- journal articles metadata
European Mathematical Society Publishing House
2024-03-28 13:17:04
61
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=ZAA&vol=20&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)
20
2001
1
A Topological Fixed-Point Index Theory for Evolution Inclusions
R.
Bader
Technische Universität München, MÜNCHEN GARCHING, GERMANY
Fixed-point index, ($U, V$)-approximation, evolution inclusions
In the paper we construct a topological fixed-point theory for a class of set-valued maps which appears in natural way in boundary value problems for differential inclusions. Our construction is based upon the notion of ($U, V$)-approximation in the sense of Ben-El-Mechaiekh and Deguire. As applications we consider initial-value problems for nonlinear evolution inclusions of the type $$x'(t) \in –A(t,x(t)) + F(t, x(t))$$ $$|x(0) = x_0$$ where the operator $A$ satisfies various monotonicity assumptions and $F$ is an upper semicontinuous set-valued perturbation.
Algebraic topology
Ordinary differential equations
General
3
15
10.4171/ZAA/1001
http://www.ems-ph.org/doi/10.4171/ZAA/1001
A Quaternionic Beltrami-Type Equation and the Existence of Local Homeomorphic Solutions
Paula
Cerejeiras
Universidade de Aveiro, AVEIRO, PORTUGAL
K.
Gürlebeck
Technische Universität Chemnitz-Zwickau, CHEMNITZ, GERMANY
Uwe
Kähler
Universidade de Aveiro, AVEIRO, PORTUGAL
H.
Malonek
Universidade de Aveiro, AVEIRO, PORTUGAL
Generalized Beltrami equation, quaternions, singular integral operators, homeomorphic solutions, monogenic functions
The paper deals with a quaternionic Beltrami-type equation, which is a very natural generalization of the complex Beltrami equation to higher dimensions. Special attention is paid to the systematic use of the embedding of the set of quaternions $\mathbb H into \mathbb C^2$ and the corresponding application of matrix singular integral operators. The proof of the existence of local homeomorphic solutions is based on a necessary and sufficient criterion, which relates the Jacobian determinant of a mapping from $\mathbb R^4$ into $\mathbb R^4$ to the quaternionic derivative of a monogenic function.
Functions of a complex variable
Partial differential equations
General
17
34
10.4171/ZAA/1002
http://www.ems-ph.org/doi/10.4171/ZAA/1002
Complex 2-Normed Linear Spaces and Extension of Linear 2-Functionals
S.N.
Lal
Banaras Hindu University, VARANASI, INDIA
S.
Bhattacharya
Banaras Hindu University, VARANASI, INDIA
C.
Sreedhar
Banaras Hindu University, VARANASI, INDIA
2-norms, linear, convex, 2-bounded and tangent 2-functionals, internal and bounding points
The known concept of 2-normed real linear spaces is extended to 2-normed complex linear spaces. This extension is not trivial. A Hahn-Banach type extension theorem for complex linear 2-functionals is established and it is shown that it is not possible to get this result from the known Hahn-Banach type extension theorem for real linear 2-functionals using the Bohnenblust-Sobczyk technique directly as is done in the case of linear functionals. As an application of our extension theorem, a 2-norm version of the Ascoli-Mazur theorem on tangent functionals is established. Several examples and counter examples illustrate the results obtained in the paper.
Functional analysis
General
35
53
10.4171/ZAA/1003
http://www.ems-ph.org/doi/10.4171/ZAA/1003
A Sequence of Integro-Differential Equations Approximating a Viscous Porous Medium Equation
Karl
Oelschläger
Universität Heidelberg, HEIDELBERG, GERMANY
Integro-differential equations, porous medium equations, asymptotic expansions
We consider a sequence of particular integro-differential equations, whose solutions $\rho_N$ converge as $N \rightarrow \infty$ to the solution $\rho$ of a viscous porous medium equation. First, it is demonstrated that under suitable regularity conditions the functions $\rho_N$ are smooth uniformly in $N \in \mathbb N$. Furthermore, an asymptotic expansion for $\rho_N$ as $N \rightarrow \inftly$ is provided, which precisely describes the convergence to $\rho$. The results of this paper are needed in particular for the numerical simulation of a viscous porous medium equation by a particle method.
Integral equations
Partial differential equations
General
55
91
10.4171/ZAA/1004
http://www.ems-ph.org/doi/10.4171/ZAA/1004
Blow-Up and Convergence Results for a One-Dimensional Non-Local Parabolic Problem
A.
Rougirel
Universität Zürich, ZÜRICH, SWITZERLAND
Parabolic problems, blow-up, convergence to steady states
Considering a one-dimensional non-local semilinear parabolic problem, it is shown that blow-up in finite time occurs for suitable large initial conditions. The asymptotic behavior of global solutions corresponding to small initial conditions is also investigated. Their convergence in $H^1$-norm to a well determinated stationary solution is proved.
Partial differential equations
General
93
113
10.4171/ZAA/1005
http://www.ems-ph.org/doi/10.4171/ZAA/1005
Parabolic Equations with Functional Dependence
A.
Bychowska
Gdansk University of Technology, GDANSK, POLAND
H.
Leszczyński
University of Gdansk, GDANSK, POLAND
Parabolic functional-differential equations, existence and uniqueness of solutions, iterative methods, Cauchy problems
We consider the Cauchy problem for nonlinear parabolic equations with functional dependence and prove theorems on the existence of solutions to parabolic differential-functional equations
Partial differential equations
General
115
130
10.4171/ZAA/1006
http://www.ems-ph.org/doi/10.4171/ZAA/1006
Boundary Integral Operators for Plate Bending in Domains with Corners
Gunther
Schmidt
Angewandte Analysis und Stochastik, BERLIN, GERMANY
Biharmonic equation, plate bending, non-smooth curves, boundary integral operators, boundary integral equations
The paper studies boundary integral operators of the bi-Laplacian on piecewise smooth curves with corners and describes their mapping properties in the trace spaces of variational solutions of the biharmonic equation. We formulate a direct integral equation method for solving interior and exterior mixed boundary value problems on non-smooth plane domains, analyze the solvability of the corresponding systems of integral equations and prove their strong ellipticity.
Potential theory
Operator theory
Numerical analysis
General
131
154
10.4171/ZAA/1007
http://www.ems-ph.org/doi/10.4171/ZAA/1007
On the Fredholm Property of the Stokes Operator in a Layer-Like Domain
Sergei
Nazarov
Institute for Problems in Mechanical Engineering RAS, ST. PETERSBURG, RUSSIAN FEDERATION
K.
Pileckas
Vilnius University, VILNIUS, LITHUANIA
Stokes equations, layer-like domains, Fredholm property, weighted spaces
The Stokes problem is studied in the domain $\Omega \subset \mathbb R^3$ coinciding with the layer $\Pi = {x = (y,z) : y = (y_1, y_2) \in (0,1)}$ outside some ball. It is shown that the operator of such problem is of Fredholm type; this operator is defined on a certain weighted function space $\mathcal D^l_{\beta} (\Omega$) with norm determined by a stepwise anisotropic distribution of weight factors (the direction of $z$ is distinguished). The smoothness exponent $l$ is allowed to be a positive integer, and the weight exponent $\beta$ is an arbitrary real number except for the integer set $\mathbb Z$ where the Fredholm property is lost. Dimensions of the kernel and cokernel of the operator are calculated in dependence of $\beta$. It turns out that, at any admissible $\beta$, the operator index does not vanish. Based on the generalized Green formula, asymptotic conditions at infinity are imposed to provide the problem with index zero.
