- journal articles metadata
European Mathematical Society Publishing House
2024-03-28 13:42:26
14
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=ZAA&vol=15&iss=3&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
3
Compactness of an Integro-Differential Operator of Cauchy-Kovalevskaya Theory
Wolfgang
Tutschke
Technische Universität Graz, GRAZ, AUSTRIA
Harkrishan
Vasudeva
Panjab University, CHANDIGARH, INDIA
Initial value problems, weighted norms, Fredholrn alternative
The integral operator connected with the Cauchy-Kovalevskaya. initial value prob-lem has been shown to be compact between suitable Frechét spaces.
Partial differential equations
Functional analysis
Operator theory
General
559
564
10.4171/ZAA/715
http://www.ems-ph.org/doi/10.4171/ZAA/715
A Continuation Method for Weakly Condensing Operators
Donal
O'Regan
National University of Ireland, GALWAY, IRELAND
Fixed points, continuation theory, measures of weak non-compactness
We present a continuation result for weakly condensing operators between Banach spaces. There are given also a new fixed point result being in the spirit of Schauder’s fixed point theorem and some applications to nonlinear operator equations.
Operator theory
General
565
578
10.4171/ZAA/716
http://www.ems-ph.org/doi/10.4171/ZAA/716
Fourier Multipliers between Weighted Anisotropic Function Spaces. Part I: Besov Spaces
P.
Dintelmann
Technische Hochschule Darmstadt, DARMSTADT, GERMANY
Fourier multipliers, weighted Besov spaces, anisotropic spaces
We determine classes $M(B^{s_0}_{p_0, q_0}(w_0),B^{s_1}_{p_1, q_1} (w_1))$ of Fourier multipliers between weighted anisotropic Besov spaces $B^{s_0}_{p_0, q_0}(w_0)$ and $B^{s_1}_{p_1, q_1}(w_1)$ where $p_0 ≤ 1$ and $w_0, w_1$ are weight functions of, polynomial growth. To this end we use a discrete characterization of the function spaces akin to the $\varphi$-transform of Frazier and Jawerth which leads to a unified approach to the multiplier problem. In this way widely generalized versions of known results of Bui, Johnson, Peetre and others are obtained from a single theorem.
Fourier analysis
General
579
601
10.4171/ZAA/717
http://www.ems-ph.org/doi/10.4171/ZAA/717
Fréchet Differentiability of the Solution of the Heat Equation with Respect to a Nonlinear Boundary Condition
Arnd
Rösch
Universität Duisburg-Essen, DUISBURG, GERMANY
Fréchet differentiability, heat equation, nonlinear boundary conditions, identification of the heat function, inverse problems
We consider the heat equation $\frac{\partial u}{\partial t} (t, x) = \Delta _x u(t,x)$ with a nonlinear function a in the boundary condition $\frac{\partial u}{\partial n} (t, x) =\alpha ((u(t,z))(\vartheta – u(t,x))$ depending on the boundary values $x$ of the solution u of the initial-boundary value problem only and belonging to a set of admissible differentiable or uniformly Lipschitz continuous functions. For this problem Lipschitz continuity and Fréchet differentiability of the mapping $\Phi : \alpha \mapsto x$ under different assumptions are derived.
Partial differential equations
General
603
618
10.4171/ZAA/718
http://www.ems-ph.org/doi/10.4171/ZAA/718
Radial Symmetry for an Electrostatic, a Capillarity and some Fully Nonlinear Overdetermined Problems on Exterior Domains
Wolfgang
Reichel
Karlsruhe Institute of Technology (KIT), KARLSRUHE, GERMANY
Overdetermined problems, exterior domains, radial symmetry, electrostatic condenser, capillary surfaces
We consider two physically motivated problems: (1) Suppose the surface of a body in $\mathbb R^2$ or $\mathbb R^3$ is charged with a constant density. If the induced single-layer potential is constant inside the body, does it have to be a ball? (2) Suppose a straight solid cylinder of unknown cross-section is dipped into a large plain liquid reservoir. If the liquid rises to the same height on the cylinder wall, does the cylinder necessarily have circular cross-section? Both questions are answered with yes, and both problems are shown to be of the type $$\mathrm {div} (g(|\triangledown u |)\triangledown u) + f(u, | \triangledown u|) = 0 \ \mathrm {in} \ \Omega, \ \ u = \mathrm {const}, \frac{\partial u}{\partial v} = \mathrm {const} \ \mathrm {on} \ \partial \Omega, \ u = 0 \ \mathrm {at} \ \infty$$ where $\partial_u f ≤ 0$ and $\Omega = \mathbb R^N \ \bar{G}$ is the connected exterior of the smooth bounded domain $G$. The overdetermined nature of this possibly degenerate boundary value problem forces $\Omega$ to be radial. This is shown by a variant of the Alexandroff-Serrin method of moving hyperplanes, as recently developed for exterior domains by the author in [19]. The results extend to Monge-Ampere equations.
