bluuebunny/crossref_metadata_2025 huggingface dataset:
The resolvent problem for the Stokes system in exterior domains: uniqueness and non-regularity in Hölder spaces
Authors: Deuring Paul
Published: November 2011

Synopsis

We consider the resolvent problem for the Stokes system in exterior domains, under Dirichlet boundary conditions:

where Ω is a bounded domain in ℝ3. It will be shown that in general there is no constant C > 0 such that for with , div u = 0, and for with . However, if a solution (u, π) of problem (*) exists, it is uniquely determined, provided that u(x) and ∇π(x) decay for large values of |x|. These assertions imply a non-existence result in Hölder spaces.
The -radical in structural matrix near rings, II
Authors: van Wyk Leon
Published: November 2011

Synopsis

Whether the -radical of a structural matrix near-ring (B, R) is the sum of two non-trivial ideals, one of which is nilpotent, is an open problem. However, it is known that ((B, R)) contains two ideals and , which are respectively precisely the two ideals, the sum of which is the Jacobson radical, in the case where the underlying near-ring is a ring. We strengthen our conjecture that and are the sought-after ideals by showing that (B, R)/≅(C, R) in the near-ring case, where C is the largest symmetric Boolean matrix such that C≦B, and by showing that is nilpotent.
The exact coupled bounds for effective tensors of electrical and magnetic properties of two-component two-dimensional composites
Authors: Cherkaev A. V. , Gibiansky L. V.
Published: November 2011

Synopsis

This paper describes the effective properties of a two-dimensional two-component dielectric composite. Each component is supposed to be isotropic and is characterised by a pair of constants of electrical and magnetic permeabilities. Effective properties of an arbitrary anisotropic composite depend on its microstructure and are characterised by a pair of symmetric tensors of electrical and magnetic permeabilities. Here we construct the so-called Gm-closure which is the set of all possible values of these pairs, corresponding to all possible microstructures with prescribed volume fractions of the given components. The exact bounds of the Gm-closure are obtained and the microstructure corresponding to each point of the Gm-closure set is determined.
Symmetry properties of positive solutions of elliptic equations in an infinite sectorial cone
Authors: Chu Chie-Ping , Wang Hwai-Chiuan
Published: November 2011

Synopsis

We prove symmetry properties of positive solutions of semilinear elliptic equations Δu + f(u) = 0 with Neumann boundary conditions in an infinite sectorial cone. We establish that any positive solution u of such equations in an infinite sectorial cone ∑α in ℝ3 is spherically symmetric if the amplitude α of ∑α is not greater than π.
Existence of multiple positive solutions for a semilinear equation with critical exponent
Authors: Yinbing Deng
Published: November 2011

Synopsis

In this paper we discuss the problem

We show that for c > 0 there exists a positive constant μ* such that (*)μ possesses at least one solution if μ ∈ (0, μ*) and no solutions if μ > μ*. Furthermore, there exists a positive constant μ**≦μ* such that (*)μ possesses at least two solutions if μ∈(0, μ**), 2<N<6. For N≧6, μ∈(0,μ**), we show that problem (*)μ possesses a unique solution if f(x) is radial with f′(r) < 0(r = |x|).
Water waves for small surface tension: an approach via normal form
Authors: Iooss Gérard , Kirchgässner Klaus
Published: November 2011

Synopsis

In this paper we determine the possible crest-forms of permanent waves of small amplitude which exist on the free surface of a two-dimensional fluid layer under the influence of gravity and surface tension when the Froude number is close to 1. The Bond number b, measuring surface tension, is assumed to satisfy b < ⅓. We find one-parameter families of periodic waves of two different types, quasiperiodic waves and solitary waves with oscillations at infinity. The existence of true solitary waves is established for a sequence of systems approximating the full Euler equations in every algebraic order of − 1.
The existence of a class of cosmic string solutions with infinite energy
Authors: Yang Yisong
Published: November 2011

Synopsis

Recent studies have indicated that in a certain critical coupling phase the Einstein–matter–gauge equations may be reduced to a Bogomol'nyi–Einstein system and the solutions are cosmic strings with finite separation. In this paper, we view the spacetime as known and treat the dynamics as being determined by the matter–gauge sector in an asymptotically Euclidean geometry. It is shown that the model admits a continuous family of infinite energy solutions which decay to the symmetric vacuum faster than any exponential functions.
Uniformly monotone dynamical systems
Authors: Nadirashvili Nikolai S.
Published: November 2011

Synopsis

We present a geometric approach for systems of ordinary differential equations which generate an order preserving flow. One of our main goals is to describe qualitatively the asymptotic behaviour of trajectories of dynamical systems enjoying a uniformly monotone principle.
The Conley index for maps in absence of compactness
Authors: Degiovanni Marco , Mrozek Marian
Published: November 2011

Synopsis

We construct the Conley index for maps. We do not assume any compactness of map or space. We prove the Ważewski property, additivity property and continuation property.
Solutions with multiple peaks for nonlinear elliptic equations
Authors: Cao Daomin , Noussair Ezzat S. , Yan Shusen
Published: November 2011

Solutions with peaks near the critical points of Q(x) are constructed for the problem

We establish the existence of 2k −1 positive solutions when Q(x) has k non-degenerate critical points in ℝN
Harmonic morphisms as a variational problem
Authors: Loubeau E.
Published: November 2011