Partial differential equations
Fluid mechanics
General
155
182
10.4171/ZAA/1008
http://www.ems-ph.org/doi/10.4171/ZAA/1008
A Discretised Nonlinear Eigenvalue Problem with Many Spurious Branches of Solutions
Charles
Stuart
Ecole Polytechnique Federale, LAUSANNE, SWITZERLAND
Gregory
Vuillaume
Ecole Polytechnique Federale, LAUSANNE, SWITZERLAND
Stokes equations, layer-like domains, Fredholm property, weighted spaces
We treat an example of a nonlinear eigenvalue problem in $L^2(0,1)$ which can be solved explicitly. It has a single branch of non-trivial solutions. Discretisation reduces the problem to a finite-dimensional one having many branches of non-trivial solutions. We investigate the convergence of these approximate solutions.
Operator theory
Numerical analysis
General
183
192
10.4171/ZAA/1009
http://www.ems-ph.org/doi/10.4171/ZAA/1009
Conditional Stability of a Real Inverse Formula for the Laplace Transform
S.
Saitoh
Gunma University, KIRYU, JAPAN
Vu
Kim Tuan
Kuwait University, SAFAT, KUWAIT
Masahiro
Yamamoto
University of Tokyo, TOKYO, JAPAN
Laplace transform, real inversion formulas, conditional stability, Bergman-Selberg space, error estimates, Mellin transform, Gauss formula, convolution, reproducing kernels
We establish a conditional stability estimate of a real inverse formula for the Laplace transform of functions under the assumption that the Bergman-Selberg norms of the Laplace transform of those functions are uniformly bounded. The rate of the stability estimate is shown to be of logarithmic order.
Integral transforms, operational calculus
Special functions
General
193
202
10.4171/ZAA/1010
http://www.ems-ph.org/doi/10.4171/ZAA/1010
On $C^1$-Regularity of Functions that Define $G$-Closure
M.
Miettinen
University of Jyväskylä, JYVÄSKYLÄ, FINLAND
Uldis
Raitums
University of Latvia, RIGA, LATVIA
Homogenization, $G$-closure, quasiconvexity
In this paper we show that the functions which are used in the characterization of the $G$-closure or the $G_\theta$-closure of sets of matrices are continuously differentiable. These regularity results are based on the observation by Ball, Kirchheim and Kristensen [1] that separate convexity and upper semidifferentiability imply continuous differentiability.
Calculus of variations and optimal control; optimization
General
203
214
10.4171/ZAA/1011
http://www.ems-ph.org/doi/10.4171/ZAA/1011
On Inequalities of Korn, Friedrichs, Magenes-Stampacchia-Necas and Babuska-Aziz
A.
Tiero
Università di Roma Tor Vergata, ROMA, ITALY
Korn’s inequalities, continuum mechanics, equations of mathematical physics
The equivalence between the inequalities of Korn, Friedrichs, Magenes-Stampacchia-Necas and Babuska-Aziz is derived using some elementary properties of the gradient, divergence and curl operators implied by these inequalities.
Partial differential equations
Mechanics of deformable solids
General
215
222
10.4171/ZAA/1012
http://www.ems-ph.org/doi/10.4171/ZAA/1012
Classification and Existence of Non-Oscillatory Solutions of Second-Order Neutral Delay Difference Equations
Yong
Zhou
Xiangtan University, XIANGTAN, HUNAN, CHINA
Binggen
Zhang
Ocean University of Qingdao, QINGDAO, CHINA
Some existence results for each kind of non-oscillatory solutions are also established
In this paper, we give a classification of non-oscillatory solutions of a second-order neutral delay difference equation of the form $$\Delta^2(x_n – c_nx_{n–\tau}) + f (n, x_{g_1(n)},..., x_{g_m(n)}) = 0 (n ≥ n_0 \in \mathbb N).$$ Some existence results for each kind of non-oscillatory solutions are also established.
Difference and functional equations
General
223
234
10.4171/ZAA/1013
http://www.ems-ph.org/doi/10.4171/ZAA/1013
Some Series over the Product of Two Trigonometric Functions and Series Involving Bessel Functions
Miomir
Stankovic
University of Nis, NIS, SERBIA
Mirjana
Vidanovic
University of Nis, NIS, SERBIA
Slobodan
Trickovic
University of Nis, NIS, SERBIA
Bessel functions, Riemann zeta and related functions
The sum of the series $$S_{\alpha} = S_{\alpha} s,a,b,f(y),g(x) = \sum^\infty_{n=1} \frac {(s)^{n–1}f (an – b)y g (an–b)x}{(an–b)^{\alpha}}$$ involving the product of two trigonometric functions is obtained using the sum of the series $$\sum^\infty_{n=1} \frac {(s)^{(n–1)} f((an–b)x)}{(an–b)^\alpha} = \frac {c\pi}{2\Gamma (\alpha) f (\frac {\pi \alpha}{2})} x^{\alpha–1} + \sum^\infty_{i=0} (–1)^i \frac {F(\alpha – 2i – \delta)}{(2i + \delta)!} x^{2i+\delta}$$ whose terms involve one trigonometric function. The first series is represented as series in terms of the Riemann zeta and related functions, which has a closed form in certain cases. Some applications of these results to the summation of series containing Bessel functions are given. The obtained results also include as special cases formulas in some known books. We further show how to make use of these results to obtain closed form solutions of some boundary value problems in mathematical physics.
Special functions
Number theory
Numerical analysis
General
235
246
10.4171/ZAA/1014
http://www.ems-ph.org/doi/10.4171/ZAA/1014
2
The Existence of Non-Trivial Bounded Functionals Implies the Hahn-Banach Extension Theorem
W.A.J.
Luxemburg
California Institute of Technology, PASADENA, UNITED STATES
Martin
Väth
Czech Academy of Sciences, PRAGUE 1, CZECH REPUBLIC
Power of the Hahn-Banach theorem, linear functionals, axiom of choice, axiom of dependent choices, Shelah’s model
We show that it is impossible to prove the existence of a linear (bounded or unbounded) functional on any $_\{infty}/C_0$ without an uncountable form of the axiom of choice. Moreover, we show that if on each Banach space there exists at least one non-trivial bounded linear functional, then the Hahn-Banach extension theorem must hold. We also discuss relations of non-measurable sets and the Hahn-Banach extension theorem.
Functional analysis
Mathematical logic and foundations
General
267
279
10.4171/ZAA/1015
http://www.ems-ph.org/doi/10.4171/ZAA/1015
Characterization of the Maximal Ideal of Operators Associated to the Tensor Norm Defined by an Orlicz Function
G.
Loaiza
Universidad EAFIT, MEDELLIN, COLOMBIA
J.
López Molina
Universidad Politecnia de Valencia, VALENCIA, SPAIN
M.
