bluuebunny/crossref_metadata_2025 huggingface dataset:
On Positive-off-Diagonal Operators on Ordered Normed Spaces
Authors: Kalauch Anke
Published: December 2009

On a normed space X ordered by a cone K we consider a continuous linear operator A from X to X of the following kind: If a positive continuous functional f attains 0 on some positive element x, then f(Ax) is greater or equal to 0. If X is a vector lattice, then such operators can be represented as sI + B, where B is a positive operator, I is the identity and s is a real number. We generalize this assertion for weaker assumptions on X, using the Riesz decomposition property.
Deriving Harmonic Functions in Higher Dimensional Spaces
Authors: Qian Tao , Sommen Franciscus
Published: December 2009

For a harmonic function, by replacing its variables with norms of vectors in some multi-dimensional spaces, we may induce a new function in a higher dimensional space. We show that, after applying to it a certain power of the Laplacian, we obtain a new harmonic function in the higher dimensional space. We show that Poisson and Cauchy kernels and Newton potentials, and even heat kernels are all deducible using this method based on their forms in the lowest dimensional spaces. Fueter's theorem and its generalizations are deducible as well from our results. The latter has been used to singular integral and Fourier multiplier theory on the unit spheres and their Lipschitz perturbations of higher dimensional Euclidean spaces.
Operator-Theoretic Positivstellensätze
Authors: Ambrozie C.-G. , Vasilescu F.-H.
Published: December 2009

We study the structure of positive polynomials with coefficients in an operator algebra as a non-commutative infinite-dimensional analogue of Hilbert's 17-th problem.
Pseudodifferential Operators on R^n with Variable Shifts
Authors: Rabinovich Vladimir S.
Published: December 2009

The aim of the paper is the study of pseudodifferential operators with shifts of the form

Au(x) = \sum_{j=1}^N a_j(x,D)V_{h_j} + \sum_{j=1}^N b_j(x,D)T_{g_j}

where

a_j(x,D) \in OPS_{1,0}^m

and

b_j(x,D) \in OPS_{1,0}^{m-\epsilon} \ \ (\epsilon > 0)

are pseudodifferential operators in the Hörmander classes, and

V_{h_j}

and

T_{g_j}

are shift operators of the form

V_{h_j}u(x) = u(x - h_j), \qquad T_{g_j}u(x) = u(x - g_j(x)), \qquad x \in \mathbb R^n

where

h_j \in \mathbb R^n

and the mappings

g_j: \, \mathbb R^n \to \mathbb R^n

have infinitely differentiable coordinate functions bounded with all their derivatives. We will investigate the Fredholm and semi-Fredholm properties of the operator

A

acting from

H^s(\mathbb R^n)

into

H^{s-m}(\mathbb R^n)

applying the limit operators method.
Solvability of a Plane Elliptic Problem for the Flow in a Channel with a Surface-Piercing Obstacle
Authors: Gatti S. , Pierotti D.
Published: December 2009

Let us consider the three-dimensional problem of steady flow of a heavy ideal fluid past a surface-piercing obstacle in a rectangular channel of constant depth. The flow is parallel at infinity upstream with constant velocity

c

. We discuss an approximate linear problem obtained in the limit of a “flat obstacle”. This is a boundary value problem for the Laplace equation in a three-dimensional unbounded domain, with a second order condition on part of the boundary – the Neumann–Kelvin condition. By a Fourier expansion of the potential function we reduce the three-dimensional problem to a sequence of plane problems for the Fourier coefficients. For every value of the velocity

c

these problems can be described in terms of a two parameter elliptic problem in a strip. We discuss a well-posed formulation of such problem by a special variational approach, relying on some a priori properties of finite energy solutions. As a result, we prove unique solvability for

c\ne c_{m,k}

, where

c_{m,k}

is a known sequence of values depending on the dimensions of the channel and on the limit length of the obstacle. Accordingly, we can prove the existence of a solution of the three-dimensional problem; the related flow has in general a non-trivial wave pattern at infinity downstream. We also investigate the regularity of the solution in a neighborhood of the obstacle. The meaning of the singular values

c_{m,k}

is discussed from the point of view of the nonlinear theory.
Fixed Points for Multi-Valued Mixed Increasing Operators in Ordered Banach Spaces with Applications to Integral Inclusions
Authors: Huang Nan-jing , Fang Ya-ping
Published: December 2009

Some new fixed point and coupled fixed point theorems for multi-valued mixed increasing operators in ordered Banach spaces are presented in this paper. As applications, we prove the existences of solutions for a class of integral inclusions.
Homogenization of Dirichlet Problems with Convex Bounded Constraints on the Gradient
Authors: Champion Thierry , De Pascale Luigi
Published: December 2009

We give a simpler proof of a conjecture of A. Bensoussan, J. L. Lions and G. Papanicolaou on the homogenization of Dirichlet problems with convex bounded constraints on the gradient. Our proof is based on a regularization technique for the constrained functionals as well as on a fine approximation technique.
A Wave Equation with Fractional Damping
Authors: Tatar Nasser-edine
Published: December 2009

We consider a wave equation with an internal damping represented by a fractional derivative of lower order than one. An exponential growth result is proved in presence of a source of polynomial type. This result improves an earlier one where the initial data are supposed to be very large in some norm. A new argument based on a new functional is proposed.
Extreme Points and Strongly Extreme Points in Orlicz Spaces Equipped with the Orlicz Norm
Authors: Cui Yunan , Hudzik Henryk , Pluciennik Ryszard
Published: December 2009

Criteria for extreme points and strongly extreme points of the unit ball in Orlicz spaces with the Orlicz norm are given. These results are applied to a characterization of extreme points of

