bluuebunny/crossref_metadata_2025 huggingface dataset:
Nonexistence of radial solutions of two elliptic boundary value problems
Authors: Ortega Rafael
Published: November 2011

Synopsis

We consider two boundary value problems (of Neumann or related type) associated with the equationin Ω. The existence of a solution was previously established assuming thatp<N/(N−s2). (Ndimension of Ω.) We prove that this exponent is critical for these problems, at least in the radially symmetric case when Ω is a ball. This is understood in the sense that the existence result does not hold whenp≧N/(N− 2).
The extension of optimisation problems containing controls in the coefficients
Authors: Lurie K. A.
Published: November 2011

Synopsis

The paper suggests a procedure for direct construction of minimal extensions of constrained optimisation problems, particularly those containing controls in coefficients of elliptic equations. The preliminary version of the procedure has been described in [1].
Almost periodic solutions in an integrodifferential equation
Authors: Hamaya Y. , Yoshizawa T.
Published: November 2011

Synopsis

We consider a system of integrodifferential equations

wheref(t,x) andF(t,s,x,y) are almost periodic intuniformly for parameters, and we assume that the system has a bounded solutionu(t). To discuss the existence of an almost periodic solution, we consider the relationship between the total stability ofu(t) with respect to a certain metric ρ and the separation condition with respect to ρ. Moreover, we discuss a sufficient condition for the existence of a positive almost periodic solution of a model of the dynamics of ann-species system.
Idempotent rank in finite full transformation semigroups
Authors: Howie John M. , McFadden Robert B.
Published: November 2011

Synopsis

The subsemigroup Singn of singular elements of the full transformation semigroup on a finite set is generated by n(n − l)/2 idempotents of defect one. In this paper we extend this result to the subsemigroup K(n, r) consisting of all elements of rank r or less. We prove that the idempotent rank, defined as the cardinality of a minimal generating set of idempotents, of K(n, r) is S(n, r), the Stirling number of the second kind.
Existence and stability of travelling wave solutions for an evolutionary ecology model
Authors: Lui Roger
Published: November 2011

Synopsis

Monotone travelling wave solutions are known to exist for Fisher's equation which models the propagation of an advantageous gene in a single locus, two alleles population genetics model. Fisher's equation assumed that the population size is a constant and that the fitnesses of the individuals in the population depend only on their genotypes. In this paper, we relax these assumptions and allow the fitnesses to depend also on the population size. Under certain assumptions, we prove that in the second heterozygote intermediate case, there exists a constant θ*>0 such that monotone travelling wave solutions for the reaction–diffusion system exist whenever θ > θ*. We also discuss the stability properties of these waves.
On the topological properties of symplectic maps
Authors: Hofer H.
Published: November 2011

Synopsis

In this paper we show that symplectic maps have surprising topological properties. In particular, we construct an interesting metric for the symplectic diffeomorphism groups, which is related, but not obviously, to the topological properties of symplectic maps and phase space geometry. We also prove a certain number of generalised symplectic fixed point theorems and give an application to a Hamiltonian system.
Compact attractor for a nonlinear wave equation with critical exponent
Authors: Lopes Orlando
Published: November 2011

Synopsis

In this paper we study the existence of a compact attractor for the solutions of the equation utt − Δu + cut + f(u) = h(t, x), x ∊ ℝ3. The phase space is H1 × L2 and periodicity in the x-variables is taken as a boundary condition. Besides the usual coercive condition, we assume f satisfies the growth condition |f′(u)|≦ a + bu2; this growth condition is critical because the embedding H1 → L6 is not compact. In the proof we use an Lp − H1.q estimate for the linear homogeneous wave equation.
Nonlinear analytic semiflows
Authors: Angenent Sigurd B.
Published: November 2011

Synopsis

In this paper a local existence and regularity theory is given for nonlinear parabolic initial value problems (x′(t) = f(x(t))), and quasilinear initial value problems (x′(t)=A(x(t))x(t) + f(x(t))). This theory extends the theory of DaPrato and Grisvard of 1979, and shows how various properties, like analyticity of solutions, can be derived as a direct corollary of the existence theorem.
On contracted semigroup rings
Authors: Munn W. D.
Published: November 2011