Rivera
Universidad Politecnia de Valencia, VALENCIA, SPAIN
Integral operators, ultraproducts of spaces and maps
Given an Orlicz function $H$ satisfying the $\Delta_2$ property at zero, one can use the Orlicz sequence space $\mathcal l_H$ to define a tensor norm $g^c_H$ and the minimal ($H^c$-nuclear) and maximal ($H^c$-integral) operator ideals associated to $g^c_H$ in the sense of Defant and Floret. The aim of this paper is to characterize $H^c$-integral operators by a factorization theorem.
Functional analysis
General
281
293
10.4171/ZAA/1016
http://www.ems-ph.org/doi/10.4171/ZAA/1016
Unbounded $C$*-Seminorms and *-Representations of Partial *-Algebras
Fabio
Bagarello
Università degli Studi di Palermo, PALERMO, ITALY
A.
Inoue
Fukuoka University, FUKUOKA, JAPAN
Camillo
Trapani
Università degli Studi di Palermo, PALERMO, ITALY
Partial *-algebras, quasi *-algebras, unbounded $C$*-seminorms, (unbounded) *- representations
The main purpose of this paper is to construct *-representations from unbounded $C$*-seminorms on partial *-algebras and to investigate their *-representations.
Functional analysis
General
295
314
10.4171/ZAA/1017
http://www.ems-ph.org/doi/10.4171/ZAA/1017
Partial Regularity of Weak Solutions to Nonlinear Elliptic Systems Satisfying a Dini Condition
Jörg
Wolf
Humboldt-Universität zu Berlin, BERLIN, GERMANY
Nonlinear elliptic systems, partial regularity, blow-up method
This paper is concerned with systems of nonlinear partial differential equations $$–D_{\alpha}a^{\alpha}_i (x, u, \bigtriangledown u) = b_i (x, u, \bigtriangledown u) (i = 1,..., N)$$ where the coefficients $a^{\alpha}_i$ are assumed to satisfy the condition $$a^{\alpha}_i (x, u, \xi) – a^{\alpha}_i (y, v, \xi) ≤ \omega |x–y| + |u–v| (1+|\xi|)$$ for all ${x, u}, {y, v} \in \Omega \times \mathbb R^N$ and all $\xi \in \mathbb R^{nN}$, and where $\int^1_0 \frac {\omega (t)}{t} dt < + \infty$ while the functions $\frac {\partial a_i^{\alpha}}{\partial \xi ^j_{\beta}}$ satisfy the standard boundedness and ellipticity conditions and the function $\xi \mapsto b_i (x, u, \xi)$ may have quadratic growth. With these assumptions we prove partial Hölder continuity of bounded weak solutions $u$ to the above system provided the usual smallness condition on $\|u \|_{L \infty ({\Omega})}$ is fulfilled.
Partial differential equations
General
315
330
10.4171/ZAA/1018
http://www.ems-ph.org/doi/10.4171/ZAA/1018
Nonlinear Diffusion Equations on Bounded Fractal Domains
Jiaxin
Hu
Tsinghua University, BEIJING, CHINA
Diffusion equations, fractals, Laplacian, Sobolev-type inequality, heat kernel, iteration scheme, maximum principle
We investigate nonlinear diffusion equations $\frac {\partial u}{\partial t} = \Delta ¢u +f(u)$ with initial data and zero boundary conditions on bounded fractal domains. We show that these equations possess global solutions for suitable $f$ and small initial data by employing the iteration scheme and the maximum principle that we establish on the bounded fractals under consideration. The Sobolev-type inequality is the starting point of this work, which holds true on a large class of bounded fractal domains and gives rise to an eigenfunction expansion of the heat kernel.
Partial differential equations
Measure and integration
General
331
345
10.4171/ZAA/1019
http://www.ems-ph.org/doi/10.4171/ZAA/1019
Convergence Rates for a Reaction-Diffusion System
Mokhtar
Kirane
Lab. et Dép. de Mathématiques, LA ROCHELLE CEDEX 1, FRANCE
Nasser-edine
Tatar
Dept. of Mathematics, DHAHRAN, SAUDI ARABIA
Reaction-diffusion systems, asymptotic behavior, biomathematics
A class of reaction-diffusion systems is investigated. This class is motivated by some diffusive epidemic models, which serve to modelise the spread of Feline Immunodeficiency Virus (FIV) in the cat population, and sexually transmitted diseases. We obtain exponential convergence rates for a system with unbounded time dependent coefficients.
Partial differential equations
Biology and other natural sciences
General
347
357
10.4171/ZAA/1020
http://www.ems-ph.org/doi/10.4171/ZAA/1020
Hyperbolic Limit of Parabolic Semilinear Heat Equations with Fading Memory
V.
Pata
Politecnico di Milano, MILANO, ITALY
Heat equation, materials with memory, non-autonomous dynamical systems, uniform absorbing sets, uniform attractors, Hausdorff semidistance, upper semicontinuity of a family of attractors
This paper is devoted to the comparison of two models describing heat conduction with memory, arising in the frameworks of Coleman-Gurtin and Gurtin-Pipkin. In particular, the second model entails an equation of hyperbolic type, where the dissipation is carried out by the memory term solely, and can be viewed as the limit of the first model as the coefficient $\omega$ of the laplacian of the temperature tends to zero. Results concerning the asymptotic behavior, with emphasis on the existence of a uniform attractor, are provided, uniformly in $\omega$. The attractor of the hyperbolic model is shown to be upper semicontinuous with respect to the family of attractors of the parabolic models, as $\omega$ tends to zero.
Partial differential equations
Integral equations
Classical thermodynamics, heat transfer
General
359
377
10.4171/ZAA/1021
http://www.ems-ph.org/doi/10.4171/ZAA/1021
On the Asymptotic Behaviour of the Integral $$\int^{\infty}_0 e^{itx} (\frac {1}{x^{\alpha}} – \frac {1}{[x^{\alpha}]+1} dx (t \rightarrow 0)$$ and Rates of Convergence to $\alpha$-Stable Limit Laws
Lothar
Heinrich
Universität Augsburg, AUGSBURG, GERMANY
Fourier-Stieltjes transform, normal domain of attraction, $\alpha$-stable distribution, exponential sums, Tauberian theorem, Fourier integrals, method of stationary phase
Approximations and expansions
Number theory
Sequences, series, summability
Probability theory and stochastic processes
379
394
10.4171/ZAA/1022
http://www.ems-ph.org/doi/10.4171/ZAA/1022
On $S$-Homogenization of an Optimal Control Problem with Control and State Constraints
Peter
Kogut
Dnipropetrovsk National University, DNIPROPETROVSK, UKRAINE
Günter
Leugering
Universität Erlangen-Nürnberg, ERLANGEN, GERMANY
Homogenization, S-convergence, optimal control
We study the limiting behavior of an optimal control problem for a linear elliptic equation subject to control and state constraints. Each constituent of the mathematical description of such an optimal control problem may depend on a small parameter $\epsilon$. We study the limit of this problem when $\epsilon \rightarrow 0$ in the framework of variational $S$-convergence which generalizes the concept of $\Gamma$-convergence. We also introduce the notion of $G*$-convergence generalizing the concept of $G$-convergence to operators with constraints. We show convergence of the sequence of optimal control problems and identify its limit. We then apply the theory to an elliptic problem on a perforated domain.