B(L^1 + L^\infty)

which corresponds to the result obtained by R. Grząślewicz and H. Schaefer [Indag. Math. 3 (1992) 173–178] and H. Schaefer [Arch. Math. 58 (1992) 160–163]. Moreover, we show that every extreme point of

B(L^1 + L^\infty)

is strongly extreme. We also get criteria for extreme points of

B(L^p \cap L^\infty)

,

1 \le p < \infty

, using Theorem 1 and for strongly extreme points of

B(L^p \cap L^\infty)

,

1 \le p <\infty

, applying Theorem 2. Although, criteria for extreme points of

B(L^1 \cap L^\infty)

were known [see H. Hudzik, A. Kamińska and M. Mastyło, Arch. Math. 68 (1997) 159–168], we can easily deduce them from our main results and we can extend those results to establish which among extreme points are strongly extreme. The descriptions of the extreme and strongly extreme points of

B(L^p \cap L^\infty)

,

1 < p < \infty

, are original. Moreover, criteria for extreme points and strongly extreme points of the unit ball in the subspace of finite elements of an Orlicz space are deduced on the basis of our main results.
Reconstructing an Analytic Function Using Truncated Lagrange Polynomials
Authors: Trong Dang Duc , Lien Tran Ngoc
Published: December 2009

Let

U

be the unit disc of the complex plane. We consider the problem of reconstructing a function

f

in the Hardy space

H^2(U)

from values

f(z^{(m)}_n)

, where

\{z^{(m)}_n\}

,

(m \in {\Bbb N}

;

1 \le n \le m)

, is a given point system in

U

. This is an ill-posed problem. The function

f

is approximated by so-called truncated Lagrange polynomials. Necessary and sufficient conditions for the convergence are established and a regularization result is given.
Integral Estimates for the Laplace-Beltrami and Green's Operators Applied to Differential Forms on Manifolds
Authors: Ding Shuseng
Published: December 2009

We obtain A_r(M)-weighted boundedness for compositions of Green's operator and the Laplace-Beltrami operator applied to differential forms on manifolds. As applications, we also prove A_r(M)-weighted Sobolev-Poincare embedding theorems for Green's operator and norm comparison theorems for solutions of the A-harmonic equation on manifolds. These results can be used in developing the Lp theory of differential forms and the Hodge decomposition.
A Class of Partial Integro-Differential Equations with Correlation-Convolution Integral I
Authors: von Wolfersdorf Lothar
Published: December 2009

By means of the iteration method with weighted norms existence and uniqueness theorems are proved for three classes of nonlinear integral equations and first order integro-differential equations in two variables. The quadratic non-linearity is given by the correlation-convolution integral. Existence and uniqueness of the solutions are shown in Lebesgue spaces with mixed norms in rectangle and strip.
Non-Analyticity in Time of Solutions to the KdV Equation
Authors: Łysik Grzegorz
Published: December 2009

It is proved that formal power series solutions to the initial value problem

{\partial_tu = \partial_x^3u+\partial_x(u^2)}

,

{u(0,x)=\varphi(x)}

, with analytic data

\varphi

belong to the Gevrey class

{G^2}

in time. However, if

{\varphi(x)=1/(1+x^2)}

, the formal solution does not belong to the Gevrey class

{G^s}

in time for

{0\le s<2}

, so it is not analytic in time. The proof is based on the estimation of a double sum of products of binomial coefficients.
Classes of Multiplication Operators and Their Limit Operators
Authors: Lindner M.
Published: December 2009

Limit operators have proven to be a device for the study of several properties of an operator including Fredholmness and invertibility at infinity, but also the applicability of approximation methods. For band-dominated operators, the question of existence and structure of their limit operators essentially reduces to the study of multiplication operators and their limit operators, which is the topic of this paper.
Existence, Uniqueness and Data Dependence for the Solutions of some Integro-Differential Equations of Mixed Type in Banach Space
Authors: Muresan V.
Published: December 2009

We study existence, uniqueness and data dependence for the solutions of some integro-differential equations of mixed type in Banach space by using Picard and weakly Picard operators' technique and suitable Bielecki norms.
F-Implicit Complementarity Problems in Banach Spaces
Authors: Huang Nan-jing , Li Jun
Published: December 2009

The F-implicit complementarity problem (F-ICP) and F-implicit variational inequality problem (F-IVIP) are introduced and studied. The equivalence between (F-ICP) and (F-IVIP) is presented under certain assumptions. Furthermore, we derive some new existence theorems of solutions for (F-ICP) and (F-IVIP) by using the Fan–Knaster–Kuratowski–Mazurkiewicz Math. Ann. 142 (1961), 305–310] and Lin's Bull. Austral. Math. Soc. 34 (1986), 107–117] under some suitable assumptions without the monotonicity.
Error Estimates for Parabolic Optimal Control Problems with Control Constraints
Authors: Rösch Arnd
Published: December 2009

An optimal control problem for the 1-d heat equation is investigated with pointwise control constraints. This paper is concerned with the discretization of the control by piecewise linear functions. The connection between the solutions of the discretized problems and the continuous one is investigated. Under an additional assumption on the adjoint state an approximation order σ^3/2 is proved for uniform discretizations. In the general case it is shown that a non-uniform control discretization ensures an approximation of order σ^3/2. Numerical tests confirm the theoretical part.
Identification of Cavities in a Three Dimensional Elastic Body
Authors: Ang Dang Dinh , Trong Dang Duc
Published: December 2009

The authors prove a uniqueness theorem for identifying cavities in a three dimensional elastic body from displacements and stresses measured on a portion of the outer surface. The cavities, finite in number, are assumed to be stress free. The surfaces of the cavities are assumed to be smooth on the complement of a set that is negligible in the sense that its fractal dimension is less than 2.
Corrigendum to: A Wave Equation with Fractional Damping
Authors: Tatar Nasser-edine
Published: December 2009