In this note, we establish a variational setting for harmonic morphisms for target spaces of any dimension. We then extend this result to horizontally weakly conformal p-harmonic maps, such maps being p-harmonic morphisms.
Stability analysis in order-preserving systems in the presence of symmetry
Authors: Ogiwara Toshiko , Matano Hiroshi
Published: November 2011

Given an equation with a certain symmetry, such as symmetry with respect to rotation or translation, one of the most fundamental questions to ask is whether or not the symmetry of the equation is inherited by its solutions. We first discuss this question in a general framework of order-preserving dynamical systems under a group action and establish a theory concerning symmetry or monotonicity properties of stable equilibrium points. We then apply this general theory to nonlinear partial differential equations. Among other things, we prove the rotational symmetry of solutions for a class of nonlinear elliptic equations and the monotonicity of travelling waves of some nonlinear diffusion equations. We also discuss the stability of stationary or periodic solutions for equations of surface motion.
Heteroclinic orbits between rotating waves in hyperbolic balance laws
Authors: Härterich Jörg
Published: November 2011

We deal with the large-time behaviour of scalar hyperbolic conservation laws with source terms

which are often called hyperbolic balance laws. Fan and Hale have proved existence of a global attractorfor this equation withx∈S1.consists of spatially homogeneous equilibria, a large number of rotating waves and of heteroclinic orbits between these objects. In this paper, we solve the connection problem and show which equilibria and rotating waves are connected by a heteroclinic orbit. Apart from existence results, our approach via generalized characteristics also gives geometric information about the heteroclinic solutions, e.g. about the shock curves and their strength.
Linear stability analysis and metastable solutions for a phase-field model
Authors: Jiménez-Casas A. , Rodríguez-Bernal A.
Published: November 2011

We study the linear stability of equilibrium points of a semilinear phase-field model, giving criteria for stability and instability. In the one-dimensional case, we study the distribution of equilibria and also prove the existence of metastable solutions that evolve very slowly in time.
The effective bulk energy of the relaxed energy of multiple integrals below the growth exponent
Authors: Bouchitté Guy , Fonseca Irene , Malý Jan
Published: November 2011

The characterisation of the bulk energy density of the relaxation inW1, P(Ω; ℝd) of a functional

is obtained forp>q–q/N, whereu∈W1,P(Ω; ℝd), andfis a continuous function on the set ofd × Nmatrices verifying

for some constantC> 0 and 1 ≦q< + ∞. Typical examples may be found in cavitation and related theories. Standard techniques cannot be used due to the gap between the exponentqof the growth condition and the exponent p of the integrability of the macroscopic strain ∇u. A recently introduced global method for relaxation and fine Sobolev trace and extension theorems are applied.
On subordinacy and spectral multiplicity for a class of singular differential operators
Authors: Gilbert D. J.
Published: November 2011

The spectral multiplicity of self-adjoint operatorsHassociated with singular differential expressions of the form

is investigated. Based on earlier work of I. S. Kac and recent results on subordinacy, complete sets of necessary and sufficient conditions for the spectral multiplicity to be one or two are established in terms of: (i) the boundary behaviour of Titchmarsh–Weylm-functions, and (ii) the asymptotic properties of solutions ofLu= λu, λ∈ℝ, at the endpointsaandb. In particular, it is shown thatHhas multiplicity two if and only ifLis in the limit point case at bothaandband the set of all λ for which no solution ofLu= λuis subordinate at eitheraorbhas positive Lebesgue measure. The results are completely general, subject only to minimal restrictions on the coefficientsp(r), q(r)andw(r), and the assumption of separated boundary conditions whenLis in the limit circle case at both endpoints.
Multibump periodic travelling waves in suspension bridges
Authors: Peletier L. A. , Troy W. C.
Published: November 2011

We investigate different types of periodic solutions of a fourth-order, nonlinear differential equation, which has been proposed as a model for travelling waves in suspension bridges. We develop a shooting argument, which enables us to prove the existence of two families of multibump periodic solutions, each containing a countably infinite number of different solutions
The finite basis problem for pseudovariety
Authors: Repnitskiǐ V. B. , Volkov M. V.
Published: November 2011

We show that the pseudovariety generated by all semigroups of order-preserving transformations of a finitechain has no finite pseudoidentity basis.
Rearrangements and polar factorisation of countably degenerate functions
Authors: Burton G. R. , Douglas R. J.
Published: November 2011

This paper proves some extensions of Brenier's theorem that an integrable vector-valued function u, satisfying a nondegeneracy condition, admits a unique polar factorisation u = u# ° s. Here u# is the monotone rearrangement of u, equal to the gradient of a convex function almost everywhere on a bounded connected open set Y with smooth boundary, and s is a measure-preserving mapping. We show that two weaker alternative hypotheses are sufficient for the existence of the factorisation; that u# be almost injective (in which case s is unique), or that u be countably degenerate (which allows u to have level sets of positive measure). We allow Y to be any set of finite positive Lebesgue measure. Our construction of the measure-preserving map s is especially simple.
Maximal operators associated to Fourier multipliers with an arbitrary set of parameters
Authors: Duoandikoetxea Javier , Vargas Ana
Published: November 2011

