bluuebunny/crossref_metadata_2025 huggingface dataset:
On the completeness of eigenvectors of right definite multiparameter problems
Authors: Volkmer Hans
Published: November 2011

Synopsis

We prove that there exists a complete system of eigenvectors of the eigenvalue problem

for self-adjoint operators Tr and Vrs on separable Hilbert spaces Hr. It is assumed that

(i) the operators Tr have discrete spectrum;

(ii) the operators Vrs are bounded and commute for each r;

(iii) the operators Vrs have the definite sign factor property.

This theorem generalizes and improves a result of Cordes for two-parameter problems. The proof of the theorem depends on an approximation of the given eigenvalue problem by simpler problems, a technique which is related to Atkinson's proof of his expansion theorem.
On conditional diffusion processes
Authors: Lyons T. J. , Zheng W. A.
Published: November 2011

Synopsis

If Xt is the diffusion process associated with a second-order uniformly elliptic operator L in divergence form, then without assuming smoothness in L we prove that for each x and y in ℝd,

where p is the fundamental solution to the heat equation associated with L. This allows one to control p when bounded drift terms are added to L; and also allows one to do Stratonovich integration with respect to the process conditioned to start at any point; previous work only dealt with quasi-every starting point.
A Hilbert space approach to distributions
Authors: Picard Rainer H.
Published: November 2011

Synopsis

A compact chain of Sobolev type Hilbert spaces , n integer, is introduced that is invariant with

respect to the Fourier transform ℱ. The spaces are related to powers of the adjoint of the so-called tempered derivative introduced in the sequential approach to distributions. It turns out that the intersection of all these Hilbert spaces coincides with the space of rapidly decaying C∞-functions and their union leads to the space of tempered distributions. Moreover, the naturally induced convergence concepts coincide with the usual ones. The approach provides not only a new and arguably more elementary approach to distributions it also provides a deeper insight into the action of the Fourier transform which is a unitary mapping in each space of the chain. Finally the Schwartz distributions are incorporated in the approach as locally tempered distributions.
On a classical two-component plasma with a logarithmic interaction
Authors: van den Berg M.
Published: November 2011

Synopsis

We prove the existence of the thermodynamic limit of the free energy per particle for a twocomponent plasma in one space dimension and with a logarithmic pair interaction.
On the growth of solutions to second order differential equations in Banach spaces
Authors: Fattorini H. O.
Published: November 2011

Synopsis

We obtain estimates for the exponential growth of the solutions tou″(t) = (A+ ζ2I)u(t) in terms of the exponential growth of the solutions tou″(t) =Au(t), where ζ is an arbitrary complex number. Estimates in exponentially weightedL2norms are also considered in Hilbert space.
Hopf bifurcation at multiple eigenvalues with zero eigenvalue
Authors: López-Gómez J.
Published: November 2011

Synopsis

We give local results related to Hopf bifurcation for parabolic equations. The linear part about the equilibrium point can have zero eigenvalues. In our results the information about the perturbation is essential and it is possible to obtain bifurcation even if some ‘i’ or zero remains on the imaginary axis for all values of the parameter.
On some infinite integrals involving logarithmic exponential and powers
Authors: Chaudhry M. Aslam , Ahmad Munir
Published: November 2011

Synopsis

In this paper we define an integral function Iμ(α; a, b) for non-negative integral values of μ by

It is proved that the function Iμ(α; a, b) satisfies a functional recurrence-relation which is then exploited to evaluate the infinite integral

Some special cases of the result are also discussed.
Blow-up time involved with perturbed Hamilton–Jacobi equations
Authors: Díaz G.
Published: November 2011

Synopsis

In this paper we consider the evolution of positive bounded uniformly continuous data u0 by perturbed equations like

Under general assumptions on u0, existence, uniqueness and regularity of the evolution u in the set

are studied, where the blow-up function is given by .

The exact blow-up rate of u is obtained. Uniqueness, regularity, decay at infinity of the function , as well a s a representation formula for the case m = 1, are also proved.
Global stability in models of population dynamics with diffusion. I. Patchy environments
Authors: Freedman H. I. , Krisztin T.
Published: November 2011

Synopsis

A class of models of single-species dynamics with diffusion within and between patches is considered. It is shown that under our prescribed conditions, there is a unique, positive, globally asymptotically stable steady state.
The validity of modulation equations for extended systems with cubic nonlinearities
Authors: Kirrmann Pius , Schneider Guido , Mielke Alexander
Published: November 2011

Synopsis

Modulation equations play an essential role in the understanding of complicated systems near the threshold of instability. Here we show that the modulation equation dominates the dynamics of the full problem locally, at least over a long time-scale. For systems with no quadratic interaction term, we develop a method which is much simpler than previous ones. It involves a careful bookkeeping of errors and an estimate of Gronwall type.