Synopsis

Sufficient conditions are given for a contracted semigroup ring, in which the two-sided ideals have a certain property, to be (a) semiprime, (b) semiprimitive, (c) prime, (d) primitive. The results are applied to the contracted semigroup rings of inverse semigroups, where they provide new proofs of theorems of Domanov.
Asymptotic theory for third-order differential equations with extension to higher odd-order equations
Authors: Al-Hammadi A. S. A.
Published: November 2011

Synopsis

An asymptotic theory is developed for linear differential equations of odd order. Theory is applied with large coefficients. The forms of the asymptotic solutions are given under general conditions on the coefficients.
Noncoercive variational inequalities with application to friction problems
Authors: Shi P. , Shillor M.
Published: November 2011

Synopsis

Noncoercive variational inequalities with sublinear functionals are considered. Necessary and sufficient conditions are given for the solvability of such problems. These conditions are in the form of compatibility conditions-for the data, as well as the boundedness of the solutions to related problems. These results are used for the obstacle problems for the membrance and the elastic contact in the presence of friction.
Symplectic connections and the linearisation of Hamiltonian systems
Authors: Marsden J. E. , Ratiu T. , Raugel G.
Published: November 2011

Synopsis

This paper uses symplectic connections to give a Hamiltonian structure to the first variation equation for a Hamiltonian system along a given dynamic solution. This structure generalises that at an equilibrium solution obtained by restricting the symplectic structure to that point and using the quadratic form associated with the second variation of the Hamiltonian (plus Casimir) as energy. This structure is different from the well-known and elementary tangent space construction. Our results are applied to systems with symmetry and to Lie–Poisson systems in particular.
On perturbations of a translationally-invariant differential equation
Authors: Magnus R.J.
Published: November 2011

Synopsis

We study certain perturbations of the differential equation Δu − u + up = 0 on all of n-dimensional Euclidean space. Conditions are obtained which ensure the existence of a solution to the perturbed equation near a given solution to the unperturbed equation. We have to overcome degeneracy of the unperturbed solution and lack of smooth dependence on the perturbation parameter. An abstract version of the argument is sketched in a functional-analytic setting related toequivariant bifurcation theory. We consider also a smooth perturbation with several parameters and study the singularities of the mapping which maps each solution to its associated parameters.
Equichordal curves
Authors: Carvalho F.J.Craveiro de , Robertson S.A.
Published: November 2011

Synopsis

In this paper we extend the notion of equichordal curve to closed simple curves in ℝn. Although it is not known if an equichordal curve can have more than one fulcrum, we show that, for plane curves, any fulcrum is inside the curve. We establish connections with the theories of transnormal and self-parallel curves and lower bounds for the length and chordal area are obtained. Such bounds are the best possible.
The steady state Navier–Stokes equations for incompressible flows with rotating boundaries
Authors: Sauer Niko
Published: November 2011

Synopsis

When a rigid body performs a rotation in a fluid, the system of governing equations consists of conservation of linear momentum of the fluid and conservation of angular momentum of the rigid body. Since the torque at the interface involves the drag due to the fluid flow, the conservation of angular momentum may be viewed as a boundary condition for the field equations of fluid motion. The familiar no-slip condition becomes an additional equation in the system which not only governs the fluid motion, but also the motion of the rigid body. The unknown functions in the system of equations are the velocity field and the pressure field of the fluid motion and the angular velocity of the rigid body.

In this paper we obtain existence and uniqueness results for the steady state problem in which a rigid body rotates about an axis of symmetry in a viscous incompressible fluid.
Output stabilisation for a class of linear parabolic differential equations
Authors: Nambu Takao
Published: November 2011

Synopsis

We study the output stabilisation for a class of linear parabolic differential equations in a Hilbert space by means of feedback controls. The output is given as a finite number of linear functionals. Stabilisationof the state, of course, implies stabilisation of the output. In the present paper, however, we give a sufficient condition (an algebraic condition on the above functionals) for the output stabilisation, which is weakerin some sense than that for the state stabilisation.
Two perturbation results for nondegenerate solutions of some semilinear Dirichlet problems
Authors: Coclite Mario Michele
Published: November 2011

Synopsis

The stability of nondegenerate solutions of some semilinear Dirichlet problems is studied. Two specific situations are considered: firstly, a singular perturbation of the differential operator; secondly, a perturbation of the nonlinear term using a term which also depends on the gradient of the solution.
Local minimisers and singular perturbations
Authors: Kohn Robert V. , Sternberg Peter
Published: November 2011