We present here some general results of boundedness on LP for maximal operators of the form , where E is a subset of the positive real numbers and Tt is a dilation of a fixed multiplier operator. The range of values of p depends only on the decay at infinity of the multiplier and the Minkowski dimension of E. For the case being the maximal operator associated to a convex body, we prove that the norm of the operator is independent of the body.
A semilinear heat equation with singular initial data
Authors: Kwak Minkyu
Published: November 2011

We first prove existence and uniqueness of non-negative solutions of the equation

in in the range 1 < p < 1 + 2/N, when initial data u(x, 0) = a|x|−2(p−1), x ≠ 0, for a > 0. It is proved that the maximal and minimal solutions are self-similar with the form

where g = ga satisfies

After uniqueness is proved, the asymptotic behaviour of solutions of

is studied. In particular, we show that

The case for a = 0 is also considered and a sharp decay rate of the above equation is derived. In the final, we reveal existence of solutions of the first and third equations above, which change sign.
A singularly perturbed elliptic problem involving supercritical Sobolev exponent
Authors: Shen Yaotian , Yan Shusen
Published: November 2011

This paper deals with −Δu + εuq−1 = u2*−1, , where q > 2*, ε > 0. We first show that the minimiser of the associated minimisation problem blows up at the global minimum point of H(x, x), where H(y, x) is the regular part of the Green's function. We then prove that for each strictly local minimum point x0 of H(x, x), this problem has a solution concentrating at x0 as ε→0.
One-dimensional Hardy-type inequalities in many dimensions
Authors: Sinnamon Gord
Published: November 2011

Weighted inequalities for certain Hardy-type averaging operators in are shown to be equivalent to weighted inequalities for one-dimensional operators. Known results for the one-dimensional operators are applied to give weight characterisations, with best constants in some cases, in the higher-dimensional setting. Operators considered include averages over all dilations of very general starshaped regions as well as averages over all balls touching the origin. As a consequence, simple weight conditions are given which imply weighted norm inequalities for a class of integral operators with monotone kernels.
A class of time periodic Hamiltonians with no bound states
Authors: Kaneta H.
Published: November 2011

Synopsis

We generalise the Paley–Wiener closedness theorem and apply it to a class of time periodic Hamiltonians to show that all solutions to the corresponding Schrodinger equation decay.
Schützenberger groups of polynomials in the semigroup of entire functions
Authors: Magill K. D.
Published: November 2011

Synopsis

Let denote the semigroup, under composition, of all entire functions of a complex variable and for , let Γ(f) denote the Schützenberger group of the ℋ-class of which contains f. The main results of this paper completely describe Γ(P) where P is a polynomial function. Γ(P) is trivial if P is a constant function. If Deg (P) (the degree of P) is one, then Γ(P) is isomorphic to the group of all matrices of the form where a and b are complex numbers with a ≠ 0. If Deg (P) > 1, then Γ(P) is isomorphic to ℂM, the multiplicative group of nonzero complex numbers if and only if P(z) = α(z +β)n + γ where α ≠ 0. If P is not of this form and Deg (P) > 1, then Γ (P)is a finite cyclic group and its order can be determined by defining (where αn−1 and α n are the coefficients in P of zn−1 and zn respectively) and then inspecting the coefficients of Q.
Transmission problems for the vector Helmholtz equation
Authors: Wilde Peter
Published: November 2011

Synopsis

Transmission problems for the vector Helmholtz equation are considered. By using boundary integral equation methods, existence and uniqueness theorems in the form of Fredholm's alternative are established.
On the asymptotics of a Wiener integral
Authors: van den Berg M. , Lewis J. T.
Published: November 2011

Synopsis

Let mt(ω) be the range of a standard brownian bridge on R with ω(0) = ω(t) = 0 and let µt(ω) be the corresponding Wiener measure. We determine the asymptotic behaviour for large t of ∫ e−G(mt(ω)) µt(dω) for an increasing convex function G.
Low energy resolvent estimates and continuity of time-delay operators
Authors: Xue-Ping Wang
Published: November 2011

Synopsis

In this paper, we are interested in the L2-continuity of the Eisenbud–Wigner time-delay operator in potential scattering theory. Using the ideas due to Jensen–Kato [5], we first establish some low energy estimates on the resolvent of the Schrödinger operator and its derivative in weighted Sobolev spaces. Then applying these results together with the global decay of the wave functions (Lemma 3.2), we show that the Eisenbud–Wigner time-delay operator extends to a bounded operator on L2(Rn) with n ≧ 4, on condition that the potential V(x) decreases as fast as 0(|x | −4−ε) at infinity and that 0 is neither the eigenvalue nor the resonance for the Schrodinger operator –Δ + V for n = 4 or 5.
Conjugacy classes in finite groups
Authors: López Antonio Vera
Published: November 2011

Synopsis

In this paper, the number of conjugacy classes in a finite group G is analysed in terms of the number of ordered pairs that generate it. Using this relation, we give a new elementary proof of one of A. Mann's results for finite groups, namely: |G| ≡ r(G) (mod. d|G|. δ|G|), where , prime and pi ≠ Pj for every i≠j, r(G) denotes the number of conjugacy classes of elements of G, d|G| = g.c.d. (p1 − 1, … pt − 1) and δ|G| = g.c.d. . The above congruence is obtained without using character theory. We also obtain new local congruences that slightly improve Mann's congruence.
On a new class of Hardy type inequalities
Authors: Pachpatte B.G.
Published: November 2011