As an example for the dissipative case, we find that the Ginzburg–Landau equation is the modulation equation for the Swift–Hohenberg problem. Moreover, the method also enables us to handle hyperbolic problems: the nonlinear Schrodinger equation is shown to describe the modulation of wave packets in the Sine–Gordon equation.
An Lp theorem for compensated compactness
Authors: Zhou Yi
Published: November 2011

Synopsis

In this paper, an Lp version of the “div-curl lemma” is generalised in a very general framework. Another form of the Lp -theorem of compensated compactness is also exploited.
The Stefan problem: continuity of the interfaces for solutions with finite lapnumber
Authors: Klaver M. H. A.
Published: November 2011

Synopsis

In this paper we consider the Stefan problem with a heating term. We study the continuity of the interfaces between the mush, the liquid and the solid for solutions of finite lapnumber.
On the commutativity of certain quasi-differential expressions II
Authors: Frentzen H. , Race D. , Zettl A.
Published: November 2011

Synopsis

We consider the question: when do two ordinary, linear, quasi-differential expressions commute? For classical differential expressions, answers to this question are well known. The set of all expressions which commute with a given such expression form a commutative ring. For quasi-differential expressions less is known and such an algebraiastructure can no longer be exploited. Using the theory of very general quasi-differential expressions with matrix-valued coefficients, we prove some general results concerning commutativity of such expressions. We show how, when specialised to scalar expressions, these results include a proof of the conjecture that if a 2nth-order scalar,J-symmetric (or real symmetric) quasi-differential expression commutes with a second order expression having the same properties, then the former must be annth-order polynomial in the latter. This result was conjectured in a paper by Race and Zettl, to which this paper is a sequel.
On the periodic Fučik spectrum and a superlinear Sturm–Liouville equation
Authors: Figueiredo Djairo G. , Ruf Bernhard
Published: November 2011

Synopsis

In the first part of the paper a variational characterisation of the periodic eigenvalues (the so-called Fučik spectrum) of a semilinear, positive homogeneous Sturm–Liouville equation is given. The proof relies on the S1-invariance of the equation.

In the second part a nonlinear Sturm–Liouville equation with, typically, an exponential nonlinearity is considered. It is proved that under certain conditions this equation is solvable for arbitrary forcing terms. The proof uses a comparison of the minimax levels of the functional associated to this equation with suitable values in the Fučik spectrum.
Boundary integral equations for the scattering of electromagnetic waves by a homogeneous dielectric obstacle
Authors: Martin P. A. , Ola Petri
Published: November 2011

Synopsis

Time-harmonic electromagnetic waves are scattered by a homogeneous dielectric obstacle. The corresponding electromagnetic transmission problem is reduced to a single integral equation over S for a single unknown tangential vector field, where S is the interface between the obstacle and the surrounding medium. In fact, several different integral equations are derived and analysed, including two previously-known equations due to E. Marx and J. R. Mautz, and two new singular integral equations. Mautz's equation is shown to be uniquely solvable at all frequencies. A new uniquely solvable singular integral equation is also found. The paper also includes a review of methods using pairs of coupled integral equations over S. It is these methods that are usually used in practice, although single integral equations seem to offer some computational advantages.
Some inequalities for arithmetic and geometric means
Authors: Alzer Horst
Published: November 2011

Let An and Gn (respectively, A′n and G′n) be the weighted arithmetic and geometric means of x1, …, xn (respectively, 1 – x1, …, 1 – xn). We present sharp upper and lower bounds for the differences and . And we determine the best possible constants r and s such that

holds for all xi ∈ [a, b] (i = 1, …, n; 0 < a < b < 1). Our theorems extend and sharpen results of Fan, Cartwright and Field, McGregor and the author.
The long-time behaviour of solutions to parabolic problems on unbounded intervals: the influence of boundary conditions
Authors: Fašangová Eva , Feireisl Eduard
Published: November 2011

For a non-negative function ū(x), we study the long-time behaviour of solutions of the heat equation

with the Dirichlet or Neumann boundary conditions at x = 0. We find a critical parameter λD > 0 such that the solution subjected to the Dirichlet boundary condition tends to a spatially localized wave travelling to infinity in the space variable. On the other hand, there exists a λN > 0 such that the corresponding solution of the Neumann problem converges to a non-trivial strictly positive stationary solution. Consequently, the dynamics is considerably influenced by the choice of boundary conditions.
On some notions of tangent space to a measure
Authors: Fragalà Ilaria , Mantegazza Carlo
Published: November 2011

We consider some definitions of tangent space to a Radon measure μ on ℝn that have been given in the literature. In particular, we focus our attention on a recent distributional notion of tangent vector field to a measure and we compare it to other definitions coming from ‘geometric measure theory’, based on the idea of blow-up. After showing some classes of examples, we prove an estimate from above for the dimension of the tangent spaces and a rectifiability theorem which also includes the case of measures supported on sets of variable dimension.
Minimizers for a double-well problem with affine boundary conditions
Authors: Allaire Grégoire , Lods Véronique
Published: November 2011

This paper is concerned with the existence of minimizers for functionals having a double-well integrand with affine boundary conditions. Such functionals are related to the so-called Kohn–Strang functional, which arises in optimal shape design problems in electrostatics or elasticity. They are known to be not quasiconvex, and therefore existence of minimizers is, in general, guaranteed only for their quasiconvex envelopes. We generalize the previous results of Allaire and Francfort, and give necessary and sufficient conditions on the affine boundary conditions for existence of minimizers. Our method relies on the computation of the quasiconvexification of these functionals by using homogenization theory. We also prove by a general argument that their rank-one convexifications coincide with their quasiconvexifications.
Fourier splitting and dissipation of nonlinear dispersive waves
Authors: Bona J. L. , Demengel F. , Promislow K.
Published: November 2011