Partial differential equations
Calculus of variations and optimal control; optimization
General
395
429
10.4171/ZAA/1023
http://www.ems-ph.org/doi/10.4171/ZAA/1023
Regularity Results for Laplace Interface Problems in Two Dimensions
Martin
Petzoldt
Angewandte Analysis und Stochastik, BERLIN, GERMANY
Elliptic equations, regularity of solutions, interface and transmission problems, singularities, discontinuous diffusion coefficients, Sturm-Liouville eigenvalue problems
We investigate the regularity of solutions of interface problems for the Laplacian in two dimensions. Our objective are regularity results which are independent of global bounds of the data (the diffusion). Therefore we use a restriction on the data, the quasi-monotonicity condition, which we show to be sufficient and necessary to provide $H^{1+\frac {1}{4}}$-regularity. In the proof we use estimates of eigenvalues of a related Sturm-Liouville eigenvalue problem. Additionally we state regularity results depending on the data.
Partial differential equations
Ordinary differential equations
General
431
455
10.4171/ZAA/1024
http://www.ems-ph.org/doi/10.4171/ZAA/1024
Eigenfunctions of Two-Scale Difference Equations and Appell Polynomials
Lothar
Berg
Universität Rostock, ROSTOCK, GERMANY
Manfred
Krüppel
Universität Rostock, ROSTOCK, GERMANY
Two-scale difference equations, distributional solutions, eigenvalue problems, Appell polynomials, basic functional equation, equivalent and reversed eigenfunctions, minimal characteristic polynomials, sums of shifted eigenfunctions
Both classical and distributional solutions of two-scale difference equations are interpreted as eigenfunctions, which are closely connected with Appell polynomials. Different generating functions are analyzed and the relations between them. Equivalent eigenfunctions as well as equivalent and minimal characteristic polynomials are defined and investigated in detail via the rational solution of a basic functional equation. Finally, reversed eigenfunctions are introduced and characterized.
Difference and functional equations
Real functions
Functional analysis
Operator theory
457
488
10.4171/ZAA/1025
http://www.ems-ph.org/doi/10.4171/ZAA/1025
On Oscillation of Equations with Distributed Delay
L.
Berezansky
Ben Gurion University of the Negev, BEER-SHEBA, ISRAEL
Elena
Braverman
Technion - Israel Institute of Technology, HAIFA, ISRAEL
Oscillation, non-oscillation, distributed delay, comparison theorems
For the scalar delay differential equation with a distributed delay $$\dot{x} (t) + \int ^t_{– \infty} x(s)d_sR(t,s) = f(t) (t > t_o)$$ a connection between the properties non-oscillation positiveness of the fundamental function existence of a non-negative solution for a certain nonlinear integral inequality is established. This enables to obtain comparison theorems and explicit non-oscillation and oscillation conditions being generalizations of some known results for delay equations and integro-differential equations and leads to oscillation results for equations with infinite number of delays.
Ordinary differential equations
General
489
504
10.4171/ZAA/1026
http://www.ems-ph.org/doi/10.4171/ZAA/1026
The Generalized Riemann Problem of Linear Conjugation for Polyanalytic Functions of Order $n$ in $W_{n,p}(D)$
Ali Seif
Mshimba
University of Dar es Salaam, DAR ES SALAAM, TANZANIA
Polyanalytic functions, Riemann problem of linear conjugation
We consider a homogeneous polyanalytic differential equation of order $n$ in a simply-connected domain $D$ with a smooth boundary $\partial D$ in the complex plane $\mathbb C$. We pose and then prove solvability of a generalized Riemann problem of linear conjugation to the differential equation. This is done by reducing the problem into $n$ classical Riemann problems of linear conjugation for holomorphic functions, the solution of which is available in the literature.
Functions of a complex variable
Partial differential equations
General
505
512
10.4171/ZAA/1027
http://www.ems-ph.org/doi/10.4171/ZAA/1027
The Generalized Riemann Problem of Linear Conjugation for Non-Homogeneous Polyanalytic Equations of Order $n$ in $W_{n,p}(D)$
Ali Seif
Mshimba
University of Dar es Salaam, DAR ES SALAAM, TANZANIA
Polyanalytic functions, generalized Cauchy-Pompeiu integral operators of higher order, Riemann problem of linear conjugation
We consider a non-homogeneous polyanalytic partial differential equation of order $n$ in a simply-connected domain $D$ with smooth boundary $\partial D$ in the complex plane $\mathbb C$. Initially we transform the given equation into an equivalent system of integro-differential equations and then find the general solution of the former in $W_{n,p} (D)$. Next we pose and prove the solvability of a generalized Riemann problem of linear conjugation to the differential equation. This is effected by first reducing the Riemann problem to a corresponding one for a polyanalytic function. The latter is solved by first transforming it into $n$ classical Riemann problems of linear conjugation for $n$ holomorphic functions expressed in terms of the analytic functions which define the polyanalytic function. The solution of the classical Riemann problem is available in the literature.
Functions of a complex variable
Partial differential equations
Operator theory
General
513
524
10.4171/ZAA/1028
http://www.ems-ph.org/doi/10.4171/ZAA/1028
An Interpolation Problem for Hilbert-Schmidt Operator-Valued Stationary Processes
L.
Klotz
Universität Leipzig, LEIPZIG, GERMANY
Hilbert-Schmidt operator-valued stationary processes, linear interpolation
The paper contains a solution of the following interpolation problem for Hilbert-Schmidt operator-valued stationary processes on the real line: Assume that the values of the process on the integers are known. Determine the best linear approximation of an unknown value on the basis of the known values and compute the approximation error. Our results generalize previous results of Yaglom and Salehi for univariate and $q$-variate processes, respectively.
Probability theory and stochastic processes
Operator theory
General
525
535
10.4171/ZAA/1029
http://www.ems-ph.org/doi/10.4171/ZAA/1029
3
On the Three “Essential” Critical Values Theorem
Berardino
Sciunzi
Università di Roma Tor Vergata, ROMA, ITALY
Essential critical values, deformation lemma, Palais-Smale condition
Global methods of the calculus of variations and the infinite dimensional critical point theory of Ljusternik and Schnirelmann are applied to give results on the existence of so-called critical values and essential critical values. The case of continuous, not necessarily differentiable functionals is considered and studied introducing a suitable variant of the Palais-Smale condition.
Global analysis, analysis on manifolds
General
553
563
10.4171/ZAA/1030
http://www.ems-ph.org/doi/10.4171/ZAA/1030
A Semilinear Furi-Martelli-Vignoli Spectrum
Jürgen
Appell
Universität Würzburg, WÜRZBURG, GERMANY
E.
De Pascale
Università della Calabria, ARCAVACATA DI RENDE (CS), ITALY
A.
Vignoli
Università di Roma 'Tor Vergata', ROMA, ITALY
Fredholm operator, continuous nonlinear operator, quasibounded nonlinear operator, measure of non-compactness, $\alpha$-contractive nonlinear operator, nonlinear spectrum, semilinear spectrum, coincidence degree, periodic solution
In this note we extend a spectrum which was introduced by Furi, Martelli and the third author in 1978 for continuous nonlinear maps $F$ to a certain new spectrum for a “semilinear pair” ($L, F$), with $L$ being a linear Fredholm operator of index zero, and $F$ being nonlinear and continuous.