This is a corrigendum to a previous paper of the author [this journal, 22 (2003) 609–617].
Wiener Algebras of Operators, and Applications to Pseudodifferential Operators
Authors: Rabinovich Vladimir S. , Roch Steffen
Published: December 2009

We introduce a Wiener algebra of operators on

L^2(\mathbb R^N)

which contains, for example, all pseudodifferential operators in the Hörmander class

OPS^0_{0,0}

. A discretization based on the action of the discrete Heisenberg group associates to each operator in this algebra a band-dominated operator in a Wiener algebra of operators on

l^2(\mathbb Z^{2N}, \, L^2(\mathbb R^N))

. The (generalized) Fredholmness of these discretized operators can be expressed by the invertibility of their limit operators. This implies a criterion for the Fredholmness on

L^2(\mathbb R^N)

of pseudodifferential operators in

OPS^0_{0,0}

in terms of their limit operators. Applications to Schrödinger operators with continuous potential and other partial differential operators are given.
Weighted Integrals of Holomorphic Functions on the Polydisc
Authors: Stevic Stevo
Published: December 2009

We show that a holomorphic function on the unit polydisc

U^n

in

{\bf C}^n

belongs to the weighted Bergman space

{\cal A}^p_{\alpha}(U^n)

, when

p\in (0,1],

if and only if all weighted derivations of order

|k|

(with positive orders of derivations) belong to the related weighted Lebesgue space

{\cal L}^p_{\alpha}(U^n).

This result extends Theorem 1.8 by Benke and Chang in their recent paper which appeared in Nagoya Math. J. 159 (2000), 25–43.
Analysis of the Operator Δ^-1div Arising in Magnetic Models
Authors: Praetorius Dirk
Published: December 2009

In the context of micromagnetics the partial differential equation

\text{div}(-\nabla u+\mathbf m)=0\text{ in }\R^d

has to be solved in the entire space for a given magnetization

\mathbf m:\Omega\to\mathbb R^d

and

\Omega\subseteq\mathbb R^d

. For an

L^p

function

\mathbf m

we show that the solution might fail to be in the classical Sobolev space

W^{1,p}(\mathbb R^d)

but has to be in a Beppo-Levi class

W_1^p(\mathbb R^d)

. We prove unique solvability in

W_1^p(\mathbb R^d)

and provide a direct ansatz to obtain

u

via a non-local integral operator

\mathcal L_p

related to the Newtonian potential. A possible discretization to compute

\nabla(\mathcal L_2\mathbf m)

is mentioned, and it is shown how recently established matrix compression techniques using hierarchical matrices can be applied to the full matrix obtained from the discrete operator.
Well-Posedness and Asymptotics for Initial Boundary Value Problems of Linear Relaxation Systems in One Space Variable
Authors: ZHANG SHU-YI , Wang Ya-Guang
Published: December 2009

We study the well-posedness and relaxation limit for the initial boundary value problem of a general linear hyperbolic system with a relaxation term in one space variable. We mainly consider the asymptotic convergence and the boundary layer behavior under the sub-characteristic condition and the stiff Kreiss condition when the relaxation rate goes to zero, which generalizes recent results of Xin and Xu [J. Diff. Eqs. 167 (2000) 388–437] for homogeneous problems to the non-homogeneous case.
Boundary Data Smoothness for Solutions of Three Point Boundary Value Problems for Second Order Ordinary Differential Equations
Authors: Henderson Johnny , Tisdell Christopher C.
Published: December 2009

Under certain conditions, solutions of the boundary value problem,

y'' = f(x,y,y')

,

a < x < b

,

y(x_1) = y_1

,

y(x_3) - y(x_2) = y_2

,

a< x_1 < x_2 < x_3 < b

, are differentiated with respect to the boundary conditions.
The Fundamental Matrix of the System of Linear Micropolar Elasticity
Authors: Gawinecki Jerzy A. , Kirchner Gerhard , Lazuka Jaroslaw
Published: December 2009

We construct the fundamental matrix of the system of partial differential operators governing the motion of linear micropolar elastic media in terms of derivatives of distributions which are convolutions of the fundamental solutions to the wave operator, the Klein-Gordon operator and the micropolar operator.
Global Nonexistence for a Quasilinear Evolution Equation with a Generalized Lewis Function
Authors: Zhou Yong
Published: December 2009

We consider the following quasilinear parabolic equation

a(x,t) u_t-\mathrm{div}\left(|\nabla u|^{m-2} \nabla u \right)=f(u),

where

a(x,t) \geq 0

is a generalized Lewis function. The main result is that the solution blows up in finite time if the initial datum

u(x,0)

possesses suitable positive energy. Moreover, we have a precise estimate for the lifespan of the solution in this case. Blowup of solutions with vanishing initial energy is considered also.
Finite Truncations of Generalized One-Dimensional Discrete Convolution Operators and Asymptotic Behavior of the Spectrum. The Matrix Case
Authors: Simonenko Igor B. , Zabroda Olga N.
Published: December 2009

We study the sequence

\{A_N(a)\}_{N\in\mathbb{N}}

of finite truncations of a generalized discrete convolution operator, which have matrices of the form

A_N(a) \sim \left(a\left(\frac{n}{E(N)}, \frac{k}{E(N)}, n-k\right)\right)_{n,k=1,\ldots,N},

where

a

is some function defined on

[0,+\infty) \times [0,+\infty)