Presented herein is a new method for analysing the long-time behaviour of solutions of nonlinear, dispersive, dissipative wave equations. The method is applied to the generalized Korteweg–de Vries equation posed on the entire real axis, with a homogeneous dissipative mechanism included. Solutions of such equations that commence with finite energy decay to zero as time becomes unboundedly large. In circumstances to be spelled out presently, we establish the existence of a universal asymptotic structure that governs the final stages of decay of solutions. The method entails a splitting of Fourier modes into long and short wavelengths which permits the exploitation of the Hamiltonian structure of the equation obtained by ignoring dissipation. We also develop a helpful enhancement of Schwartz's inequality. This approach applies particularly well to cases where the damping increases in strength sublinearly with wavenumber. Thus the present theory complements earlier work using centre-manifold and group-renormalization ideas to tackle the situation wherein the nonlinearity is quasilinear with regard to the dissipative mechanism.
Liouville theorems on some indefinite equations
Authors: Zhu Meijun
Published: November 2011

In this note, we present some Liouville type theorems about the non-negative solutions to some indefinite elliptic equations.
Closed geodesies on compact Lorentzian manifolds of splitting type
Authors: Antonacci Flavia , Sampalmieri Rosell
Published: November 2011

In this paper, we consider the problem of the existence of a spacelike closed geodesic on compact Lorentzian manifolds. Tipler and Galloway proved that, under suitable topological properties of the manifold, there exists a closed timelike geodesic. In their proofs, they use the hypothesis that the time coordinate of one timelike geodesic has derivative always different from zero. This clearly fails for spacelike geodesies. Using variational methods and applying the relative category theory, we prove the existence of a closed spacelike geodesic on a compact manifoldof splitting type. Observe that, thanks to the previous results, the existence of at least two geometrically distinct closed geodesies onfollows.
A multi-phase Mullins–Sekerka system: matched asymptotic expansions and an implicit time discretisation for the geometric evolution problem
Authors: Bronsard Lia , Garcke Harald , Stoth Barbara
Published: November 2011

We propose a generalisation of the Mullins–Sekerka problem to model phase separation in multi-component systems. The model includes equilibrium equations in bulk, the Gibbs–Thomson relation on the interfaces, Young's law at triple junctions, together with a dynamic law of Stefan type. Using formal asymptotic expansions, we establish the relationship to a transition layer model known as the Cahn-Hilliard system. We introduce a notion of weak solutions for this sharp interface model based on integration by parts on manifolds, together with measure theoretical tools. Through an implicit time discretisation, we construct approximate solutions by stepwise minimisation. Under the assumption that there is no loss of area as the time step tends to zero, we show the existence of a weak solution.
Positivity results for a nonlocal elliptic equation
Authors: Freitas Pedro , Sweers Guido
Published: November 2011

In this paper we consider a second-order linear nonlocal elliptic operator on a bounded domain in ℝn (n ≧ 3), and give conditions which ensure that this operator has a positive inverse. This generalises results of Allegretto and Barabanova, where the kernel of the nonlocal operator was taken to be separable. In particular, our results apply to the case where this kernel is the Green's function associated with second-order uniformly elliptic operators, and thus include the case of some linear elliptic systems. We give several other examples. For a specific case which appears when studying the linearisation of nonlocal parabolic equations around stationary solutions, we also consider the associated eigenvalue problem and give conditions which ensure the existence of a positive eigenfunction associated with the smallest real eigenvalue.
On the interior spike solutions for some singular perturbation problems
Authors: Wei Juncheng
Published: November 2011

For some singular perturbed Dirichlet and Neumann problems in a bounded smooth domain, we study solutions which have a spike in the interior. We obtain both necessary and sufficient conditions for the existence of interior spike solutions. We use, among others, the methods of projections and viscosity solutions, weak convergence of measures and Liapunov–Schmidt reduction.
Heat flow with tangent penalisation converging to mean curvature motion
Authors: Zhai Jian
Published: November 2011

In this paper, we prove that mean curvature motion can be regarded as the singular limit of the following model:

where ε > 0 is a small parameter and . This model is related to the Landau–Lifshitz equation of ferromagnetism. Local existence of classical solutions of the Dirichlet problem and global existence of the travelling wave solutions are also obtained.
An elliptic boundary-value problem with a discontinuous nonlinearity, II
Authors: Keady G. , Kloeden P. E.
Published: November 2011

Synopsis

Let Ω be a bounded domain in ℝ2. The study, begun in Keady [13], of the boundary-value problem, for (λ/k, ψ),

is continued. Here Δ denotes the Laplacian, H is the Heaviside step function and one of λ or k is a given positive constant. The solutions considered always have ψ > 0 in Ω and λ/k > 0, and have cores

In the special case Ω = B(0, R), a disc, the explicit exact solutions of the branch τe have connected cores A and the diameter of A tends to zero when the area of A tends to zero. This result is established here for other convex domains Ω and solutions with connected cores A.