Synopsis

We construct local minimisers to certain variational problems. The method is quite general and relies on the theory of Γ-convergence. The approach is demonstrated through the model problem

It is shown that in certain nonconvex domains Ω ⊂ ℝn and for ε small, there exist nonconstant local minimisers uε satisfying uε ≈ ± 1 except in a thin transition layer. The location of the layer is determined through the requirement that in the limit uε →u0, the hypersurface separating the states u0 = 1 and u0 = −1 locally minimises surface area. Generalisations are discussed with, for example, vector-valued u and “anisotropic” perturbations replacing |∇u|2.
Non-resonance for linear differential systems
Authors: Eastham M.S.P. , McLeod J.B.
Published: November 2011

Synopsis

The existence of a set of real numbers σ, depending only on ∧0, is established such that if λ ∉ σ, the system Y'=(i∧0 + R)Y can be transformed into a system Z' = (i∧ + S)Z of the Levinson form. Here ∧0 and ∧ are real diagonal matrices, with ∧0 constant, and R(x) = ξ(x)T(λx). The scalar factor ξ(x) is o(l) (x →∞) and T belongs to a certain class of periodic matrices. The consequences for non-resonance, and the necessity of this class, are discussed.
Vector bundles over (8k + 5)-dimensional manifolds
Authors: Ng Tze-Beng
Published: November 2011

Suppose that M is a closed, connected and smooth manifold of dimension n = 8k + 5, with k ≧1. Let η be an n-plane bundle over M. Under suitable conditions on M, we derive necessary and sufficient conditions for the span of η to be ≧j, j = 5 or 6. We then apply the results to the tangent bundle of M. In particular, we prove a conjecture of E. Thomas, namely, if M is 3-connected mod 2, then span M ≧ 5 if, and only if, χ2(M) = 0. We prove that if also w8k(M) = 0, then span M≧6. We also derive some immersion theorems for M.
Lifting homotopies through fixed points II
Authors: Armstrong M. A.
Published: November 2011

Synopsis

This note complements an earlier paper of the same title. Let G be a discontinuous group of homeomorphisms of a connected, locally path connected, Hausdorff space X, and let ∏:X → X/G denote the associated projection. We work relative to a G-invariant subgroup H of the fundamental group of X and investigate the quotient group ∏1(X/G)/∏*(H). By choosing H appropriately, we can calculate ∏1(X/G) and show that ∏1(X/G)/∏*(∏1(X)) is isomorphic to G/F, where F is the normal subgroup of G generated by those elements which have fixed points. In a final section, we give analogous results for actions of a compact Lie group.
Integrable spreads
Authors: Farran H. R. , Robertson S. A.
Published: November 2011

Synopsis

For any integer k such that 0≦k≦m, Mk denotes the Grassmann bundle of tangent k-planes on the m-manifold M. A k-spread on M is a field Φ of tangent k-planes on Mk such that the derivative of the projection maps Φ(λ) to λ. Previous work by Douglas and others studied the local properties of such spreads. Here we develop the global theory, with special emphasis on the case in which Φ is integrable.
The Dirichlet index under minimal conditions
Authors: Brown R. C.
Published: November 2011

Synopsis

We extend several known properties of the Dirichlet index to the case of minimal conditions and prove that the index is invariant under positive t bounded perturbations of the associated quadratic form. The index is also shown to be minimal for fourth order operators with certain growth conditions on some of their coefficients.
Nonselfadjoint Schrödinger operators with singular first-order coefficients
Authors: Kato Tosio
Published: November 2011

Synopsis

Schrödinger operators of the form T = (i grad + b(x))2 + a(x) · grad + q(x) in Rm are considered, where a, b ate real vector-valued functions and q is a scalar complex-valued function. It is shown that T is essentially quasi-m-accretive in L2(Rm) if (1 + #x2223;∣)−1a ∈ L4 + L∞, div a ∈ L∞, , and Re q ≧ 0. The proof is elementary.
Double MS -algebras
Authors: Blyth T. S. , Varlet J. C.
Published: November 2011