,

E(\cdot)

is defined on

\mathbb{N}

and

E(N) \to \infty

,

\frac{N}{E(N)} \to \infty

as

N \to \infty

. For this sequence we get a generalization of the Szegő limit theorem.
The Schwarz Problem for Polyanalytic Functions
Authors: Begehr Heinrich , Schmersau Dieter
Published: December 2009

The analytic Schwarz kernel function the real part of which is the harmonic Poisson kernel was generalized to a polyanalytic Schwarz kernel in a joint work with G. N. Hile [Rocky Mountain J. Math. 27 (1997) 669–706], see also H. Begehr [Complex Analytic Methods for Partial Differential Equations. An Introductory Text, Singapore: World Scientific 1994]. It is here used to give some higher order Cauchy–Schwarz–Pompeiu representations and to solve the Schwarz problem for the Poisson, the inhomogeneous Bitsadze and the inhomogeneous polyanalytic equation.
Integral Inequalities in Higher Dimensional Spaces
Authors: TSENG SHIOJENN , WANG PEN-CHI , CHEUNG WING-SUM , Chen Chur-Jen
Published: December 2009

Integral inequalities play an important role in many different areas including differential equations, integral equations, variational calculus, etc. In this work, we present some new higher dimensional integral inequalities involving monotonic or convex functions in higher dimensional spaces. These are then applied to solve directly some Calculus of Variations problems for optimal solutions, effectively.
Global Asymptotic Stability of High-Order Delay Hopfield Neural Networks with Time-Varying Coefficients
Authors: WANG PEIGUANG , LIAN HAIRONG , Wu Yonghong
Published: December 2009

The problem of global asymptotic stability of the high-order delay neural networks with time-varying coefficients is investigated. Sufficient conditions are obtained for the existence and global asymptotic stability of the equilibrium of such neural networks by using Brouwer's fixed point theorem and Liapunov method.
Compactness and Sobolev-Poincaré Inequalities for Solutions of Kinetic Equations
Authors: Lecumberry Myriam
Published: December 2009

In this paper, we prove a regularity result on the velocity averages of the solution of a kinetic equation whose data have a Sobolev regularity \vspace{-0.05cm}

W^{s,p}

,

0<s<\frac{1}{p}

,

p\in[1,+\infty)

, in the velocity variable. Namely, the velocity averages have the same Sobolev regularity in the time-space variable.
The Factorization Method for Real Elliptic Problems
Authors: Gebauer Bastian
Published: December 2009

The Factorization Method localizes inclusions inside a body from measurements on its surface. Without a priori knowing the physical parameters inside the inclusions, the points belonging to them can be characterized using the range of an auxiliary operator. The method relies on a range characterization that relates the range of the auxiliary operator to the measurements and is only known for very particular applications. In this work we develop a general framework for the method by considering symmetric and coercive operators between abstract Hilbert spaces. We show that the important range characterization holds if the difference between the inclusions and the background medium satisfies a coerciveness condition which can immediately be translated into a condition on the coefficients of a given real elliptic problem. We demonstrate how several known applications of the Factorization Method are covered by our general results and deduce the range characterization for a new example in linear elasticity.
Nonexistence of Solutions to a Hyperbolic Equation with a Time Fractional Damping
Authors: Kirane Mokhtar , Tatar Nasser-edine
Published: December 2009

We consider the nonlinear hyperbolic equation

u_{tt}-\Delta u+D_{+}^{\alpha }u=h(t,x)\left| u\right| ^{p}

posed in

Q:=(0,\infty )\times \mathbb{R}^{N},

where

D_{+}^{\alpha }u

,

0<\alpha <1

is a time fractional derivative, with given initial position and velocity

u(0,x)=u_{0}(x)

and

u_{t}(0,x)=u_{1}(x).

We find the Fujita's exponent which separates in terms of

p,\alpha

and

N,

the case of global existence from the one of nonexistence of global solutions. Then, we establish sufficient conditions on

u_{1}(x)

and

h(x,t)

assuring non-existence of local solutions.
A Gauss-Bonnet Formula for Metrics with Varying Signature
Authors: Steller Michael
Published: December 2009

A Gauss–Bonnet formula for compact orientable connected Riemannian or Lorentz-ian 2-manifolds is well-known. We investigate singular metrics on 2-manifolds with varying signature. Such metrics are necessarily degenerate at some points of

M

where most of the usual definitions for geometric quantities break down. We prove that under some additional assumptions there is a Gauss–Bonnet formula for compact orientable connected 2-manifolds with a singular metric. Some examples are given.
The Plancherel and Hausdorff–Young Type Theorems for an Index Transformation
Authors: Yakubovich Semyon B.
Published: December 2009

The Plancherel and Hausdorff-Young type theorems are proved for an integral transformation, which is associated with the product of the modified Bessel functions of different arguments. The transform essentially generalizes Lebedev's transformation involving squares of the modified Bessel functions as kernels.
Conditions for Correct Solvability of a Simplest Singular Boundary Value Problem of General Form. I
Authors: Chernyavskaya N.A. , Shuster L.A.
Published: December 2009

We consider the singular boundary value problem

\begin{align*}
-r(x)y'(x)+q(x)y(x)&=f(x),\quad x\in R\\
\lim_{|x|\to\infty}y(x)&=0,
\end{align*}

where

f \in L_p(\mathbb R)

,

p\in[1,\infty]

(

L_\infty(\mathbb R):=C(\mathbb R))

,

r

is a continuous positive function on

\mathbb R

,

0\le q \in L_1^{\mathrm{loc}}

. A solution of this problem is, by definition, any absolutely continuous function

y

satisfying the limit condition and almost everywhere the differential equation. This problem is called correctly solvable in a given space

L_p(\mathbb R)

if for any function

f\in L_p(\mathbb R)

it has a unique solution

y\in L_p(\mathbb R)

and if the following inequality holds with an absolute constant

c_p\in (0,\infty):

\|y\|_{L_p(\mathbb R)}\le c_p\|f\|_{L_p(\mathbb R)},\quad \ f\in L_p(\mathbb R) .