An adaptation of the maximum principles and of the domain folding arguments of Gidas, Ni and Nirenberg [9] is an important step in establishing the above result.
Semilinear evolution equations and fractional powers of a closed pair of operators
Authors: Dalsen Marié Grobbelaar-Van
Published: November 2011

Synopsis

The nonlinear evolution problem [Bu(t)]′ =A(t,Bu)u+f(t,Bu) withBa constant linear operator andA=A(t,Bu) a time-dependent nonlinear operator from one Banach space to another, is studied. Existence and uniqueness results are obtained by making use of the theory of B-evolutions and the fractional powers ofAandB. Two examples are presented in which the theory is applied to nonlinear equations with dynamic boundary conditions.
Bistable subvarieties of MS-algebras
Authors: Noor A. S. A.
Published: November 2011

Synopsis

A solution is given to an open question of Blyth and Varlet concerning bistable subvarieties of MS-algebras.
An elementary proof of Gorny's inequality
Authors: Fabry C.
Published: November 2011

Synopsis

Gorny's inequality provides upper bounds for the sup-norms ∥f(k) of a function f over an interval [a, b] for k = 1, …, n − 1, assuming the sup-norms of f and f(n) to be given. We present a simple proof of that inequality and obtain sharper estimates of the constants contained in that inequality, compared with the original verison of Gorny.
On a quasilinear wave equation with nonlinear damping
Authors: Hai Dang Dinh
Published: November 2011

Synopsis

We prove the global existence and uniqueness of the solution of the initial and boundary value problem for the equation

by using the classical Galerkin method when the forcing term and the initial data are in some sense small. The asymptotic behaviour of the solution as t → ∞ is also considered.
Unique factorisation of normal elements in polynomial rings
Authors: Jordan D.A.
Published: November 2011

Synopsis

We consider the class of Noetherian UFN-rings, that is, Noetherian prime rings such that every non-zero ideal contains a non-zero normal element and such that the monoid of non-zero normal elements is a unique factorisation monoid. We ask whether this class of rings, a generalisation of the commutative Noetherian unique factorisation domains (UFD), is closed under polynomial extensions. The general question is apparently difficult and remains open. However, we give positive answers in special cases, in particular, for algebras over an infinite field and for domains.
Global spatially periodic solutions to the Ginzburg–Landau equation
Authors: Yang Yisong
Published: November 2011

Synopsis

In this paper we study the global existence and nonexistence of spatially periodic solutions to the initial value problem of the Ginzburg–Landau equation.
On the uniqueness and ordering of steady states of predator-prey systems
Authors: Li Lige
Published: November 2011

Synopsis

This paper discusses the relationship between the uniqueness and the ordering of strictly positive solutions of elliptic predator-prey interacting systems. If (ū, v) and (u#, v#) are two such solutions with ū ≧ u# or v ≧ v#, then ū ≡= u#, v ≡v#. When the positive solutions are numerically close to the extreme case, the solution is unique.
L2-norm inequalities for the differentiation operator on compact intervals
Authors: Gillman David W. , Kaper Hans G. , Kwong Man Kam
Published: November 2011

Synopsis

In this paper we establish the best possible value of the constant K in the inequality ∥f ′ ∥2 ≦ K ∥f∥ ∥f″∥ for functions f that are defined and twice continuously differentiable on a compact interval [a, b] ⊂ ℝ and whose first derivative vanishes at some point in [a, b].
On the k-invariants of iterated loop spaces
Authors: Arlettaz Dominique
Published: November 2011

Synopsis

The purpose of this paper is to give universal bounds for the order of the Postnikov k-invariants of infinite loop spaces. This is done by giving universal bounds for the order of the k-invariants of m-connected r-fold loop spaces in dimensions ≦ r + 2m. An application of the result provides information on the Hurewicz homomorphism between the algebraic K-theory of aring and the homology of its general linear group.
A proof of characterisation of oscillation for higher-order neutral differential equations of mixed type by the Laplace transform
Authors: Arino O. , Attar M. A. El
Published: November 2011

Consider the general expression of such equations in the form

whereAi, Bj, ∊ ℝ, δo= 0dn/ 0,dnaren-derivatives,n≧ l, the σj'S and δj,'s respectively, are ordered as an increasing family with possibly positive and negative terms. These are the deviating arguments. In this paper, we provide a proof of this result based on the use of the Laplace transform. Our method involves new results regarding the exponential growth of positive solutions for such equations.
Relaxation of quasi-convex integrals of arbitrary order
Authors: Amar Micol , Cicco Virginia De
Published: November 2011

An integral representation result is given for the lower semicontinuous envelope of the functional ʃΩf(∇ku)dxon the spaceBVk(Ω:ℝm) of the integrable functions, whose thef-th derivative in the sense of distributions is a Radon measure with bounded total variation.
Generating functions and semi-classical orthogonal polynomials
Authors: Maroni Pascal , Van Iseghem Jeannette
Published: November 2011

An orthogonal family of polynomials is given, and a link is made between the special form of the coefficients of their recurrence relation and a first-order linear homogenous partial differential equation satisfied by the associated generating function. A study is also made of the semiclassical character of such families.
On the uniqueness of arithmetic structures
Authors: Johnson F. E. A.
Published: November 2011

Margulis has given conditions under which a lattice in a semisimple Lie group admits the structure of an arithmetic subgroup. We show that these arithmetic structures are unique. The result is not subject to the condition“rkR(G)≧ 2” required by the Margulis result. In the lowest dimensions, the result has previously been observed by Takeuchi, Maclachlan and Reid.
A polyconvexity condition in dimension two
Authors: Hartwig H.
Published: November 2011

We state a necessary and sufficient polyconvexity condition inR2X2for functions of classC1. This condition is applied tof(X) = |X|2(|X|2−2 detX) for obtaining a convex representation inR2X2xR.
On positive entire solutions of the elliptic equation Δu + K(x)up = 0 and its applications to Riemannian geometry
Authors: Gui Changfeng
Published: November 2011

We study the existence and asymptotic behaviour of positive solutions of a semilinear elliptic equation in entire space. A special case of this equation is the scalar curvature equation which arises in Riemannian geometry.
Dilation method and smoothing effects of solutions to the Benjamin–Ono equation
Authors: Hayashi Nako , Kato Keiichi , Ozawa Tohru
Published: November 2011