Synopsis

We introduce the notion of a double MS-algebra (L, 0, +) as an MS-algebra (L, 0) whose dual isan MS-algebra (Ld, +), with certain linking conditions concerning the operations x↦x0 and x↦x+. We determine necessary and sufficient conditions whereby an MS-algebra can be made into a double MS-algebra and show that this, when possible, can be done in one and only one way. We also consider the notion of a bistable subvariety of MS-algebras, namely a subvariety R with the property that, for every double MS-algebra (L, 0, +), whenever L, 0 ↦ R, we have (Ld,+)↦ R. Finally, we determine those subvarieties R of MS that are dense (in the sense that every MS-algebra L ↦ R can be made into a double MS-algebra), and those that are sparse (in the sense that if L ↦ R can be made into a double MS-algebra then it belongs to a proper subclass of R).
Asymptotic behaviour of solutions of some general nonlinear differential equations and integral inequalities
Authors: Beesack Paul R.
Published: November 2011

Synopsis

We deal with the asymptotic behaviour, as t→∞, of complex-valued solutions of nonlinear differential equations

Upper bounds for ∣x(l)(t)∣, 0≦j≦n, are obtained by obtaining upper bounds for solutions u(t) of Bihari-type integral inequalities of the form
Fibrewise P-universal nilpotent fibrations
Authors: Scheerer H.
Published: November 2011

Synopsis

Let P be a set of primes. A nilpotent space X is called “P-universal”, if its P-localization canbe obtained as a direct limit over a sequence of selfmaps of X. A nilpotent fibration is called “fibrewiseP-universal”, if its fibrewise P-localization can be obtained in a similar way as a direct limit of fibrewise maps. In this paper, the following results are proved. Let π:E → B be a nilpotent fibration of connected finite or cofinite spaces. If π is fibrewise P-universal for some proper subset P of the set of primes, then its minimal model (in the sense of D. Sullivan's rational homotopy theory) admits a certaintype of weight decomposition. But the existence of such a weight decomposition implies only that there exists a finite set of primes, such that IT is fibrewise P-universal for all P with . In the absolute case however, i.e.B is a point, the set P can be chosen as the empty set. Thus a result announced by R. Body and D. Sullivan is recovered.
The form of blow-up for nonlinear parabolic equations
Authors: Lacey A. A.
Published: November 2011

Synopsis

Semilinear parabolic equations of the form u1 = ∇2u + δf(u), where f is positive and is finite, are known to exhibit the phenomenon of blow-up, i.e. for sufficiently large S, u becomes infinite after a finite time t*. We consider one-dimensional problems in the semi-infinite region x>0 and find the time to blow-up (t*). Also, the limiting behaviour of u as t→t*- and x→∞ is determined; in particular, it is seen that u blows up at infinity, i.e. for any given finite x, u is bounded as t→t*. The results are extended to problems with convection.

The modified equation xu, = uxx +f(u) is discussed. This shows the possibility of blow-up at x =0 even if u(0, f) = 0. The manner of blow-up is estimated.

Finally, bounds on the time to blow-up for problems in finite regions are obtained by comparing u with upper and lower solutions.
Asymptotic behaviour of finite energy solutions of polyharmonic equations
Authors: Yu Lao Sen
Published: November 2011

Synopsis

Asymptotic behaviour of solutions of polyharmonic equations

on exterior domains in Rn is considered under suitable conditions on f. It is shown that finite energysolutions u have the asymptotic property

This partially extends results of Egnell [6] for m = 1.
Strong asymptotic stability for a beam equation with weak damping
Authors: Feireisl Eduard
Published: November 2011

Synopsis

Any solution to the problem

is shown to decay to zero in the strong topology of the energy space as t→ ∞. The function β is allowed to be nonmonotone and d is a nonnegative function strictly positive on a nonvoid subset of (0, л).
The theory of Legendrian unfoldings and first-order differential equations
Authors: Izumiya Shyuichi
Published: November 2011

Synopsis

We consider some properties of completely integrable first-order differential equations for real-valued functions. In order to study this subject, we introduce the theory of Legendrian unfoldings. We give a characterisation of equations with classical complete solutions in terms of Legendrian unfoldings, and also assert that the set of equations with singular solutions is an open set in the space of completely integrable equations even though such a set is thin in the space of all equations.
Permanence in ecological systems with spatial heterogeneity
Authors: Cantrell Robert Stephen , Cosners Chris , Hutson Vivian
Published: November 2011