We find minimal requirements for

r

and

q

under which the above problem is correctly solvable in

L_p(\mathbb R)

.
An Existence Result for a Class of Extended Inclusion Problems with Applications to Equilibrium Problems
Authors: Fang Ya-ping , Huang Nan-jing
Published: December 2009

Let

X

be a real reflexive Banach space,

K\subset X

a nonempty, closed and convex set, and

F:K\times K\to 2^X

(the family of all the subsets of

X

) be a multi-valued mapping. In this paper, we consider the following extended inclusion problem: find

x^*\in K

such that

K\subset F(x^*,x^*)

. Under suitable conditions, we establish an existence result for the extended inclusion problem. As applications, we give some existence theorems for equilibrium problems.
Local Growth Envelopes of Besov Spaces of Generalized Smoothness
Authors: Caetano António M. , Farkas Walter
Published: December 2009

The concept of local growth envelope

(\mathcal{E}_{LG}A,u)

of the quasi-normed function space

A

is applied to the Besov spaces of generalized smoothness

B_{p,q}^{\sigma,N}(\mathbb{R}^n)

.
Asymptotic and Pseudo Almost Periodicity of the Convolution Operator and Applications to Differential and Integral Equations
Authors: Bugajewski Dariusz , Diagana Toka , Mahop Crépin M.
Published: December 2009

We examine conditions which do ensure the asymptotic almost periodicity (respectively, pseudo almost periodicity) of the convolution function

f \ast h

of

f

with

h

whenever

f

is asymptotically almost periodic (respectively, pseudo almost periodic) and

h

is a (Lebesgue) measurable function satisfying some additional assumptions. Next we make extensive use of those results to investigate on the asymptotically almost periodic (respectively, pseudo almost periodic) solutions to some differential, functional, and integral equations.
What do we Learn from the Discrepancy Principle?
Authors: Mathé Peter
Published: December 2009

The author analyzes the discrepancy principle when smoothness is given in terms of general source conditions. As it turns out, this framework is particularly well suited to reveal the mechanism under which this principle works. For general source conditions there is no explicit way to compute rates of convergence. Instead arguments must be based on geometric properties. Still this approach allows to generalize previous results. The analysis is accomplished with a result showing why this discrepancy principle inherently has the early saturation for a large class of regularization methods of bounded qualification.
Best Possible Maximum Principles for Fully Nonlinear Elliptic Partial Differential Equations
Authors: Porru G. , Safoui A. , Vernier-Piro S.
Published: December 2009

We investigate a class of equations including generalized Monge–Ampere equations as well as Weingarten equations and prove a maximum principle for suitable functions involving the solution and its gradient. Since the functions which enjoy the maximum principles are constant for special domains, we have a so called best possible maximum principle that can be used to find accurate estimates for the solution of the corresponding Dirichlet problem. For these equations we also give a variational form which may have its own interest.
Lipschitz Stability of Solutions to Some State-Constrained Elliptic Optimal Control Problems
Authors: Griesse Roland
Published: December 2009

In this paper, optimal control problems with pointwise state constraints for linear and semilinear elliptic partial differential equations are studied. The problems are subject to perturbations in the problem data. Lipschitz stability with respect to perturbations of the optimal control and the state and adjoint variables is established initially for linear-quadratic problems. Both the distributed and Neumann boundary control cases are treated. Based on these results, and using an implicit function theorem for generalized equations, Lipschitz stability is also shown for an optimal control problem involving a semilinear elliptic equation.
Colombeau Generalized Functions and Solvability of Differential Operators
Authors: Benmeriem Khaled , Bouzar Chikh
Published: December 2009

The aim of this paper is to prove that the well known non solvable Mizohata type partial differential equations have Colombeau generalized solutions which are distributions if and only if they are solvable in the space of Schwartz distributions. Therefore the Colombeau generalized solvability includes both a new solution concept and new mathematical objects as solutions.
Integral Equations with Diagonal and Boundary Singularities of the Kernel
Authors: Pedas Arvet , Vainikko Gennadi
Published: December 2009

We study the smoothness and the singularities of the solution to Fredholm and Volterra integral equations of the second kind on a bounded interval. The kernel of the integral operator may have diagonal and boundary singularities, information about them is given through certain estimates. The weighted spaces of smooth functions with boundary singularities containing the solution of the integral equation are described. Examples show that the results cannot be improved.
Maximal Characterization of Locally Summable Functions
Authors: Andreano F. , Grande R.
Published: December 2009

We prove a characterization of locally summable functions with bounded Stepanoff norm through the maximal function

M_\phi f(x)=\sup_{t>0} |(f*\phi_t)(x) |,

where

\phi

is a suitable function in the class of Schwartz.
Two-Scale Convergence of First-Order Operators
Authors: Visintin Augusto
Published: December 2009

Nguetseng's notion of
it two-scale convergence
and some of its main properties are first shortly reviewed. The (weak) two-scale limit of the gradient of bounded sequences of

W^{1,p}(\mathbb R^N)

is then studied: if

u_\varepsilon \to u

weakly in

W^{1,p}(\mathbb R^N)

, a sequence

\{u_{1\varepsilon}\}

is constructed such that

u_{1\varepsilon}(x)\to u_1(x,y)

and

\nabla u_\varepsilon(x)\to \nabla u(x) + \nabla_y u_1(x,y)

weakly two-scale. Analogous constructions are introduced for the weak two-scale limit of derivatives in the spaces