In this paper we study smoothing effects of solutions to the Benjamin–Ono equation

where H is the Hilbcrt transform defined by

We prove that ifφ∈H2and (x∂x)4φ then the solutionuof(BO) belongs to

, where
A fixed-point theorem for weakly condensing operators
Authors: O'Regan Donal
Published: November 2011

A new fixed-point theorem is presented for weakly condensing operators. Existence is then established for the nonlinear abstract operator y = Ny.
Singular limits for travelling waves for a pair of equations
Authors: Hutson Vivian , Mischaikow Konstantin
Published: November 2011

The singular limit as one diffusion coefficient approaches zero is considered for travelling wave solutions to a pair f reaction diffusion equations. An explicit criterion determining the sign of the wave speed is obtained. The limit behaviour turns out to be of a different nature for positive and negative wave speed. Different techniques, which may be applicable to a range of examples, are needed in the two cases.
Critical diffusion exponents for self-similar blow-up solutions of a quasilinear parabolic equation with an exponential source
Authors: Budd Chris J. , Galaktionov Victor A.
Published: November 2011

We study the self-similar solutions of the quasilinear parabolic equation

We show that there is an exponent

such that if σ> then the equation admits a countable set {uk(x, t)} of self-similar blow-up solutions. These solutions have the form

where T> 0 is a finite blow-up time, θ(ξ) solves a nonlinear ODE and each function uk(x, t) is nonconstant in a neighbourhood of the origin and has exactly k maxima and minima for x ≧ 0. There is a further critical exponent σ = ф such that if σ > ф there is a second set of self-similar solutions which are constant (in x) in a neighbourhood of the origin. We conjecture (and provide formal arguments and numerical evidence for) the existence of an infinite sequence σk→σ∞ of critical values, such that σ1 = 0 and uk exists only in the range σ>σk (when σ> 0 the equation has no nontrivial self-similar solutions). The proof of existence when σ>σ∞(σ>ф) is obtained by a combination of comparison and dynamical systems arguments and relates the existence of the self-similar solutions to a homoclinic bifurcation in an appropriate phase-space.
Multiple positive solutions of nonhomogeneous semilinear elliptic equations in ℝN
Authors: Dao-Min Cao , Huan-Song Zhou
Published: November 2011

We consider the following problem

where for all ≦f(x,u)≦c1up-1 + c2u for all x ∈ℝN,u≧0 with c1>0,c2∈(0, 1), 2<p<(2N/(N – 2)) if N ≧ 3, 2 ≧ + ∝ if N = 2. We prove that (*) has at least two positive solutions if

and h≩0 in ℝN, where S is the best Sobolev constant and
A characterisation of tangential exceptional sets for αp = n
Authors: Cascante Carme , Ortega Joaquín
Published: November 2011

In this paper we characterise some closed tangential exceptional sets for Hardy–Sobolev spaces , αp = n, 1 < p ≦ 2, in terms of the annihilation of a nonisotropic Hausdorff measure.
On the decay of higher-order norms of the solutions of Navier–Stokes equations
Authors: Schonbek Maria E. , Wiegner Michael
Published: November 2011

We show that an energy decay ∥u(t)∥2 = O(t−µ) for solutions of the Navier–Stokes equations on ℝn, n ≦ 5, implies a decay of the higher order norms, e.g. ∥Dα u(t)∥2 = O(t−µ −|α|/2) and ∥u(t)|∞ = O(t−µ −n/4).
Cauchy problem for hyperbolic conservation laws with a relaxation term
Authors: Klingenberg Christian , Lu Yun-guang
Published: November 2011

This paper considers the Cauchy problem for hyperbolic conservation laws arising in chromatography:

with bounded measurable initial data, where the relaxation term g(δ, u, v) converges to zero as the parameter δ > 0 tends to zero. We show that a solution of the equilibrium equation

is given by the limit of the solutions of the viscous approximation

of the original system as the dissipation ε and the relaxation δ go to zero related by δ = O(ε). The proof of convergence is obtained by a simplified method of compensated compactness [2], avoiding Young measures by using the weak continuity theorem (3.3) of two by two determinants.
Disjointly strictly singular operators and interpolation
Authors: del Amo A. García , Hernández F. L. , Ruiz C.
Published: November 2011

Interpolation properties of the class of disjointly strictly singular operators on Banach lattices are studied. We also give some applications to compare the lattice structure of two rearrangement invariant function spaces. In particular, we obtain suitable analytic characterisations of when the inclusion map between two Orlicz function spaces is disjointly strictly singular.
Homoclinic orbits to small periodic orbits for a class of reversible systems
Authors: Lombardi Eric
Published: November 2011