Partial differential equations
Potential theory
Fluid mechanics
General
619
635
10.4171/ZAA/719
http://www.ems-ph.org/doi/10.4171/ZAA/719
A Non-Degeneracy Property for a Class of Degenerate Parabolic Equations
Carsten
Ebmeyer
Universität Bonn, BONN, GERMANY
Free boundary problems, finite speed of propagation, porous medium equations
We deal with the initial and boundary value problem for the degenerate parabolic equation $u_t = \Delta \beta (u)$ in the cylinder $\Omega \times (0,T)$, where $\Omega \subset \mathbb R^n$ is bounded, $\beta (0) = \beta’(0) = 0$, and $\beta' ≥ 0$ (e.g., $\beta (u) = u |u|^{m–1} \ (m > 1))$. We study the appearance of the free boundary, and prove under certain hypothesis on $\beta$ that the free boundary has a finite speed of propagation, and is Holder continuous. Further, we estimate the Lebesgue measure of the set where $u > 0$ is small and obtain the non-degeneracy property $|\{ 0 < \beta' (u(x,t)) < \epsilon \} | ≤ c \epsilon^{\frac{1}{2}}$.
Partial differential equations
Fluid mechanics
General
637
650
10.4171/ZAA/720
http://www.ems-ph.org/doi/10.4171/ZAA/720
Asymptotics of the Solution of an Integral Equation to Transmission Problems with Singular Perturbed Boundary
Ralf
Mahnke
Universität Rostock, ROSTOCK, GERMANY
Asymptotics, boundary integral equations, transmission problems
The integral equation to a transmission problem of the Laplacian is considered on a smooth boundary of a plane domain. The contour depends on a positive parameter e and the domain has a corner in the limit case $\epsilon = 0$. The main terms of an asymptotic expansion showing the influence of the parameter are given. The remaining part is estimated in a weak norm.
Partial differential equations
General
651
660
10.4171/ZAA/721
http://www.ems-ph.org/doi/10.4171/ZAA/721
On the Convergence of the Goerisch Method for Self-Adjoint Eigenvalue Problems with Arbitrary Spectrum
U.
Mertins
Technische Universität Clausthal, CLAUSTHAL-ZELLERFELD, GERMANY
Eigenvalue problems, variational methods, upper and lower bounds to eigenvalues, approximation of eigenelements, convergence of the Goerisch method, curve veering
It was shown recently in [13] that the Goerisch method provides upper and lower bounds to eigenvalues of variationally posed self-adjoint eigenvalue problems with arbitrary spectrum. In the present paper the approximation of eigenelements is established. In addition, the convergence of the eigenvalue and eigenelement approximations is shown in a pure func-tional analytic procedure. A numerical example is given where the curve veering phenomenon occurs.
Calculus of variations and optimal control; optimization
Partial differential equations
Numerical analysis
General
661
686
10.4171/ZAA/722
http://www.ems-ph.org/doi/10.4171/ZAA/722
Second Order Sufficient Optimality Conditions for a Nonlinear Elliptic Boundary Control Problem
E.
Casas
Universidad de Cantabria, SANTANDER, SPAIN
Fredi
Tröltzsch
Technische Universität Berlin, BERLIN, GERMANY
A.
Unger
Technische Universität Chemnitz, CHEMNITZ, GERMANY
Optimal control, semilinear elliptic equations, second order conditions, sufficient optimality conditions
In this paper sufficient second order optimality conditions are established for opti-mal control problems governed by a linear elliptic equation with nonlinear boundary condition, where pointwise constraints on the control are given. The second order condition requires coercivity of the Lagrange function on a suitable subspace together with first order sufficient conditions on a certain set of strongly active points.