Operator theory
Ordinary differential equations
General
565
577
10.4171/ZAA/1031
http://www.ems-ph.org/doi/10.4171/ZAA/1031
On Fourier Transforms of Wavelet Packets
R.
Kumar
Jamia Millia Islama, NEW DELHI, INDIA
K.
Ahmad
Jamia Millia Islama, NEW DELHI, INDIA
L.
Debnath
University of Central Florida, ORLANDO, UNITED STATES
Wavelet packets, multi-resolution analysis, Fourier transform, quadrature mirror filter
This paper deals with the Fourier transform $\hat{\omega}_n$ of wavelet packets $\omega_n \in L^2 (\mathbb R)$ relative to the scaling function $\varphi = \omega_0$. Included there are proofs of the following statements: (i) $\hat{\omega}_n (0)$) = 0 for all $n \in \mathbb N$. (ii) $\hat{\omega}_n (4nk\pi) = 0$ for all $k \in \mathbb Z, n = 2j$ for some $j \in \mathbb N_0$, provided $|\hat{\varphi}|, |m_0|$ are continuous. (iii)$|\hat{\omega}_n (\xi)|^2 = \sum^{2^r–1}_{s=0} |\hat{\omega}_{2^r n+s} (2^r \xi)|^2$ for $r 2\in \mathbb N$. (iv) $\sum^\infty_{j=1} \sum^{2^r–1}_{s=0} \sum_{k \in \mathbb Z} |\hat{\omega}_n (2^{j+r} (\xi + 2k\pi))|^2 = 1$ for a.a. $\xi \in \mathbb R$ where $r = 1, 2,...,j$. Moreover, several theorems including a result on quadrature mirror filter are proved by using the Fourier transform of wavelet packets.
Fourier analysis
Difference and functional equations
Approximations and expansions
General
579
588
10.4171/ZAA/1032
http://www.ems-ph.org/doi/10.4171/ZAA/1032
A Note on Degenerate Variational Problems with Linear Growth
M.
Bildhauer
Universität des Saarlandes, SAARBRÜCKEN, GERMANY
Degenerate problems, linear growth, duality, regularity
Given a class of strictly convex and smooth integrands $f$ with linear growth, we consider the minimization problem $\int_{\Omega} f \bigtriangledown u) dx \rightarrow$ min and the dual problem with maximizer $\sigma$. Although degenerate problems are studied, the validity of the classical duality relation is proved in the following sense: there exists a generalized minimizer $u* \in BV (\Omega; \mathbb R^N)$ of the original problem such that $\sigma (x) = \bigtriangledown f (\bigtriangledown^a u*)$ holds almost everywhere, where $\bigtriangledown^a u*$ denotes the absolutely continuous part of $\bigtriangledown u*$ with respect to the Lebesgue measure. In particular, this relation is also true in regions of degeneracy, i.e. at points $x$ such that $D^2f(\bigtriangledown ^a u*(x)) = 0$. As an application, we can improve the known regularity results for the dual solution.
Calculus of variations and optimal control; optimization
General
589
598
10.4171/ZAA/1033
http://www.ems-ph.org/doi/10.4171/ZAA/1033
Entropy Solution for a Hyperbolic Equation
Sévérine
Bernard
Université des Antilles et de la Guyane, POINTE-A-PITRE, GUADELOUPE (FRENCH)
Conservation laws, discontinuous coefficients, product of distributions, entropy solutions
Nonlinear hyperbolic systems of conservation laws play a central role in Science and Engineering, and their mathematical theory as well as their numerical approximation have made recent significative progress. This paper deals with the existence and uniqueness of an entropy solution of the Cauchy problem for the quasi-linear equation $u-t + a(f(u))_x = 0$ in one space dimension, where $a$ is a non-smooth coefficient.
Partial differential equations
General
599
615
10.4171/ZAA/1034
http://www.ems-ph.org/doi/10.4171/ZAA/1034
$C^{1,\alpha}$ Local Regularity for the Solutions of the $p$-Laplacian on the Heisenberg Group for $2≤p
Silvana
Marchi
Università di Parma, PARMA, ITALY
Degenerate elliptic equations, weak solutions, regularity of solutions, higher differentiability
We prove local Hölder continuity of the homogeneous gradient for weak solutions $u \in W^{1,p}_{loc}$ of the $p$-Laplacian on the Heisenberg group $\mathbb H^n$ for $2≤p
Partial differential equations
General
617
636
10.4171/ZAA/1035
http://www.ems-ph.org/doi/10.4171/ZAA/1035
Semilinear Hyperbolic Systems with Singular Non-Local Boundary Conditions: Reflection of Singularities and Delta Waves
Irina
Kmit
Mechanics and Mathematics, LVIV, UKRAINE
Günther
Hörmann
Universität Wien, WIEN, AUSTRIA
Semilinear hyperbolic equations, Colombeau algebras, non-local boundary conditions, delta waves
In this paper we study initial-boundary value problems for first-order semilinear hyperbolic systems where the boundary conditions are non-local. We focus on situations involving strong singularities, of the Dirac delta type, in the initial data as well as in the boundary conditions. In such cases we prove an existence and uniqueness result in an algebra of generalized functions. Furthermore, we investigate the existence and structure of delta waves, i.e., distributional limits of solutions to the regularized systems. Due to the additional singularities in the boundary data the search for delta waves requires a delicate splitting of the solution into a linearly evolving singular part and a regular part satisfying a nonlinear equation. A new feature in the splitting procedure used here, compared to delta waves in pure initial value problems, is the dependence of the singular part also on part of the regular part due to singularities enetering from the boundary. Finally, we include simple examples where the existence of delta waves breaks down.
Functional analysis
General
637
659
10.4171/ZAA/1036
http://www.ems-ph.org/doi/10.4171/ZAA/1036
Wave Solutions to Reaction-Diffusion Systems in Perforated Domains
S.
Heinze
Mathematik in den Naturwissenschaften, LEIPZIG, GERMANY
Reaction-Diffusion equations, front propagation, homogenization
Traveling waves in periodically perforated domains are shown to exist for large classes of reaction-diffusion systems, provided the homogenized equation admits a non-degenerate traveling wave. This can be applied e.g. to a single equation with bistable non-linearity and to bistable monotone systems. The proof uses the implicit function theorem of a suitably transformed problem in the space $H^1$. Furthermore, corrector-type estimates are given for the wave profile and the wave velocity.
Partial differential equations
General
661
676
10.4171/ZAA/1037
http://www.ems-ph.org/doi/10.4171/ZAA/1037
On the Cauchy Problem for a Degenerate Parabolic Equation
Michael
Winkler
Universität Paderborn, PADERBORN, GERMANY
Degenerate diffusion, large-time behaviour
Existence and uniqueness of global positive solutions to the degenerate parabolic problem $$u_t = f(u)\Delta u \ \ \mathrm {in} \ \ \mathbb R^n \times (0, \infty)$$ $$u|_{t=0} = u–0$$ with $f \in C^0 ((0, \infty)) \cap C^1 ((0, \infty))$ satisfying $f(0) = 0$ and $f(s) > 0$ for $s > 0$ are investigated. It is proved that, without any further conditions on $f$, decay of $u_0$ in space implies uniform zero convergence of $u(t)$ as $t \rightarrow \infty$. Furthermore, for a certain class of functions $f$ explicit decay rates are established.