In this paper a one-parameter class of four-dimensional, reversible vector fields is investigated near an equilibrium. We call the parameter μ and place the equilibrium at 0. The differential at 0 is supposed to have ±iq, q > 0, as simple eigenvalues and 0 as a double, nonsemisimple eigenvalue. Our ultimate goal is to construct homoclinic connections of periodic orbits of arbitrary small size, in fact we shall show that the oscillations of the homoclinic orbits at infinity are bounded by a flat function of μ. This result receives its significance from the still unsolved question as to whether solutions exist which are homoclinic to the equilibrium or whether the amplitudes of the oscillations at infinity have a positive infimum. First we construct the periodic solutions. In contrast to previous work, we find these in a full rectangle [0, K0] × ]0,μ0], where K measures the amplitude of the periodic orbits. Then we show that for each n ∈ ℕ there is a μn and a family of periodic solutions X(μ), μ ∈]0,μn[, of Size μn. To each of these solutions, we can find two homoclinic orbits, which are distinguished by their phase shift at infinity. One example of such a vector field occurs when describing the flow of an inviscid irrotational fluid layer under the influence of gravity and small surface tension (Bond number b < ⅓), for a Froude number F close to 1. In this context a homoclinic solution to a periodic orbit is called a generalised solitary wave. Our work shows that there exist solitary waves with oscillations at infinity of order less than |μ|n for every n.
A regularity theorem for minimisers of quasiconvex integrals: the case 1 <p<2
Authors: Carozza Menita , di Napoli Antonia Passarelli
Published: November 2011

We prove partial regularity for minimisers of quasiconvex integrals of the form ∫Ωf(Du(x))dx. More precisely, we consider an integrandf(ξ) having subquadratic growth, i.e. |f(ξ)|≦L(1+|ξ|p) withp< 2. The case of a general integrand depending also onxanduis also considered.
Commutative orders
Authors: Easdown David , Gould Victoria
Published: November 2011

A subsemigroup S of a semigroup Q is a left (right) order in Q if every q ∈ Q can be written as q = a*b(q = ba*) for some a, b ∈S, where a* denotes the inverse of a in a subgroup of Q and if, in addition, every square-cancellable element of S lies in a subgroup of Q. If S is both a left order and a right order in Q, we say that S is an order in Q. We show that if S is a left order in Q and S satisfies a permutation identity xl…xn = x1π…xnπ where 1 < 1π and nπ<n, then S and Q are commutative. We give a characterisation of commutative orders and decide the question of when one semigroup of quotients of a commutative semigroup is a homomorphic image of another. This enables us to show that certain semigroups have maximum and minimum semigroups of quotients. We give examples to show that this is not true in general.
A probabilistic approach to blow-up of a semilinear heat equation
Authors: Etheridge Alison M.
Published: November 2011

We introduce a probabilistic approach to the study of blow-up of positive solutions to a class of semilinear heat equations. This then gives a representation of the coefficients in the power series expansion of the solutions. In a special case, this approach leads to a path-valued Markov process which can also be understood via the theory of Dawson-Watanabe superprocesses. We demonstrate the utility of the approach by proving a result on ‘complete blow-up’ of solutions.
A phase transition speed estimate in the Stefan problem
Authors: Zheng Weian
Published: November 2011

We give an estimate of the time needed for a phase transition to be completed in the Stefan problem. Our method is to compare the density of the diffusion process associated with the Stefan problem, with that of Brownian motion.
Boundedness of operators of Hardy type in ΛP,q spaces and weighted mixed inequalities for singular integral operators
Authors: Martín-Reyes F. J. , Salvador P. Ortega , Gavilán M. D. Sarrión
Published: November 2011

We consider certain n-dimensional operators of Hardy type and we study their boundedness in These spaces were introduced by M. J. Carro and J. Soria and include weighted Lp, q spaces and classical Lorentz spaces. As an application, we obtain mixed weak-type inequalities for Calderón—Zygmund singular integrals, improving results due to K. Andersen and B. Muckenhoupt.
On connected subsets of M2×2 without rank-one connections
Authors: Zhang Kewei
Published: November 2011

We prove that connected subsets of M2×2 without rank-one connections are Lipschitz graphs of mappings from subsets of a fixed two-dimensional subspace to its orthogonal complement. Under a weaker condition that the set does not have rank-one connections locally, we are able to establish some global results on the set. We also establish some results on Lipschitz extensions of the functions thus obtained.
Optimal existence and uniqueness in a nonlinear diffusion–absorption equation with critical exponents
Authors: Chaves M. , Vazquez J. L. , Walias M.
Published: November 2011

We study the existence and uniqueness of non-negative solutions of the nonlinear parabolic equation

posed inQ=RN× (0, ∞) with general initial datau(x, 0) =u0(x) ≧ 0. We find optimal exponential growth conditions for existence of solutions. Similar conditions apply for uniqueness, but the growth rate is different. Such conditions strongly depart from the linear casem= 1,ut= Δu–u, and also from the purely diffusive caseut= Δum.
Eigenvalue problems for the wave equation with strong damping
Authors: Freitas Pedro
Published: November 2011

This paper presents a study of linear operators associated with the linearisation of general semilinear strongly damped wave equations around stationary solutions. The structure of the spectrum of such operators is considered in detail, with an emphasis on stability questions. Necessary and sufficient conditions for the stability of the trivial solution of the linear equation are given, together with conditions for this solution to become unstable. In the latter case, the mechanisms which are responsible for the change of stability are analysed. These results are then applied to obtain stability and instability conditions for the semilinear problem. In particular, a condition is given which ensures that the dimensions of the centre and unstable manifolds of a stationary solution are the same as when that solution is considered as a stationary solution of an associated parabolic problem.
Stability and instability of steady-state solutions for a hydrodynamic model of semiconductors
Authors: Hattori Harumi
Published: November 2011

We discuss the stability and instability of steady-state solutions for a hydrodynamic model of semiconductors. We study the case where the doping profile is close to a positive constant and depends on the special variable x. We shall show that a given steady-state solution is asymptotically stable or unstable, depending on whether or not the density of the initial data satisfies P = 0, where P is defined in (3.12).
Almost-Lp-projections and Lp isomorphisms
Authors: Cambern Michael , Jarosz Krzysztof , Wodinski Georg
Published: November 2011