Synopsis

In this paper we establish a new class of integral inequalities which originate from the well-known Hardy's inequality. The analysis used in the proofs is quite elementary and is based on the idea used by Levinson to obtain generalisations of Hardy's inequality.
Global existence and convergence to a singular steady state for a semilinear heat equation
Authors: Lacey A.A. , Tzanetis D.
Published: November 2011

Synopsis

With certain initial and boundary conditions the solution u* to the semilinear heat equation ∂u*/∂t = ∂u* + λ * f(u*), where f is a positive superlinear function and λ is the supremum of the open spectrum for the steady state problem Δw + λf(w) = 0, is found to exist for all time and to be unbounded. Moreover u* approaches w* a singular steady state, as / tends to infinity.
Conformal maps in methods of asymptotic continuation
Authors: Gingold H.
Published: November 2011

Synopsis

Take the coefficients of a Taylor series expansion of a holomorphic function about its regular point zR. It is known that the holomorphic function possesses an asymptotic expansion about a possibly singular point zs. We show how to construct and calculate the coefficients in the asymptotic expansion from the coefficient of the Taylor series. The main theorem demonstrates that a suitable conformal map is a decisive step in dealing with the problem above. Therefore, a suitable conformal map is critical to a successful summation of divergent series. Some other methods which utilise orthogonal polynomial and Cesaro summability are also discussed. The paper may serve as a theoretical basis for a new computational method.
Periodic solutions of forced Liénard equations with jumping nonlinearities under nonuniform conditions
Authors: Iannacci R. , Nkashama M.N. , Omari P. , Zanolin F.
Published: November 2011

Synopsis

This paper is devoted to the existence of periodic solutions for the scalar forced Lienard differential equation

The key assumptions relate the asymptotic behaviour asx→± ∞ofg(t;x)/xto the “critical values” of the positively 1-homogeneous problem

No condition onf, except continuity, is assumed. Our approach is based on Leray–Schauder degree techniques anda prioriestimates.
Some triple trigonometric series
Authors: Melrose G. , Tweed J.
Published: November 2011

Synopsis

In this paper the authors investigate two classes of triple series equations of trigonometric type. Closed form solutions are given for eight triple series in each class.
On certain permutation representations of the Hall–Janko group
Authors: Prince Alan R.
Published: November 2011

Synopsis

Certain permutation representations of the Hall–Janko group J2 are studied. These representations are of interest in connection with the problem of whether J2 can act as astrongly irreducible collineation group of a finite projective plane.
A dual method in the calculation of homogenisation and applications
Authors: Qi Tang
Published: November 2011

Synopsis

We give a new method for calculating the Γ-limit functional encountered in the problems of homogenisation. We use the Legendre–Lagrange transform in the convex analysis and regularisation method to obtain the explicit expression of the Γ-limit functional. The result can be applied to some nonlocal function spaces.
Restrictions on microstructure
Authors: Bhattacharya Kaushik , Firoozye Nikan B. , James Richard D. , Kohn Robert V.
Published: November 2011

We consider the following question: given a set of matriceswith no rank-one connections, does it support a nontrivial Young measure limit of gradients? Our main results are these: (a) a Young measure can be supported on four incompatible matrices; (b) in two space dimensions, a Young measure cannot be supported on finitely many incompatible elastic wells; (c) in three or more space dimensions, a Young measure can be supported on three incompatible elastic wells; and (d) ifsupports a nontrivial Young measure with mean value 0, then the linear span ofmust contain a matrix of rank one.
Idempotent depth in semigroups of order-preserving mappings
Authors: Higgins Peter M.
Published: November 2011

We introduce algorithms for calculating minimum length factorisations of order-preserving mappings on a finite chain into products of idempotents, and into products of idempotents of defect one. The least upper bounds for these lengths are given.
Asymptotic convergence to dipole solutions in nonlinear parabolic equations
Authors: Galaktionov Victor A. , Posashkov Sergey A. , Vazquez Juan Luis
Published: November 2011

We study the asymptotic behaviour as t → ∞ of the solution u = u(x, t) ≧ 0 to the quasilinear heat equation with absorption ut = (um)xx − f(u) posed for t > 0 in a half-line I = { 0 < x < ∞}. For definiteness, we take f(u) = up but the results generalise easily to more general power-like absorption terms f(u). The exponents satisfy m > 1 and p >m. We impose u = 0 on the lateral boundary {x = 0, t > 0}, and consider a non-negative, integrable and compactly supported function uo(x) as initial data. This problem is equivalent to solving the corresponding equation in the whole line with antisymmetric initial data, uo(−x) = −uo(x).
Equations for biological evolution
Authors: Calsina Angel , Perelló Carles
Published: November 2011

In this paper we consider mathematical models inspired by the mechanisms of biological evolution. We take populations which are subject to interaction and mutation. In the cases we consider, the interaction is through competition or through a prey-predator relationship. The models consider the specific characteristics as taking values in real intervals and the equations are of the integro—differential type. In the case of competition, thanks to the fact that some of the equations have solutions which are quite explicit, we succeed in proving the existence of attracting stationary solutions. In the case of prey and predator, using techniques of dynamical systems in infinite-dimensional spaces, we succeed in showing the existence of a global attractor, which in some instances reduces to a point. Our analysis takes into account having δ distributions, corresponding to all individuals having the same characteristics, as possible populations.
Complexity in spatio-temporal dynamics
Authors: Rod D. L. , Sleeman B. D.
Published: November 2011