Partial differential equations
General
677
690
10.4171/ZAA/1038
http://www.ems-ph.org/doi/10.4171/ZAA/1038
Parametric Weighted Integral Inequalities for A-Harmonic Tensors
Shuseng
Ding
Seattle University, SEATTLE, UNITED STATES
$A_r$-weights, inequalities, $A$-harmonic equation, differential forms
We prove the $A_r(\Omega$)-weighted Hardy-Littlewood inequality, the $A_r(\Omega$)-weighted weak reverse Hölder inequality and the $A_r(\Omega$)-weighted Caccioppoli-type estimate for $A$-harmonic tensors all being generalizations of classical results.
Potential theory
Real functions
Partial differential equations
Global analysis, analysis on manifolds
691
708
10.4171/ZAA/1039
http://www.ems-ph.org/doi/10.4171/ZAA/1039
Floquet Boundary Value Problems for Differential Inclusions: a Bound Sets Approach
J.
Andres
Palacky University, OLOMOUC-HEIJIN, CZECH REPUBLIC
Luisa
Malaguti
Università di Modena e Reggio Emilia, MODENA, ITALY
Valentina
Taddei
Università di Modena e Reggio Emilia, MODENA, ITALY
Floquet problems, bound sets, differential inclusions, viability arguments, existence results
A technique is developed for the solvability of the Floquet boundary value problem associated to a differential inclusion. It is based on the usage of a not necessarily $C^1$-class of Liapunov-like bounding functions. Certain viability arguments are applied for this aim. Some illustrating examples are supplied.
Ordinary differential equations
General
709
725
10.4171/ZAA/1040
http://www.ems-ph.org/doi/10.4171/ZAA/1040
Quadratic Forms and Nonlinear Non-Resonant Singular Second Order Boundary Value Problems of Limit Circle Type
R.P.
Agarwal
National University of Singapore, SINGAPORE, SINGAPORE
Donal
O'Regan
National University of Ireland, GALWAY, IRELAND
V.
Lakshmikantham
Florida Institute of Technology, MELBOURNE, UNITED STATES
Singular and non-resonant problems, points of limit circle type, existence criteria for solutions
New existence results are presented for non-resonant second order singular boundary value problems $$\frac {1}{p(t)}(p(t)y'(t))' + \tau (t)y(t) = \lambda f(t,y(t)) \ \ \mathrm {a.e. on \ \ [0,1]}$$ $$\mathrm {lim}_{t\to 0^+} p(t)y'(t) = y (1) = 0$$ where one of the endpoints is regular and the other may be singular or of limit circle type.
Ordinary differential equations
General
727
737
10.4171/ZAA/1041
http://www.ems-ph.org/doi/10.4171/ZAA/1041
The Upper and Lower Functions Method for Second Order Systems
A. Ja.
Lepin
University of Latvia, RIGA, LATVIA
Felix
Sadyrbaev
University of Latvia, RIGA, LATVIA
Monotone iterative techniques, upper and lower functions, maximal and minimal solutions, second order systems
Two-point boundary value problems for $m$-dimensional second order systems are considered. The method of upper and lower functions is applied to problems of the Dirichlet type and problems with nonlinear boundary conditions. The conditions on upper and lower functions are substantially relaxed comparing with the classical $C^2$-class and properties of them are studied for systems with monotone in $x$ right sides. Consequences for even order differential equations with mixed monotonicities are given.
Ordinary differential equations
General
739
753
10.4171/ZAA/1042
http://www.ems-ph.org/doi/10.4171/ZAA/1042
Crack Detection in Plane Semilinear Elasticity
Dang Duc
Trong
National University, HOCHIMINH CITY, VIETNAM
Crack detection, Lamé coefficient, plane stress, semilinear elastic body
Let $\Omega$ be a two-dimensional semilinear elastic body limited by a known outer boundary $\Gamma$ represented by a Jordan curve and an unknown inner boundary $\gamma$ represented by a finite disjoint union of piecewise $C^1$ Jordan curves. Plane stress is considered. We assume that the Lamé coefficient depends on the spacial variables $x, y$ and the displacements $u, v$. Our main result asserts that $\gamma$ is uniquely determined by the displacements and stresses prescribed on an open portion $\Gamma_0$ of $\Gamma$.
Partial differential equations
General
755
760
10.4171/ZAA/1043
http://www.ems-ph.org/doi/10.4171/ZAA/1043
Orthogonality and Completeness of $q$-Fourier Type Systems
Mourad
Ismail
University of Central Florida, ORLANDO, UNITED STATES
Continuous $q$-ultraspherical polynomials, $q$-exponential functions, orthogonality, completeness, discrete orthogonal polynomials, dual orthogonality
We establish orthogonality and completeness of the system of $q$-exponential functions {$\mathcal E_q(\cdot; i\omega_n)$} using orthogonality and dual orthogonality of a $q$-analogue of Lommel polynomials. We also set up a very general procedure by which one can produce similar orthogonal systems using bilinear generating functions formed by products of two complete orthogonal function systems.
Fourier analysis
Special functions
General
761
775
10.4171/ZAA/1044
http://www.ems-ph.org/doi/10.4171/ZAA/1044
Some Distributional Products of Mikusiński Type in the Colombeau Algebra $\mathcal G(R^m)$
B.
Damyanov
Bulgarian Academy of Sciences, SOFIA, BULGARIA
Schwartz distributions, Colombeau generalized functions, multiplication of distributions
Particular products of Schwartz distributions on the Euclidean space $\mathbb R^m$ are derived when the latter have coinciding point singularities and the products are ’balanced’ so that their sum to give an ordinary distribution. These products follow the pattern of a known distributional product published by Jan Mikusiński in 1966. The results are obtained in the Colombeau algebra $\mathcal G (\mathbb R^m)$ of generalized functions. $\mathcal G (\mathbb R^m)$ is a relevant algebraic construction, with the distribution space linearly embedded, which by the notion of ’association’ allows the results to be evaluated on the level of distributions.
Functional analysis
General
777
785
10.4171/ZAA/1045
http://www.ems-ph.org/doi/10.4171/ZAA/1045
4
Linear Combinations of Frames and Frame Packets
Ole
Christensen
Technical University of Denmark, LYNGBY, DENMARK
Frames, frame packets, Gabor frames
We find coefficients $c_{mn} (m, n \in \mathbb Z)$ such that for an arbitrary frame $\lbrace f_n \rbrace_{n \in \mathbb Z}$ the set of vectors $\lbrace \phi_m \rbrace_{m \in \mathbb Z} = \lbrace \sum _{n \in \mathbb Z} c_{mn}f_n \rbrace_{m \in \mathbb Z}$ will again be a frame. Appropriate coefficients can always be chosen as function values $c_{mn} = g(\frac {n}{\beta} – m\alpha)$), where $g$ belongs to a broad class of functions generating a Gabor frame $\lbrace E_{\beta m} T_{\alpha n}g \rbrace_{m, n \in \mathbb Z}$ for $L^2(\mathbb R)$. We also prove a version of the splitting trick, which allows to construct a large family of frames based on a single (wavelet or Gabor) frame.
Fourier analysis
General
805
815
10.4171/ZAA/1046
http://www.ems-ph.org/doi/10.4171/ZAA/1046
Lipschitz Continuity of Polyhedral Skorokhod Maps
Pavel
Krejcí
Academy of Sciences, PRAHA, CZECH REPUBLIC
A.A.