Calculus of variations and optimal control; optimization
General
687
707
10.4171/ZAA/723
http://www.ems-ph.org/doi/10.4171/ZAA/723
On Linear Integro-Differential Equations with Integral Impulsive Conditions
H.
Akça
Akdeniz University, ANTALYA, TURKEY
L.
Berezansky
Ben Gurion University of the Negev, BEER-SHEBA, ISRAEL
Elena
Braverman
Technion - Israel Institute of Technology, HAIFA, ISRAEL
Integro-differential equations, impulsive conditions
Linear integro-differential equations with linear integral impulsive conditions are considered. Existence results and representations of solutions are obtained. Stability of these equations is investigated.
Ordinary differential equations
General
709
727
10.4171/ZAA/724
http://www.ems-ph.org/doi/10.4171/ZAA/724
Fibonacci Polynomials their Properties and Applications
Z.W.
Trzaska
University of Warsaw, WARSAW, POLAND
Fibonacci sequence, Fibonacci polynomials, recurrence relations, zeros of polynomials, stability of dynamical systems
The paper deals with polynomials characterized by coefficients determined by successive elements of the Fibonacci sequence. Basic properties and applications of the Fibonacci polynomials are demonstrated. The index of concentration of Fibonacci polynomials at $k$-th degree, locations of their zeros and optimization procedures for such polynomials are discussed. Illustrative examples are presented.
Abstract harmonic analysis
General
729
746
10.4171/ZAA/725
http://www.ems-ph.org/doi/10.4171/ZAA/725
Asymptotic Inequalities Related to the Maximum Modulus of a Polynomial
C.
Frappier
École Polytechnique de Montréal, MONTREAL, CANADA
M.A.
Qazi
École Polytechnique de Montréal, MONTREAL, CANADA
Polynomials, inequalities, asymptotic
Let $\mathcal P_n$ be the class of all polynomials of degree at most $n$. If $\| \cdot \|$ denotes the supremum norm on $| z | =1$ and $M_p(R) = max_{|x|=R} | P(z) |$, then for an arbitrary polynomial $P(z) = \sum ^n_{v=0} a_v z^v$ in $\mathcal P_n$ the inequality $M_P(R) ≤ R^n \| P \|$ holds, with equality if and only if $a_0 = … = a_{n–1} = 0$. Given $n,k \in \mathbb N$ with$ 0 ≤ k ≤ n–1$, let $\varphi _{n,k} (R)$ be the largest number such that $M_P (R)+ \varphi_{n,k}(R)|a_k| ≤ R^n \|P\| (R ≥ 1)$ for all $P \in \mathcal P_n$. Values of $\varphi_{n,k} (R)$ for $k=0$ and $k = 1$ are known since some time. We study the case $k ≥ 2$.
Functions of a complex variable
Real functions
General
474
758
10.4171/ZAA/726
http://www.ems-ph.org/doi/10.4171/ZAA/726
A Generalization of the Weierstrass Theorem
A.
Drwalewska
Technical University Lodz, LODZ, POLAND
Multiobjective optimization, maximal points, ($p,p$)-maximal points, monotonically semicontinuous functions, sequentially compact sets
The well-known Weierstrass theorem stating that a real-valued continuous function $f$ on a compact set $K \subset \mathbb R$ attains its maximum on $K$ is generalized. Namely, the space of real numbers is replaced by a set $Y$ with arbitrary preference relation $p$ (in place of the inequality ≤), and the assumption of continuity of $f$ is replaced by its monotonic semicontinuity (with respect to the relation $p$).
Calculus of variations and optimal control; optimization
Functional analysis
Operations research, mathematical programming
General
759
763
10.4171/ZAA/727
http://www.ems-ph.org/doi/10.4171/ZAA/727
On the Method of Backward Steps of Carathéodory-Tonelli
E.
De Pascale
Università della Calabria, ARCAVACATA DI RENDE (CS), ITALY
G.
Marino
Università della Calabria, ARCAVACATA DI RENDE (CS), ITALY
Giorgio
Metafune
Università del Salento, LECCE, ITALY
Ordinary differential equations, Cauchy problems, Backward steps method, Carathéodory-Tonelli solution, Peano phenomenon
We study the convergence of a sequence of functions which has been introduced by Carathéodory and Tonelli in connection with the solvability of the Cauchy problem for ordinary differential equations.
Ordinary differential equations
General
765
770
10.4171/ZAA/728
http://www.ems-ph.org/doi/10.4171/ZAA/728