Synopsis

Lp -summands and Lp -projections in Banach spaces have been studied by E. Behrends, who showed that for a fixed value of p, l ≦ p ≦ ∞, p ≠ 2, any two Lp -projections on a given Banach space E commute. Here we introduce the notion of almost-Lp -projections, and we establish a result which generalises Behrends' theorem, while also simplifying its proof. Almost-Lp-projections are then applied to the study of small-bound isomorphisms of Bochner LP -spaces. It is shown that if the Banach space E satisfies a geometric condition which, in the finite-dimensional case, reduces to the absence of non-trivial Lp-summands, then for separable measure spaces, the existence of a small-bound isomorphism between Lp (λ1, E) and LP(λ2, E) implies that these Bochner spaces are, in fact, isometric.
Existence of conjugate points for self-adjoint linear differential equations
Authors: Došlý Ondřej
Published: November 2011

Synopsis

The conjecture of Muller-Pfeiffer [4] concerning the oscillation behaviour of the differential equation (–l)n(p(x)y(n))(n) + q(x)y = 0 is proved, and a similar conjecture concerning the more general differential equation ∑nk=0(−l)k(Pk(x)y(k)(k + q(x)y= 0 is formulated.
Multiple solutions of a nonlinear boundary value problem
Authors: Troy William C.
Published: November 2011

Synopsis

We consider a system of nonlinear differential equations derived from a model for the self-trapping of light. We prove that there is a parameter range for which there is a fundamental mode solution. Further, given any positive integer N, there also exists a parameter range over which the boundary value problem has N solutions. The first of these is a fundamental mode solution while the next N — 1 solutions represent additional modes.
The Dirichlet problem for harmonic maps from the disc into the 2-sphere
Authors: Soyeur Alain
Published: November 2011

Synopsis

We consider the Dirichlet problem for harmonic maps from the discD2into the sphereS2, with prescribed boundary values γ:∂D2→S2, and we prove that if γ is not a rational function, one can find infinitely many nonhomotopic harmonic maps which agree with γon ∂D2.
Presentation of groups acting on trees with inversions
Authors: Mahmud R. M. S.
Published: November 2011

Synopsis

The Bass-Serre theory of groups acting on trees without inversions is extended to the case in which the action with inversions is possible.
Instability of a class of dispersive solitary waves
Authors: Souganidis P. E. , Strauss W. A.
Published: November 2011

Synopsis

This paper studies the stability and instability properties of solitary wave solutions φ(x – ct) of a general class of evolution equations of the form Muttf(u)x=0, which support weakly nonlinear dispersive waves. It turns out that, depending on their speed c and the relation between the dispersion (i.e. the order of the pseudodifferential operator) and the nonlinearity, travelling waves maybe stable or unstable. Sharp conditions to that effect are given.
Elliptic equations on manifolds and isoperimetric inequalities
Authors: Cianchi Andrea
Published: November 2011

Synopsis

We consider linear and nonlinear elliptic equations in divergence form on Riemannian manifolds with or without boundary. In the former case we impose a homogeneous Neumann boundary condition. By making use of isoperimetric inequalities for manifolds, we obtain a priori sharp estimates for the decreasing rearrangement of the solutions to such equations. These estimates enable us to derive bounds for suitable norms of the solutions and of their gradients.
A rigorous proof of an exponentially small estimate for a boundary value arising from an ordinary differential equation
Authors: Byatt-Smith J. G. B. , Davie A. M.
Published: November 2011

Synopsis

The equation

has a solution y(t) which is non-oscillating on the interval (0, ∞) and has the asymptotic expansion

Each term of this expansion is even in t so that formally is zero to all orders of ɛ. The estimate of has been obtained by Byatt-Smith [3] who corrects (2) in the complex plane near t = i where the series ceases to be valid. This requires asolution of the equation

the equation for the first Painlevé transcedent. Here we prove rigorously that this method gives the correct asymptotic estimate

where

The proof involves converting (1) and (3) to integral equations. The existence and uniqueness of these integral equations are established by use of the contraction mapping theorem. We also prove that the appropriate solution to (3) provides a uniformly valid approximation to (2) over a suitably defined region of the complex plane.

We also consider the connection problem for the oscillatory solutions of (1) which have asymptotic expansions

where Ã+, Ã− φ+, and φ− are constants. The connection problem is to determine the asymptotic expansion at +∞ of a solution which has a given asymptotic expansion at −∞. In other words, we wish to find (Ã+,φ+) as a function of Ã−and φ−. We prove that there is a unique solution to the connection problem, provided Ã− is small enough, and obtain bounds on the estimate of Ã+
On the Titchmarsh-Weyl M(λ)-coefficient and spectral density for a Dirac system
Authors: Clemence Dominic P.
Published: November 2011

Synopsis

For a Dirac system on the line, we study the Jost solutions and the existence of half-bound states. A characterisation of this phenomenon is given in terms of the M(λ)-function, which supplements a well-known spectral characterisation for the system. As a corollary, we deduce the spectral density behaviour at the spectral gap endpoints, which is shown to depend on the possible existence of half-bound states.
Boundary value problems of the Ginzburg–Landau equations
Authors: Yang Yisong
Published: November 2011