Complex and chaotic structures in certain dynamical systems in biology arise as a consequence of noncomplete integrability of two-degree-of-freedom Hamiltonian systems. A study of this problem is made using Ziglin theory and implemented with the aid of the Kovacic algorithm.
Asymptotic behaviour and attractors for a semilinear damped wave equation with supercritical exponent
Authors: Feireisl Eduard
Published: November 2011

We prove the existence of a global attractor for the problem

where f is ‘coercive at infinity’ and satisfies the growth condition

while g ϵL2(R3).
Trace functions on the algebras of certain E-unitary inverse semigroups
Authors: Crabb M. J. , Munn W. D.
Published: November 2011

A construction is given for a trace function on the semigroup algebra of a certain type of E-unitary inverse semigroup over any subfield of the complex field that is closed under complex conjugation. In particular, the method applies to the semigroup algebras of free inverse semigroups of arbitrary rank.
On the existence of solitary waves in rotating fluids
Authors: Sun S. M.
Published: November 2011

This paper considers the existence of axisymmetric solitary waves in an inviscid and incompressible rotating fluid bounded by a rigid cylinder. It has been obtained by many experiments and formal derivations that this flow has internal solitary waves in the fluid when equilibrium state at infinity satisfies certain conditions. This paper gives a rigorous proof of the existence of solitary wave solutions for the exact equations governing the flow under such conditions at infinity, and shows that the first-order approximations of the solitary wave solutions for the exact equations are solitary wave solutions derived formally using long-wave approximation. The ideas in the proof of the existence of solitary waves in two-dimensional stratified fluids are used and a main difficulty from the singularity at axis of rotation is overcome.
Multiscale convergence and reiterated homogenisation
Authors: Allaire G. , Briane M.
Published: November 2011

This paper generalises the notion of two-scale convergence to the case of multiple separated scales of periodic oscillations. It allows us to introduce a multi-scale convergence method for the reiterated homogenisation of partial differential equations with oscillating coefficients. This new method is applied to a model problem with a finite or infinite number of microscopic scales, namely the homogenisation of the heat equation in a composite material. Finally, it is generalised to handle the homogenisation of the Neumann problem in a perforated domain.
L1-approximation of Fourier series of complex-valued functions
Authors: Xie T. F. , Zhou S. P.
Published: November 2011

V. B. Stanojevic suggested in her recent paper that it would be of interest to prove a corresponding L1-convergence theorem for Fourier series with complex O-regularly varying quasimonotonc coefficients. The present paper will discuss this question and establish L1-convergence and. furthermore. L1-approximation theorems for complex-valued integrable functions.
On the convolution operators arising in the study of abstract initial boundary value problems
Authors: Alonso-Mallo I. , Palencia C.
Published: November 2011

We consider convolution operators arising in the study of abstract initial boundary value problems. These operators are of the form

where {S(t)}t ≧0 is a C0-semigroup in a Banach space X,, with infinitesimal generator A0,: D(A0), ⊂ X, → X, and K(z): Y → X is a linxear, continuous mapping defined in another Banach space Y., We study the continuity of T between the spaces Lp([0, + ∞), Y), and Lq([0, + ∞), X), 1 ≦ p, q, ≦ + ∞. We give several examples of the applicability of the results to some familiar initial boundary value problems, including both parabolic and hyperbolic cases.
Singularly perturbed ordinary differential equations with dynamic limits
Authors: Artstein Zvi , Vigodner Alexander
Published: November 2011

Coupled slow and fast motions generated by ordinary differential equations are examined. The qualitative limit behaviour of the trajectories as the small parameter tends to zero is sought after. Invariant measures of the parametrised fast flow are employed to describe the limit behaviour, rather than algebraic equations which are used in the standard reduced order approach. In the case of a unique invariant measure for each parameter, the limit of the slow motion is governed by a chattering type equation. Without the uniqueness, the limit of the slow motion solves a differential inclusion. The fast flow, in turn, converges in a statistical sense to the direct integral, respectively the set-valued direct integral, of the invariant measures.
The initial-value problem for the forced Korteweg-de Vries equation
Authors: Bona Jerry L. , Zhang Bing-Yu
Published: November 2011

The initial-value problem for the Korteweg-de Vries equation with a forcing term has recently gained prominence as a model for a number of interesting physical situations. At the same time, the modern theory for the initial-value problem for the unforced Korteweg-de Vries equation has taken great strides forward. The mathematical theory pertaining to the forced equation is currently set in narrow function classes and has not kept up with recent advances for the homogeneous equation. This aspect is rectified here with the development of a theory for the initial-value problem for the forced Korteweg-de Vries equation that entails weak assumptions on both the initial wave configuration and the forcing. The results obtained include analytic dependence of solutions on the auxiliary data and allow the external forcing to lie in function classes sufficiently large that a Dirac δ-function or its derivative is included. Analyticity is proved by an infinite-dimensional analogue of Picard iteration. A consequence is that solutions may be approximated arbitrarily well on any bounded time interval by solving a finite number of linear initial-value problems.
Positivity of solutions of elliptic equations with nonlocal terms
Authors: Allegretto W. , Barabanova A.
Published: November 2011

