional variational problems in parametric form.
General
373
383
10.4171/ZAA/160
http://www.ems-ph.org/doi/10.4171/ZAA/160