Synopsis

For a domain Ω in the Euclidean space Rd (d = 2, 3) existence of weak solutions for both interior and exterior Dirichlet boundary value problems of the Ginzburg-Landau equations are established without any restriction on the range of the coupling constant λ, thesize of Ω, or the boundary data. For the critical choice λ = 1, we prove the existence of confined multivortices in a bounded domain by a constructive monotone iteration method.
On blow-up and degeneracy for the semilinear heat equation with source
Authors: Galaktionov V. A.
Published: November 2011

Synopsis

The asymptotic behaviour of the solution of the semilinear parabolic equation ut = uxx + (1 + u)ln2(l + u) for t > 0, x ∊[−π, π ], ux(t, ± π) = 0 for t > 0 and u(0, x) = u0(x) ≧ 0 in [−π, π], which blows up at a finite time T0, is investigated. It is proved that for some two-parametric set of initial functions u0 the behaviour of u(t, x) near t = T0 is described by the approximate self-similar solution va(t, x) = exp {(T0 −t)−1 cos2 (x/2)} − 1, satisfying the first order nonlinear Hamilton–Jacobi equation vt, = (vx)2 /(1 + v) + (1 + v) ln2 (1 + v). Some open problems of degeneracy near a finite blow-up time for other semilinear or quasilinear parabolic equations with source ut, = Δu + (1 + u) lnβ (1 + u) (β >1), ut, = Δu + uβ(β > l), ut = Δu + eu; ut = ∇. (lnσ(1 + u)∇u)+ (1 + u)lnβ(1 + u) (σ > 0, β > 1) are discussed.
Stability of discontinuous steady states in shearing motions of a non-Newtonian fluid
Authors: Nohel John A. , Pego Robert L. , Tzavaras Athanasios E.
Published: November 2011

Synopsis

We study the nonlinear stability of discontinuous steady states of a model initial-boundary value problem in one space dimension for incompressible, isothermal shear flow of a non-Newtonian fluid driven by a constant pressure gradient. The non-Newtonian contribution to the shear stress is assumed to satisfy a simple differential constitutive law. The key feature is a non-monotone relation between the total steady shear stress and shear strain-rate that results in steady states having, in general, discontinuities in the strain rate. We show that every solution tends to a steady state as t → ∞, and we identify steady states that are stable.
On dichotomic maps for a class of differential-difference equations
Authors: Carvalho L. A. V. , Cooke K. L.
Published: November 2011

Synopsis

Stability and asymptotic stability of the null solution of the differential-difference equation (E)x′(t) = f(x(t), x(t − r)), f: RNxRN → RN, f(0, 0) = 0, are studied by means of an extension of the Liapunov–Razumikhin method. Let V: RN → R be a differentiate map, let C = C(+ −r, 0=, RN), and let x(t, ψ) denote the solution of (E) with initial condition ψ in C at t = 0. For t ≧ 0 let xt(ψ) be defined by xt,(ψ)(θ) = x(t + θ, ψ), −r ≦θ ≦0. Let V′ (ψ) be the variation of V along the solution x(t, ψ). We say that V is dichotomic with respect to (E) if there exist T ≧0 and Ω, a neighbourhood of the origin in C, such that if ψ is in the closure of the set where V′ (xT(ψ)) >; 0, then V(x(T, ψ)) ≦ V(x(s, ψ)) for some s, −r ≦ sT. It is proved that if V is positive definite, continuously differentiable, and dichotomic, then the null solution of (E) is stable. A concept of strict dichotomic map is introduced and used to prove asymptotic stability. A number of examples are given to illustrate the applications of the method.
Examples for obstructions to action-angle coordinates
Authors: Bates Larry M.
Published: November 2011

Synopsis

We give examples of symplectic manifolds which are also non-trivial principal torus-bundles with Lagrangian fibres. These bundles are examples of spaces with an obstruction to the global existence of action-angle variables.
A note on a paper of Atkinson concerning the asymptotics of an eigenvalue problem with interior singularity
Authors: Harris B.J.
Published: November 2011

Synopsis

In [2] Atkinson considers the asymptotic form of the eigenvalues of the linear differentialequation

where a < 0 < b and y satisfies appropriate conditions at a andb. In particular Atkinson considers where q is singular at 0. In this case q(x) = xK, his results cover the case l ≦ K < . We extend Atkinson's results to cover more singular q, in the power case 1 ≦K <
A note on the Stieltjes transformation
Authors: Sinnamon Gordon
Published: November 2011

Synopsis

A necessary and sufficient condition for the boundedness of the generalised Stieltjes transformation on weighted Lebesgue spaces is obtained and applied to extend the Hilbert double series theorem.
Bounded solutions of linear periodic abstract parabolic equations
Authors: Lunardi Alessandra
Published: November 2011

Synopsis

We study the asymptotic behaviour of the parabolic evolution operatorG(t,s) associated with a family of generators of analytic semigroups {A(t);t∊ ℝ} in general Banach space, under the periodicity assumptionA(t) =A(t+T). We find new maximal regularity results for the bounded solutions of abstract nonhomogeneous parabolic equations in unbounded time intervals.
A singular perturbation problem for nonlinear damped hyperbolic equations
Authors: Biler Piotr
Published: November 2011

Synopsis

We consider damped nonlinear hyperbolic equations utt + Aut + αAu + βA2u + G(u) = 0, where A is a positive operator and G is the Gateaux derivative of a convex functional. We examine the asymptotic behaviour of solutions and the convergence of strong solutions to these equations when the parameter β tends to zero.