In this paper we study a nonlocal problem for a second-order partial differential equation which depends on a parametern. We prove the existence of critical values 0 such that for all≦ɳ≦and for all non-negative right-hand sides, our problem has nonnegative solutions. We obtain a formula forɳ0, the maximal possible value of, and find the exact value of ɳ for spherical ɳ. We also study the corresponding eigenvalue problem. At the end of the paper, we consider the application of our results to the nonlinear system describing the distribution of temperature and potential in a microsensor.
Quasimonotone versus pseudomonotone
Authors: Landes Rüdiger
Published: November 2011

Under some natural hypotheses, we show that if the (Nemitsky-) operator associated with an elliptic system is pseudomonotone, then the system has to be quasimonotone. Conversely, if the system satisfies a strict quasimonotonicity condition, then an existence proof of K.-W. Zhang contains the arguments to verify the pseudomonotonicity of the operator. We present a simplified proof of this fact under more general assumptions.
A necessary and sufficient condition for finite speed of propagation in the theory of doubly nonlinear degenerate parabolic equations
Authors: Gilding B. H. , Kersner R.
Published: November 2011

A degenerate parabolic partial differential equation with a time derivative and first- and second-order derivatives with respect to one spatial variable is studied. The coefficients in the equation depend nonlinearly on both the unknown and the first spatial derivative of a function of the unknown. The equation is said to display finite speed of propagation if a non-negative weak solution which has bounded support with respect to the spatial variable at some initial time, also possesses this property at later times. A criterion on the coefficients in the equation which is both necessary and sufficient for the occurrence of this phenomenon is established. According to whether or not the criterion holds, weak travelling-wave solutions or weak travelling-wave strict subsolutions of the equation are constructed and used to prove the main theorem via a comparison principle. Applications to special cases are provided.
Uniqueness, multiplicity and stability for positive solutions of a pair of reaction–diffusion equations
Authors: Du Yihong
Published: November 2011

We study the number and stability of the positive solutions of a reaction–diffusion equation pair. When certain parameters in the equations are large, the equation pair can be viewed as singular or regular perturbations of some single (or essentially single) equation problems, for which the number and stability of their solutions can be well understood. With the help of these simpler equations, we are able to obtain a rather complete understanding of the number and stability of the positive solutions for the equation pair for the cases that certain parameters are large. In particular, we obtain a fairly satisfactory description of the positive solution set of the equation pair.
Spectral asymptotics with a remainder estimate for Schrödinger operators with slowly growing potentials
Authors: Levendorskiĭ S. Z.
Published: November 2011

We compute the principal term of the asymptotics with a remainder estimate for Schrödinger operators with slowly growing potentials q, a typical example being q(x) = In … In |x| outside some compact.
Three-phase boundary motions under constant velocities. I: The vanishing surface tension limit
Authors: Reitich Fernando , Soner H. Mete
Published: November 2011

In this paper, we deal with the dynamics of material interfaces such as solid–liquid, grain or antiphase boundaries. We concentrate on the situation in which these internal surfaces separate three regions in the material with different physical attributes (e.g. grain boundaries in a polycrystal). The basic two-dimensional model proposes that the motion of an interface Гij between regions i and j (i, j = 1, 2, 3, i ≠ j) is governed by the equation

Here Vij, kij, μij and fij denote, respectively, the normal velocity, the curvature, the mobility and the surface tension of the interface and the numbers Fij stand for the (constant) difference in bulk energies. At the point where the three phases coexist, local equilibrium requires that

In case the material constants fij are small, and ε ≪ 1, previous analyses based on the parabolic nature of the equations (0.1) do not provide good qualitative information on the behaviour of solutions. In this case, it is more appropriate to consider the singular case with fij = 0. It turns out that this problem, (0.1) with fij = 0, admits infinitely many solutions. Here, we present results that strongly suggest that, in all cases, a unique solution—‘the vanishing surface tension (VST) solution’—is selected by letting ε→0. Indeed, a formal analysis of this limiting process motivates us to introduce the concept of weak viscosity solution for the problem with ε = 0. As we show, this weak solution is unique and is therefore expected to coincide with the VST solution. To support this statement, we present a perturbation analysis and a construction of self-similar solutions; a rigorous convergence result is established in the case of symmetric configurations. Finally, we use the weak formulation to write down a catalogue of solutions showing that, in several cases of physical relevance, the VST solution differs from results proposed previously.
The distance function and defect energy
Authors: Aviles Patricio , Giga Yoshikazu
Published: November 2011

Several energies measuring jump discontinuities of a unit length gradient field are considered and are called defect energies. The main example is a total variation I(φ) of the hessian of a function φ in a domain. It is shown that the distance function is the unique minimiser of I(φ) among all non-negative Lipschitz solutions of the eikonal equation |grad φ| = 1 with zero boundary data, provided that the domain is a two-dimensional convex domain. An example shows that the distance function is not a minimiser of I if the domain is noncovex. This suggests that the selection mechanism by I is different from that in the theory of viscosity solutions in general. It is often conjectured that the minimiser of a defect energy is a distance function if the energy is formally obtained as a singular limit of some variational problem. Our result suggests that this conjecture is very subtle even if it is true.
Pseudo-orbit tracing property for random diffeomorphisms
Authors: Peidong Liu , Minping Qian , Fuchang Tang
Published: November 2011