W^{1,p}(\mathbb R^N)^N

,

L^2_{\mathrm{rot}}(\mathbb R^3)^3

,

L^2_{\mathrm{div}}(\mathbb R^N)^N

,

L^2_{\mathrm{div}}(\mathbb R^N)^{N^2}

. The application to the two-scale limit of some classical equations of electromagnetism and continuum mechanics is outlined. These results are then applied to the homogenization of quasilinear elliptic equations like

\nabla \!\times\! \big[A(u_\varepsilon(x), x,\frac{x}{\varepsilon}) \!\cdot\! \nabla \!\times\! u_\varepsilon\big] = f

.
Pointwise Inequalities in Variable Sobolev Spaces and Applications
Authors: Almeida Alexandre , Samko Stefan
Published: December 2009

Pointwise estimates for variable exponent Sobolev functions are derived to obtain several results on Sobolev spaces with variable exponent. Hypersingular operators acting in these spaces are considered and the corresponding boundedness and pointwise statements are given over bounded open sets with Lipschitz boundary. Moreover, classical Sobolev embeddings into Hölder spaces are generalized to the variable exponent setting.
A Nonlinear Case of the 1-D Backward Heat Problem: Regularization and Error Estimate
Authors: Trong Dang Duc , Quan Pham Hoang , Khanh Tran Vu , Tuan Nguyen Huy
Published: December 2009

We consider the problem of finding, from the final data

u(x,T)=\varphi(x)

, the temperature function

u(x,t),\ x\in (0,\pi),\ t\in [0,T]

satisfies the following nonlinear system

\begin{align*}
u_t-u_{xx}&= f(x,t,u(x,t)), &\quad &(x,t)\in (0,\pi)\times (0,T) \\
u(0,t)&= u(\pi,t)=0, &\quad &t\in (0,T).
\end{align*}

The nonlinear problem is severely ill-posed. We shall improve the quasi-boundary value method to regularize the problem and to get some error estimates. The approximation solution is calculated by the contraction principle. A numerical experiment is given.
Completely Positive Invariant Conjugate-Bilinear Maps in Partial *-Algebras
Authors: Bagarello Fabio , Inoue A. , Trapani Camillo
Published: December 2009

The notion of completely positive invariant conjugate-bilinear map in a partial *-algebra is introduced and a generalized Stinespring theorem is proven. Applications to the existence of integrable extensions of *-representations of commutative, locally convex quasi*-algebras are also discussed.
Asymptotically Almost Periodic and Almost Periodic Solutions for Partial Neutral Integrodifferential Equations
Authors: Henríquez Hernán R. , Hernández M. Eduardo , dos Santos José C.
Published: December 2009

In this paper we study the existence of \ap\ and \aap\ solutions for a class of partial neutral functional integro-differential equation with unbounded delay.
Multiplicative Perturbation by Contractions and Uniform Stability
Authors: Vieira P. C. M. , Kubrusly C. S.
Published: December 2009

Let

T

be an arbitrary bounded linear transformation of a Hilbert space into itself. We investigate classes of contractions

S

for which the spectral radius

r(ST)

of the product

ST

is less than one. The main result gives a collection of necessary and sufficient conditions for

{r(ST)<1}

when

T

is multiplicatively perturbed by compact contractions

S

. We also give either necessary or sufficient conditions for perturbation by other classes of Hilbert space contractions, such as those that include the symmetries (e.g., involutions, unitary operators, self-adjoint, normal and normaloid contractions) or the orthogonal projections (e.g., nonnegative contractions).
Stochastic Comparison of Solutions of Stochastic Functional Differential Equations
Authors: Luo Jiaowan
Published: December 2009

A stochastic comparison of solutions of nonlinear stochastic functional differential equations with different drift and diffusion coefficients is obtained. Some known results are generalized.
Linear $q$-Difference Equations
Authors: Abu Risha M.H. , Annaby M.H. , Ismail Mourad E. H. , Mansour Z.S.
Published: December 2009

We prove that a linear

q

-difference equation of order
n
has a fundamental set of

n

-linearly independent solutions. A

q

-type Wronskian is derived for the

n

th order case extending the results of Swarttouw–Meijer (1994) in the regular case. Fundamental systems of solutions are constructed for the

n

-th order linear

q

-difference equation with constant coefficients. A basic analog of the method of variation of parameters is established.
Estimates of Approximate Solutions and Well-Posedness in Vector Optimization
Authors: Fang Ya-ping , Huang Nan-jing
Published: December 2009

In this paper we estimate the sizes of approximate solution sets for vector optimization from outside and inside, respectively. In terms of an important scalarization function, we obtain some estimates of approximate solutions for well-posed vector optimization, and some estimates of approximate solutions for well-posed vector optimization under perturbations.
A Sharp Stability Criterion for Soliton-Type Propagating Phase Boundaries in Korteweg's Model
Authors: Zumbrun Kevin
Published: December 2009

Recently, Benzoni–Gavage, Danchin, Descombes, and Jamet have given a sufficient condition for linear and nonlinear stability of solitary wave solutions of Korteweg's model for phase-transitional isentropic gas dynamics in terms of convexity of a certain "moment of instability" with respect to wave speed, which is equivalent to variational stability with respect to the associated Hamiltonian energy under a partial subset of the constraints of motion; they conjecture that this condition is also necessary. Here, we compute a sharp criterion for spectral stability in terms of the second derivative of the Evans function at the origin, and show that it is equivalent to the variational condition obtained by Benzoni–Gavage et al., answering their conjecture in the positive.
On Two-Dimensional Immersions of Prescribed Mean Curvature in $\mathbb R^n$
Authors: Bergner Matthias , Fröhlich Steffen
Published: December 2009