Vladimirov
Institute for Information Transmission Problems, Moscow, RUSSIAN FEDERATION
Polyhedral Skorokhod problem, oblique reflections, Lipschitz continuity
We show that a special stability condition of the associated system of oblique projections (the so-called $\ell$-paracontractivity) guarantees that the corresponding polyhedral Skorokhod problem in a Hilbert space $X$ is solvable in the space of absolutely continuous functions with values in $X$. If moreover the oblique projections are transversal, the solution exists and is unique for each continuous input and the Skorokhod map is Lipschitz continuous in both spaces $C([0, T];X)$ and $W^{1,1} (0, T; X)$. Also, an explicit upper bound for the Lipschitz constant is derived.
Ordinary differential equations
Operator theory
Convex and discrete geometry
Operations research, mathematical programming
817
844
10.4171/ZAA/1047
http://www.ems-ph.org/doi/10.4171/ZAA/1047
Existence and Regularity Results for Non-Negative Solutions of some Semilinear Elliptic Variational Inequalities via Mountain Pass Techniques
Mario
Girardi
Università degli studi Roma Tre, ROMA, ITALY
Loretta
Mastroeni
Università degli Studi di Roma Tre, ROMA, ITALY
Michele
Matzeu
Università di Roma Tor Vergata, ROMA, ITALY
Semilinear variational inequalities, penalization method, Mountain-Pass theorem
The main result stated in the present paper is the existence of a non-negative solution for a semilinear variational inequality through the use of some estimates for the Mountain-Pass critical points obtained for the penalized equations associated with the variational inequality. The positivity of the solution is achieved through a regularity result and the strong maximum principle.
Calculus of variations and optimal control; optimization
Partial differential equations
Global analysis, analysis on manifolds
General
845
857
10.4171/ZAA/1048
http://www.ems-ph.org/doi/10.4171/ZAA/1048
Free Boundary Problem for a One-Dimensional Transport Equation
Christina
Kuttler
Universität Tübingen, TÜBINGEN, GERMANY
Telegraph and transport equations, free boundary, correlated random walk
For a linear transport equation in one space dimension with speeds in a compact interval and a general symmetric kernel for the change of velocity a problem with free boundary (Stefan problem) is stated. The case of constant speed corresponds to a Stefan problem for the damped wave equation (telegraph equation). Existence and uniqueness of the free boundary is shown, and the connection to the classical Stefan problem (parabolic limit) is exhibited.
Partial differential equations
Classical thermodynamics, heat transfer
General
859
881
10.4171/ZAA/1049
http://www.ems-ph.org/doi/10.4171/ZAA/1049
Automatic Control of the Temperature in Phase Change Problems with Memory
S.
Gatti
Università di Ferrara, FERRARA, ITALY
Parabolic Stefan problem, Heat conduction with memory, thermostats control, hysteresis operator of Preisach type, existence, uniqueness
We study a parabolic two-phase system with memory occupying a bounded and smooth domain. The heat exchange at part of the boundary is controlled by a thermostat. Assuming on the phase variable either a relaxation dynamics or a Stefan condition, we prove existence and uniqueness results for feedback control problems corresponding to two different types of thermostat: the relay switch and the Preisach operator. These results are strictly related to the continuous dependence of the solution on the boundary datum, which is investigated in advance.
Partial differential equations
Classical thermodynamics, heat transfer
Integral equations
Systems theory; control
883
914
10.4171/ZAA/1050
http://www.ems-ph.org/doi/10.4171/ZAA/1050
Asymptotical Behavior of Solutions of Nonlinear Elliptic Equations in $R^N$
Michèle
Grillot
Université d'Orléans, ORLÉANS CEDEX 2, FRANCE
Philippe
Grillot
Université d'Orléans, ORLÉANS CEDEX 2, FRANCE
Laplacian, non-linearity, asymptotical behavior
In this paper we study the behavior near infinity of non-negative solutions $u \in C^2(\mathbb R^N)$ of the semi-linear elliptic equation $$– \Delta u+u^q – u^p =0$$ where $q \in (0, 1), p > q$ and $N ≥2$. Especially, for a non-negative radial solution of this equation we prove the following alternative: either $u$ has a compact support or $u$ tends to one at infinity. Moreover, we prove that if a general solution is sufficiently small in some sense, then it is compactly supported. To prove this result we use some inequalities between the solution and its spherical average at a shift point and consider a differential inequality. Finally, we prove the existence of non-trivial solutions which converge to one at infinity.
Partial differential equations
General
915
928
10.4171/ZAA/1051
http://www.ems-ph.org/doi/10.4171/ZAA/1051
Boundary Layer Correctors for the Solution of Laplace Equation in a Domain with Oscillating Boundary
Y.
Amirat
Université Blaise Pascal, AUBIÈRE CEDEX, FRANCE
O.
Bodart
Université Blaise Pascal, AUBIÈRE CEDEX, FRANCE
Asymptotic behaviour, oscillating boundary, boundary layers
We study the asymptotic behaviour of the solution of Laplace equation in a domain with very rapidly oscillating boundary. The motivation comes from the study of a longitudinal flow in an infinite horizontal domain bounded at the bottom by a plane wall and at the top by a rugose wall. The rugose wall is a plane covered with periodic asperities which size depends on a small parameter $\epsilon > 0$. The assumption of sharp asperities is made, that is the height of the asperities does not vanish as $\epsilon \to 0$. We prove that, up to an exponentially decreasing error, the solution of Laplace equation can be approximated, outside a layer of width 2$\epsilon$, by a non-oscillating explicit function.
Partial differential equations
General
929
940
10.4171/ZAA/1052
http://www.ems-ph.org/doi/10.4171/ZAA/1052
Hausdorff Convergence and Asymptotic Estimates of the Spectrum of a Perturbed Operator
T.A.
Mel'nyk
Kyiv University, KYIV, UKRAINE
Spectrum, asymptotic estimates, perturbed operators
A family of self-adjoint compact operators $A_{\epsilon} (\epsilon > 0)$ acting in Hilbert spaces $\mathcal H_{\epsilon}$ is considered. The asymptotic behaviour as $\epsilon \to 0$ of eigenvalues and eigenvectors of the operators $A_{\epsilon}$ is studied; the limiting operator $A_0 : \mathcal H_0 \mapsto \mathcal H_0$ is non-compact. Asymptotic estimates of the differences between eigenvalues of $A_{\epsilon}$ and points of the spectrum $\sigma (A_0)$ (both of the discrete spectrum and the essential one) are obtained. Asymptotic estimates for eigenvectors of $A_{\epsilon}$ are also proved.
Operator theory
Partial differential equations
General
941
957
10.4171/ZAA/1053
http://www.ems-ph.org/doi/10.4171/ZAA/1053
A Priori Gradient Bounds and Local $C^{1, \alpha}$-Estimates for (Double) Obstacle Problems under Non-Standard Growth Conditions
M.