In this paper we consider the pseudo-orbit tracing property for dynamical systems generated by iterations of random diffeomorphisms. We first define a type of hyperbolicity by means of a ‘random’ multiplicative ergodic theorem, and then prove our shadowing result by employing the graph transformation methods. That result applies to, for example, the case of small random diffeomorphisms type perturbations of hyperbolic sets of deterministic dynamical systems.
Some remarks on quasiconvexity and rank-one convexity
Authors: Pedregal Pablo
Published: November 2011

We explore some necessary conditions for quasiconvexity in an attempt to show that rank-one convexity does not imply quasiconvexity when the target space for deformations is two- dimensional. An interesting construction is presented, showing how rank-one directions may fit with each other, making the task harder than in higher dimensions.
Dense point spectrum for the one-dimensional Dirac operator with an electrostatic potential
Authors: Schmidt Karl Michael
Published: November 2011

For the one-dimensional Dirac operator, examples of electrostatic potentials with decay behaviour arbitrarily close to Coulomb decay are constructed for which the operator has a prescribed set of eigenvalues dense in the whole or part of its essential spectrum. A simple proof that the essential spectrum of one-dimensional Dirac operators with electrostatic potentials is never empty is given in the appendix.
Multiplicity of homoclinic solutions for singular second-order conservative systems
Authors: Bertotti Maria Letizia , Jeanjean Louis
Published: November 2011

We consider a class of second-order systems, withq(t) ∊ℝn, for which the potential energyV: ℝn\S→ℝ admits a (possibly unbounded) singular setS⊂ℝnand has a unique absolute maximum at 0 ∈ℝn. Under some conditions on S andV, we prove the existence of several solutions homoclinic to 0.
The limit of the anisotropic double-obstacle Allen–Cahn equation
Authors: Elliott Charles M. , Schätzle Reiner
Published: November 2011

In this paper, we prove that solutions of the anisotropic Allen–Cahn equation in doubleobstacle form

where A is a strictly convex function, homogeneous of degree two, converge to an anisotropic mean-curvature flow

when this equation admits a smooth solution in ℝn. Here VN and R respectively denote the normal velocity and the second fundamental form of the interface, and More precisely, we show that the Hausdorff-distance between the zero-level set of φ and the interface of the above anisotropic mean-curvature flow is of order O(ε2).
On classical solutions of the Prandtl-Reuss equations of perfect elastoplasticity
Authors: Kuksin Sergej B.
Published: November 2011

We study the elasticity domain for an antiplane deformation of a perfect elastoplastic medium, which is described by the Prandtl-Reuss equations. We prove that a boundary of this domain can be found by solving a system of nonlinear functional equations. In the simplest case of simple shear deformations, this system of equations is studied in detail.
Resolution of the spherical piston problem for compressible isentropic gas dynamics via a self-similar viscous limit
Authors: Slemrod M.
Published: November 2011

This paper proves the existence of solutions to the spherical piston problem for isentropic gas dynamics with equation of state p(p) = Apγ, γ≧ 1. The method of analysis is to replace the usual viscosity ε with εt, thus permitting a search for self-similar viscous limits. The main result of the paper is that self-similar viscous limits are proved to exist and converge to a solution to the piston problem when

N = 1, 2, 3 space dimensions.
Compactness and single-point blowup of positive solutions on bounded domains
Authors: Filippas Stathis , Merle Frank
Published: November 2011

This paper is concerned with the blowup of positive solutions of the semilinear heat equation

with zero boundary conditions. The domainΩis supposed to be smooth, convex and bounded. We first show that, under the assumption that the initial data are uniformly monotone near the boundary, solutions that exist on the time interval (0,Tform a compact family in a suitable topology. We then derive some localisation properties of these solutions. In particular, we discuss a general criterion, independent of the initial data, which in some cases ensures single-point blowup.
On rational de Rham cohomology associated with the generalised confluent hypergeometric functions I, ℙ1 case
Authors: Kimura Hironobu
Published: November 2011

We define the rational de Rham cohomology associated with the generalised confluent hypergeometric functions. Purity of the cohomology is proved and an explicit ℂ-basis of the nontrivial cohomology is computed.
Diffusion-mediated permanence problem for a heterogeneous Lotka–Volterra competition model
Authors: Furter J. E. , López-Gómez J.
Published: November 2011

We analyse the dynamics of a prototype model for competing species with diffusion coefficients (d1d2) in a heterogeneous environment Ω. When diffusion is switched off, at each pointx∊ Ω we have a pair of ODE's: thekinetic. If for somex∊ Ω kinetic has a unique stable coexistence state, we show that there existsuch that for everythe RD-model ispersistent, in the sense that it has a compact global attractor within the interior of the positive cone and has a stable coexistence state. The same result is true if there existxu,xv∊ Ω such that the semitrivial coexistence states (u, 0) and (0,v) of thekineticare globally asymptotically stable atx=xuandx=xv, respectively. More generally, our main result shows that, for most kinetic patterns, stable coexistence of xspopulations can be found for some range of the diffusion coefficients.