We consider two-dimensional immersions of disc-type in

\mathbb R^n

. We focus on well known classical concepts and study the nonlinear elliptic systems of such mappings. Using an Osserman-type condition we give a-priori estimates of the principle curvatures for graphs with prescribed mean curvature fields and derive a theorem of Bernstein type for minimal graphs.
Weighted Norm Inequality for a Maximal Operator on Homogeneous Space
Authors: Fernandes Iara A. A. , Tozoni Sergio A.
Published: December 2009

Let

X=G/H

be a homogeneous space,

X=X\times [0,\infty)

,

\mu

a doubling measure on

X

induced by a Haar measure on the group

G

,

\beta

a positive measure on

X

and

W

a weight on

X

. Consider the maximal operator given by

{\cal M} f(x,r)=\sup _{s\geq r} \frac{1}{\mu (B(x,s))}\int_{B(x,s)} |f(y)|\,d\mu (y), \quad (x,r)\in X.

In this paper, we obtain, for each

p, q, 1<p\leq q<\infty

, a necessary and sufficient condition for the boundedness of the maximal operator

{\cal M}

from

L^p(X, Wd\mu)

to

L^q( X, d\beta)

. As an application, we obtain a necessary and sufficient condition for the boundedness of the Poisson integral of functions defined on the unit sphere

S^n

of the Euclidian space

\mathbb{R}^{n+1}

, from

L^p(S^n,Wd\sigma)

to

L^q(\mathbb{B}, d\nu)

, where

\sigma

is the Lebesgue measure on

S^n

,

W

is a weight on

S^n

and

\nu

is a positive measure on the unit ball

\mathbb{B}

of

\mathbb{R}^{n+1}

.
On a Singular Perturbation Problem for a Class of Variational Inequalities
Authors: Chipot Michel , García J. C. , Muñiz M.C. , Álvarez-Dios J.A.
Published: December 2009

The goal of this paper is to study the asymptotic behavior of a degenerate singular perturbation problem for a class of variational inequalities depending on a positive parameter

\varepsilon

. We also give an existence and uniqueness result.
Growth Envelopes of Anisotropic Function Spaces
Authors: Moura Susana D. , Neves Júlio S. , Piotrowski Mariusz
Published: December 2009

The present paper is devoted to the study of growth envelopes of anisotropic function spaces. An affirmative answer is given to the question of H. Triebel [Wavelet bases in anisotropic function spaces. In:
Function Spaces, Differential Operators and Nonlinear Analysis
(FSDONA-04). Praha: Math. Inst. Acad. Czech Rep. 2005, pp. 370–387; Conjecture 13], whether the growth envelopes are independent of anisotropy. As an application, related anisotropic Hardy inequalities are presented and we also discuss a connection to some anisotropic fractal sets.
Artificial Boundary Conditions for the Stokes and Navier–Stokes Equations in Domains that are Layer-Like at Infinity
Authors: Nazarov Sergei A. , Specovius-Neugebauer Maria
Published: December 2009

Artificial boundary conditions are presented to approximate solutions to Stokes- and Navier-Stokes problems in domains that are layer-like at infinity. Based on results about existence and asymptotics of the solutions

v^\infty

,

p^\infty

to the problems in the unbounded domain

\Omega

the error

v^\infty-v^R

,

p^\infty-p^R

is estimated in

H^1(\Omega_R)

and

L^2(\Omega_R)

, respectively. Here

v^R

,

p^R

are the approximating solutions on the truncated domain

\Omega_R

, the parameter

R

controls the exhausting of

\Omega

. The artificial boundary conditions involve the Steklov-Poincar\'{e} operator on a circle together with its inverse and thus turn out to be a combination of local and nonlocal boundary operators. Depending on the asymptotic decay of the data of the problems, in the linear case the error vanishes of order

O(R^{-N})

, where

N

can be arbitrarily large.
Error Estimates for the Regularization of Optimal Control Problems with Pointwise Control and State Constraints
Authors: Cherednichenko S. , Rösch Arnd
Published: December 2009

We discuss a linear-quadratic optimal control problem with pointwise control and state constraints. The state constraints are regularized by a Lavrentiev type regularization. The main results of the paper are estimates for the regularization error and the stability with respect to noisy data.
About Solutions of Poisson's Equation with Transition Condition in Non-Smooth Domains
Authors: Schlömerkemper Anja
Published: December 2009

Starting from integral representations of solutions of Poisson's equation with transition condition, we study the first and second derivatives of these solutions for all dimensions

d\geq 2

. This involves derivatives of single layer potentials and Newton potentials, which we regularize smoothly. On smooth parts of the boundary of the non-smooth domains under consideration, the convergence of the first derivative of the solution is uniform; this is well known in the literature for regularizations using a sharp cut-off by balls. Close to corners etc.\ we prove convergence in

L^1

with respect to the surface measure. Furthermore we show that the second derivative of the solution is in

L^1

on the bulk.

The interface problem studied in this article is obtained from the stationary Maxwell equations in magnetostatics and was initiated by work on magnetic forces.
Numerical Survey on Hyperbolicity of the Homogenized Equation for Van Der Waals Gas in Eulerian Coordinates
Authors: Oh Myunghyun
Published: December 2009

We describe numerical methods for the location of the zero set of the periodic Evans function

D(\xi, \lambda)

for

\lambda, \xi

sufficiently small, or equivalently the spectrum of a linearized operator

L

with periodic coefficients through the homogenization. We demonstrate these methods for an example system, van der Waals gas in eulerian coordinates. We observe the hyperbolicity of the system. The hyperbolicity is necessary to show asymptotic behavior of a multi-dimensional single periodic wave of systems of conservation laws with viscosity under small perturbation.
Uniform Rectifiability from Mean Curvature Bounds
Authors: Luckhaus Stephan
Published: December 2009