Bildhauer
Universität des Saarlandes, SAARBRÜCKEN, GERMANY
Martin
Fuchs
Universität des Saarlandes, SAARBRÜCKEN, GERMANY
Giuseppe
Mingione
Università di Parma, PARMA, ITALY
Non-standard growth, (double) obstacle problems, a priori estimates, regularity of minimizers
We prove local gradient bounds and interior Hölder estimates for the first derivatives of functions $u \in W^1_{1, loc} (\Omega)$ which locally minimize the variational integral $I(u) = \int _{\Omega} f (\bigtriangledown u) dx$ subject to the side condition $\psi _1 ≤ u ≤ \psi_2$. We establish these results for various classes of integrands $f$ with non-standard growth. For example, in the case of smooth $f$ the $(s, \mu, q)$-condition is sufficient. A second class consists of all convex functions $f$ with $(p, q)$-growth.
Calculus of variations and optimal control; optimization
Partial differential equations
General
959
985
10.4171/ZAA/1054
http://www.ems-ph.org/doi/10.4171/ZAA/1054
On Bernis’ Interpolation Inequalities in Multiple Space Dimensions
Günther
Grün
Universität Erlangen-Nünberg, ERLANGEN, GERMANY
Interpolation inequalities, fourth order degenerate parabolic equations, thin films
In spatial dimensions $d < 6$, we derive estimates of the form $$\int_{\Omega} u^{n–4} |\bigtriangledown u|^6 + \int_{\Omega} u^{n–2}|\bigtriangledown u|^2|D^2u|^2 ≤ C \int_{\Omega} u^n|\bigtriangledown \Delta u|^2$$ for functions $u \in H^2 (\Omega)$ with vanishing normal derivatives on the boundary $\partial \Omega$. These inequalities imply that $\int_{\Omega}|\bigtriangledown \Delta u \frac {n+2}{2} |^2$ can be controlled by $\int –{\Omega} u^n| \bigtriangledown \Delta u|^2$. This observation will be a key ingredient for the proof of certain qualitative results – e.g. finite speed of propagation or occurrence of a waiting time phenomenon – for solutions to fourth order degenerate parabolic equations like the thin film equation. Our result generalizes – in a slightly modified way – estimates in one space dimension which were obtained by F. Bernis.
Partial differential equations
Functional analysis
Fluid mechanics
General
987
998
10.4171/ZAA/1055
http://www.ems-ph.org/doi/10.4171/ZAA/1055
$L_1$-Norms of Exponential Sums and the Corresponding Additive Problem
M.Z.
Garaev
Academia Sinica, TAIPEI, TAIWAN
Ka-Lam
Kueh
Academia Sinica, TAIPEI, TAIWAN
$L_1$-norms, exponential sums, cardinality of sumsets
In this note, a new estimate of $L_1$-norm of certain exponential sum is obtained. At the same time, we establish a sharp lower bound for the cardinality of corresponding sumsets. In some cases this lower bound gives the true order of the cardinality.
Number theory
General
999
1006
10.4171/ZAA/1056
http://www.ems-ph.org/doi/10.4171/ZAA/1056
On a New Type of Eisenstein Series in Clifford Analysis
Soeren
Krausshar
Universiteit Gent, GENT, BELGIUM
Eisenstein series, Clifford analysis, Riemann zeta function, multiple divisor sums, permutational products
In this paper we deduce a recursion formula for the partial derivatives of the fundamental solution of the generalized Cauchy-Riemann operator in $\mathbb R^{k+1}$ in terms of permutational products. These functions generalize the classical negative power functions to Clifford analysis. We exploit them to introduce a new generalization of the classical complex analytic Eisenstein series on the half-plane to higher dimensions satisfying the generalized Cauchy-Riemann differential equation. Under function-theoretical and number-theoretical aspects we investigate their Fourier series expansion in which multiple divisor sums and certain generalizations of the Riemann zeta function play a crucial role.
Functions of a complex variable
General
1007
1029
10.4171/ZAA/1057
http://www.ems-ph.org/doi/10.4171/ZAA/1057
Iteration Procedures of Shuttle Iteration Type in Continuous Non-Monotone Problems
D.
Rachinskii
Russian Academy of Sciences, Moscow GSP-4, RUSSIAN FEDERATION
Cone semiordering, monotone operator, robust stable solution, continuous branch of solutions, second order elliptic operator
We suggest and study iteration procedures converging from below and above to robust stable solutions and to robust stable continuous branches of solutions for quasilinear boundary-value problems with continuous non-monotone non-linearities. The iterations are constructed by modifications of the shuttle iteration method, which is used in problems with monotone operators leaving invariant a cone interval.
Ordinary differential equations
Numerical analysis
General
1031
1054
10.4171/ZAA/1058
http://www.ems-ph.org/doi/10.4171/ZAA/1058
Differential-Functional Inequalities for Bounded Vector-Valued Functions
Gerd
Herzog
Karlsruher Institut für Technologie (KIT), KARLSRUHE, GERMANY
Ordered vector spaces, differential-functional inequalities, quasimonotonicity
For the space $\mathbb R^n$ ordered by a cone and some functions $f : \mathbb R^{n+mn} \to \mathbb R^n$ and $h_1, ..., h_m : \mathbb R \to \mathbb R$ we consider differential-functional inequalities of the type $$v'' + cv' + f v,v(_1), ..., v(h_m) ≤ u'' + cu' + f u, u(h_1), ..., u(h_m)$$ and conclude $u ≤ v$ under suitable conditions on $u, v, h_k$ and $f$. The result can be applied to obtain existence and uniqueness results for differential-functional boundary value problems of the form $$u'' + cu' + f u, u(h_1), ..., u(h_m) = q$$ with $u \in C^2 (\mathbb R, \mathbb R^n$ bounded.
Ordinary differential equations
General
1055
1063
10.4171/ZAA/1059
http://www.ems-ph.org/doi/10.4171/ZAA/1059
Existence of Non-Oscillatory Solutions of Second-Order Neutral Delay Difference Equations
Yong
Zhou
Xiangtan University, XIANGTAN, HUNAN, CHINA
Y.
Huang
Xiangtan University, HUNAN, CHINA
Neutral difference equations, non-oscillatory solutions, existence of solutions
In this paper, we consider the second-order neutral delay difference equation with positive and negative coefficients $$\Delta r_n \Delta (x_n + cx_{n–k} + p_{n+1}x_{n+1–l} = 0$$ where $c \in \mathbb R, k ≥ 1$ and $m, l ≥ 0$ are integers, $\lbrace r_n\rbrace^\infty _{n=n0}, \lbrace p_n\rbrace ^\infty _{n=n0}$ and $\lbrace q_n \rbrace ^\infty _{n=n0}$ are sequences of non-negative real numbers. We obtain global results (with respect to $c$) which are some sufficient conditions for the existences of non-oscillatory solutions.
Difference and functional equations
General
1065
1074
10.4171/ZAA/1060
http://www.ems-ph.org/doi/10.4171/ZAA/1060
On Topological Structure of Solution Sets for Delay and Functional-Differential Equations
D.
Bugajewska
Adam Mickiewicz University, POZNAN, POLAND
Aronszajn’s property, global solutions, delay and functional-differential equations
In this paper we characterize the topological structure of global solution sets for classical delay and functional-differential equations in terms of $R_{\delta} sets.
Ordinary differential equations
General
1075
1080
10.4171/ZAA/1061
http://www.ems-ph.org/doi/10.4171/ZAA/1061