Synopsis

A basic problem in population dynamics is that of finding criteria for the long-term coexistence of interacting species. An important aspect of the problem is determining how coexistence is affected by spatial dispersal and environmental heterogeneity. The object of this paper is to study the problem of coexistence for two interacting species dispersing through a spatially heterogeneous region. We model the population dynamics of the species with a system of two reaction–diffusion equations which we interpret as a semi-dynamical system. We say that the system is permanent if any state with all components positive initially must ultimately enter and remain within a fixed set of positive states that are strictly bounded away from zero in each component. Our analysis produces conditions that can be interpreted in a natural way in terms of environmental conditions and parameters, by combining the dynamic idea of permanence with the static idea of studying geometric problems via eigenvalue estimation.
The existence of principal eigenvalues for problems with indefinite weight function on ℝk
Authors: Brown K. J. , Tertikas A.
Published: November 2011

Synopsis

We investigate the existence of positive principal eigenvalues of the problem - ∆u(x) = λg(x)u(x) for x ∈ ℝk where the weight function g changes sign in ℝk and is negative for |x| sufficiently large.
The distance between unitary orbits of normal elements in the Calkin algebra
Authors: Davidson Kenneth R.
Published: November 2011

Synopsis

Let s and t be normal elements of the Calkin algebra, and let (s) denote the unitary orbit of s. A formula ρ(s,t) is defined to measure the distance between unitary orbits, and satisfies
A weighted norm inequality for the Hankel transformation
Authors: Heywood P. , Rooney P. G.
Published: November 2011

Synopsis

We give conditions on pairs of non-negative weight functions U and V which are sufficient that for 1<p≤<∞

where Hλ is the Hankel transformation.

The technique of proof is a variant of Muckenhoupt's recent proof for the boundedness of the Fourier transformation between weighted Lp spaces, and we can also use this variant to prove a somewhat different boundedness theorem for the Fourier transformation.
The number of solutions to an equation from catalysis
Authors: Hastings S. P. , McLeod J. B.
Published: November 2011

Synopsis

In catalysis theory there is interest in the number of solutions to the equation

with the boundary conditions

the parameters λ,β, γ being all positive and p a non-negative integer. The paper answers this question when γ is large, which is the interesting situation physically. Although the treatment is somewhat different in the cases p = 0 and p ≠ 0, the final answer is the same, that is, given β, there exist two positive functions λ1(γ) and λ2(γ) such that the problem has one solution if λ<λ1(γ), or λ>λ2(γ), three solutions if λ1(γ)<λ <λ2(γ), and two solutions if λ=λ1(γ) or λ=λ2(γ).
Generic properties of equilibria of reaction-diffusion equations with variable diffusion
Authors: Rocha Carlos
Published: November 2011

Synopsis

It is shown that, generically, scalar one-dimensional parabolic equations ut = (a2(x)ux)x + f(u), x ∈ [0, 1], with Neumann boundary conditions, have all the equilibrium solutions hyperbolic.

Moreover, the bifurcations of these equilibria are generically of the saddle-node type.
Boundary value problems for systems of second order differential equations
Authors: Erbe L. H. , Knobloch H. W.
Published: November 2011

Synopsis

We consider boundary value problems for second order differential systems of the form (1)x” = A(t)x′ + f(t, x) and (2) x” = A(t)x′ + f(t, x) + q(t, x). By assuming the existence of a solution to (1) with a given region in (t, x) space, we derive conditions under which there exists a solution to (2) which stays in a certain neighbourhood of and satisfies given boundary conditions.
Uniform operator σ-additivity of indefinite integrals induced by scalar-type spectral operators
Authors: Okada S. , Ricker W.
Published: November 2011

Synopsis

This note characterises those Banach space valued, scalar-type spectral operators T = ∫ z dP(z), where P is the resolution of the identity for T, whose indefinite spectral integral E→∫EzdP(z) as a set function of the Borel sets of the complex plane is countably additive with respect to the uniform operator topology.
Boundary integral equations for magnetic screens in ℝ3
Authors: Stephan Ernst P.
Published: November 2011

Synopsis

A boundary integral method is developed for the scattering of electromagnetic waves at thin obstacles. The exterior boundary value problem for the vector Helmholtz equation with given Neumann data on an open surface piece (screen S) is converted into a system of integral equations for the jumps of the tangential component of the field and its divergence across the screen. A slight modification of the Cauchy data yields a strongly elliptic system of pseudodifferential equations on S which can therefore be used for numerical computations using Galerkin's procedure. The resulting boundary integral equations are analysed using pseudodifferential operator calculus. The principal symbol concept, together with the Wiener–Hopf technique, are used to derive existence and regularity results for the solutions to the boundary integral equations. Quasi-optimal error estimates in the energy norm are given for the numerical scheme.
The G-invariant implicit function theorem in infinite dimensions II
Authors: Dancer E. N.
Published: November 2011