A version of Allard's rectifiability theorem with explicit bounds is given. The condition on the mean curvature would correspond to a bound in Sobolev spaces with fractional negative exponents.
Superposition Operator in a Space of Infinitely Differentiable Functions
Authors: Romeo Mario
Published: December 2009

In this paper we prove a degeneration result for the superposition operator in

V(\mathbb{R}^d)

, a particular space of infinitely differentiable functions which have all derivatives uniformly bounded by a constant that does not depend on the order of derivation.
Some Limit Results on the Haar–Fisz Transform for Inhomogeneous Poisson Signals
Authors: Schmidt Thorsten , Xu Ling
Published: December 2009

One method to estimate the intensity of inhomogeneous Poisson processes, suggested by Fryzlewicz and Nason in [
J. Comput. Graph. Statist.
13 (2004)(3)], is first to preprocess the data using the so-called Haar–Fisz transform and then to apply wavelet methods to the outcome of the first step. For this procedure it is necessary, that the outcomes of the preprocessing step can be approximated by a normal distribution. In this paper we establish the necessary weak convergence results for the case of inhomogeneous Poisson processes which show that the outcome of the preprocessed data can be approximated by Gaussian random variables and wavelet shrinkage with a global threshold may be applied. A small simulation studies the application to shot-noise models. It suggests that this method is able to detect small peaks while at the same time it does not over-smooth large peaks in comparison with kernel estimators or standard wavelet estimators.
$W^{2,p}$- and $W^{1,p}$-Estimates at the Boundary for Solutions of Fully Nonlinear, Uniformly Elliptic Equations
Authors: Winter Niki
Published: December 2009

In this paper we extend Caffarelli's result on interior

W^{2,p}

-estimates for viscosity solutions of uniformly elliptic equations and prove

W^{2,p}

-estimates at a flat boundary. Moreover we extend a result of A. Świech and prove

W^{1,p}

-estimates at the boundary. Thereafter we combine these results and prove global

W^{2,p}

-estimates for equations with dependence on

Du

and

u

. Finally, we show that the previous estimates lead to an existence result for

W^{2,p}

-strong solutions.
Wong–Zakai Type Approximations for Stochastic Differential Equations Driven by a Fractional Brownian Motion
Authors: Tudor Constantin
Published: December 2009

We consider Wong–Zakai type approximations for a class of Itô–Volterra equations related to the fractional Brownian motion. The quadratic mean convergence, uniformly on compact time intervals, of the approximations to the solution of an Itô–Volterra equation with a modified drift is obtained.
Error Estimates for the Cardinal Spline Interpolation
Authors: Vainikko Gennadi
Published: December 2009

For the Sobolev class

W_{\textrm{per}}^{m,\infty}(\mathbb{R})

of 1-periodic functions, an unimprovable error estimate for the spline interpolants of order

m

on the uniform grid is known. In the present paper, this error estimate is extended to the Sobolev class

V^{m,\infty}(\mathbb{R})

of (nonperiodic) functions on

\mathbb{R}

having bounded

m

th derivative. Some further error estimates are established including the error estimates for derivatives of the spline interpolant.
Strongly Extreme Points in Musielak–Orlicz Spaces with the Orlicz Norm
Authors: Wang Ping , Yu FeiFei , Cui Yunan
Published: December 2009

In this paper, criteria of strongly extreme points in Musielak–Orlicz spaces endowed with the Orlicz norm are given.
On Nontrivial Solutions of Variational-Hemivariational Inequalities with Slowly Growing Principal Parts
Authors: Le Vy Khoi , Motreanu Dumitru
Published: December 2009

This paper is concerned with the inclusion

−\mathrm{div}(a(|∇u|)∇u) + ∂_u G(x,u) \ni 0 \quad \text{in } Ω,

with Dirichlet boundary condition

u = 0

on

∂Ω

, in the case where the higher order part has slow growth and the lower order part is locally Lipschitz. By using a Mountain Pass theorem for variational-hemivariational inequalities without the Palais–Smale condition in Orlicz–Sobolev spaces, we show the existence of nontrivial solutions of the above inclusion.
Asymptotic Behavior of Approximate Solutions to Evolution Equations in Banach Spaces
Authors: Aizicovici Sergiu , Reich Simeon , Zaslavski Alexander J.
Published: December 2009

We study evolution equations in Banach spaces governed by a class of mappings associated with continuous descent methods for the minimization of convex functions. In our previous work we showed that for most of these mappings (in the sense of Baire category) the corresponding solutions converged. In the present paper we show that this remains true even for approximate solutions.
Local Tb Theorem on Spaces of Homogeneous Type
Authors: Tan Chaoqiang , Yan Lixin
Published: December 2009

In this article, we obtain a local Tb theorem for singular integral operators on spaces of homogeneous type by using tree selection algorithm of the dyadic model and the BCR algorithm, which extends an earlier result of M. Christ [Colloq. Math. 60/61 (1990), 601–628].
Global Bifurcation for a Reaction-Diffusion System with Inclusions
Authors: Eisner Jan , Kučera Milan , Väth Martin
Published: December 2009

We consider a reaction-diffusion system exhibiting diffusion driven instability if supplemented by Dirichlet–Neumann boundary conditions. We impose unilateral conditions given by inclusions on this system and prove that global bifurcation of spatially non-homogeneous stationary solutions occurs in the domain of parameters where bifurcation is excluded for the original mixed boundary value problem. Inclusions can be considered in one of the equations itself as well as in boundary conditions. The proof is based on the degree theory for multivalued mappings (jump of the degree implies bifurcation). We show how the degree for a class of multivalued maps including those corresponding to a weak formulation of our problem can be calculated.