Singular perturbation techniques, monotone schemes, fixed point index, global analysis ofpersistence curves, global continuation and singularity theory are some of the technical tools employed to get the previous results, among others. These techniques give us necessary and/or sufficient conditions for the existence and uniqueness of coexistence states, conditions which can be explicitly evaluated by estimating some principal eigenvalues of certain elliptic operators whose coefficients are solutions of semilinear boundary value problems.

We also discuss counterexamples to the necessity of the sufficient conditions through the analysis of the local bifurcations from the semitrivial coexistence states at the principal eigenvalues. An easy consequence of our analysis is the existence of models having exactly two coexistence states, one of them stable and the other one unstable. We find that there are also cases for which the model hasthree or morecoexistence states.
On non-negative quasiconvex functions with unbounded zero sets
Authors: Zhang Kewei
Published: November 2011

We construct nontrivial, non-negative quasiconvex functions denned on M2×2 with p-th order growth such that the zero sets of the functions are Lipschitz graphs of mappings from subsets of a fixed two-dimensional subspace to its orthogonal complement. We assume that the graphs do not have rank-one connections with the Lipschitz constants sufficiently small. In particular, we are able to construct quasiconvex functions which are homogeneous of degree p (p > 1) and ‘conjugating’ invariant.
Positive solutions for the p-Laplacian: application of the fibrering method
Authors: Drábek Pavel , Pohozaev Stanislav I.
Published: November 2011

Using the fibrering method, we prove the existence of multiple positive solutions of quasilinear problems of second order. The main part of our differential operator is p-Laplacian and we consider solutions both in the bounded domain Ω⊂ℝN and in the whole of ℝN. We also prove nonexistence results.
Bifurcation from homoclinic to periodic solutions in singularly perturbed differential inclusions
Authors: Fečkan Michal
Published: November 2011

The existence of periodic solutions is studied for certain singularly perturbed differential inclusions. Applications are given to dry friction problems.
Invariant radicals
Authors: Gardner B. J. , Kelarev A. V.
Published: November 2011

We describe all pairs of semigroups S and radicals ρ, such that ρ is invariant in S-graded rings. This generalises several known results due to Amitsur and Sands.
Travelling fronts in nonlocal models for phase separation in an external field
Authors: Orlandi Enza , Triolo Livio
Published: November 2011

We consider the one-dimensional, nonlocal, evolution equation derived by De Masi et al. (1995) for Ising systems with Glauber dynamics, Kac potentials and magnetic field. We prove the existence of travelling fronts, their uniqueness modulo translations among the monotone profiles and their linear stability for all the admissible values of the magnetic field for which the underlying spin system exhibits a stable and metastable phase.
On the period maps of projective hypersurfaces
Authors: Ran Z.
Published: November 2011

This paper gives an explicit infinitesimal (to all orders) description of the period map associated to a smooth projective hypersurface, as well as related objects such as the full Hodge filtration on the middle cohomology, the local moduli space and the Gauss–Manin connection and its iterates.
Weak asymptotic decay via a “relaxed invariance principle” for a wave equation with nonlinear, non-monotone damping
Authors: Slemrod M.
Published: November 2011

Synopsis

This paper considers the problem of asymptotic decay as t → ∞ of solutions of the wave equation utt − Δu = −a(x)β(ut, ∇u), (t,x) ∊ ℝ+ × Ω (a bounded, open, connected set in ℝN, N≧ 1, with smooth boundary), u =0 on ℝ+ × ∂Ω. The nonlinear function β is not assumed to be globally Lipschitz continuous, β(0, y2, …, yN+1) = 0, y1 β(y1…,yN+1) ≧ 0 for all y ∊ ℝN+1; β is not assumed to be monotone in y1. Under additional restrictions on the kernel of β, conditions are given which imply that [u, ut,] converges to [0,0] weakly in H = H10(Ω) × L2(Ω) as t → ∞. The work generalises earlier results of Dafermos and Haraux where strong decay in H as t → ∞ was obtained in the case β(y1 …, yN+1) = q(y1), q monotone on ℝ.
The displacement problem for elastic crystals
Authors: Fonseca I. , Tartar L.
Published: November 2011

Synopsis

In this paper we obtain necessary and sufficient conditions for the existence of Lipschitz minimisers of a functional of the type

where h is a convex function converging to infinity at zero and u is subjected to displacement boundary conditions. We provide examples of body forces f for which the infimum of J(.) is not attained.
Stability and regularity in bifurcations of planar vector fields
Authors: Sotomayor J.
Published: November 2011

Synopsis

Bifurcations appearing in generic one-parameter families of vector fields on a compact planar region are conceptually characterised in terms of their stability and regularity properties.
Weak solutions to the initial-boundary value problem for the shearing of non-homogeneous thermoviscoplastic materials
Authors: Charalambakis Nicolas , Murat François
Published: November 2011

Synopsis

We prove the existence of a weak solution for the system of partial differential equations describing the shearing of stratified thermoviscoplastic materials with temperature-dependent non-homogeneous viscosity.
A differential equation in the theory of resonant oscillations of water waves
Authors: Amick C. J. , Toland J. F.
Published: November 2011

Synopsis

We show that the problem

has a unique solution and that the asymptotic behaviour of this solution as x → −∞ is given by w(x) = x + B ∣x∣−1/4 cos (b ∣x∣3/2 + φ) + O(x−1), for some constant φ. Here B ≠ 0 depends on ε and In particular there are no even solutions of problem (*) for any ε > 0.