Synopsis

In this paper, we study the perturbation of zeros of maps of Banach spaces where the maps are invariant under continuous groups of symmetries. In some cases, we allow the perturbed maps partially to break the symmetries. Our results improve earlier results of the author by removing smoothness conditions on the group action. The key new idea is a regularity theorem for the zeros of invariant Fredholm maps.
On Hamilton–Jacobi equations in bounded domains
Authors: Engler Hans
Published: November 2011

Synopsis

Initial-boundary value problems for nonlinear first order partial differential equations ∂tu + H(x, t, u, Dxu) = 0 and corresponding boundary value problems H(x, u, Dxu) = 0 are studied in bounded sets, using Crandal's and Lions' notion of viscosity solutions. We give pointwise conditions on the boundary data that guarantee the existence of such solutions and estimate their moduli of continuity in terms of continuity properties of the data. The results are applied to properties of the value function for certain differential games.
Strict inequalities for integrals of decreasingly rearranged functions
Authors: Friedman Avner , McLeod Bryce
Published: November 2011

Synopsis

It is well known that iff,g,hare nonnegative functions andf*,g*,h* their symmetrically decreasing rearrangements, then

also ifu* is a spherically decreasing rearrangement of a functionu,

In this paper it is proved under suitable assumptions (including the assumption thathis already rearranged) that equality holds in (i) if and only iffandgare already rearranged, and, if 1 <p< ∞ equality holds in (ii) if and only ifuis already rearranged. We discuss (ii) both in ℝnand on the unit sphere.
A limiting solution for an ill-posed membrane problem
Authors: Lacey A. A.
Published: November 2011

Synopsis

We consider a perturbed version of a membrane problem when the latter has no solution. The modified problem has a solution which depends on a parameter ε > 0. As ε →0 this solution is seen to tend towards a “generalised solution” of the original problem.
Homoclinic and heteroclinic orbits of reversible vectorfields under perturbation
Authors: Churchill Richard C. , Rod David L.
Published: November 2011

Synopsis

Averaging techniques on Hamiltonian dynamical systems can often be used to establish the existence of hyperbolic periodic orbits. In equilibrium situations, it is then often difficult to show that there are homoclinic/heteroclinic connections between these hyperbolic orbits in the original unaveraged system. This existence problem is solved in this paper for a class of Hamiltonian systems admitting a sufficient number of symmetries (including reversing symmetries). Under isoenergetic reduction, the problem is reduced to one involving reversible vector fields under time-dependent perturbations admitting the same reversing symmetries. Applications are made to the one-parameter Hénon-Heiles family. The paper concludes with remarks on the problem of showing transversality of these homoclinic/heteroclinic orbits.
General boundary eigenvalue problems for systems of differential equations with multiple roots of the characteristic equation
Authors: Freiling Gerhard
Published: November 2011

Synopsis

We consider a class of non-self adjoint multipoint eigenvalue problems. Using necessary conditions for the regularity of these problems, we obtain a theorem on the expansion of certain functions into a series of eigen- and associated functions.
Hölder gradient estimates for fully nonlinear elliptic equations
Authors: Trudinger Neil S.
Published: November 2011

Synopsis

In this paper we prove interior and global Hölder estimates for Lipschitz viscosity solutions of second order, nonlinear, uniformly elliptic equations. The smoothness hypotheses on the operators are more general than previously considered for classical solutions, so that our estimates are also new in this case and readily extend to embrace obstacle problems. In particular Isaac's equations of stochastic differential game theory constitute a special case of our results, and moreover our techniques, in combination with recent existence theorems of Ishii, lead to existence theorems for continuously differentiable viscosity solutions of the uniformly elliptic Isaac's equation.
On the instability of non-semi-Fredholm closed operators under compact perturbations with applications to ordinary differential operators
Authors: Labuschagne L. E.
Published: November 2011

Synopsis

The stability of several natural subsets of the bounded non-semi-Fredholm operators undercompact perturbations were studied by R. Bouldin [2] in separable Hilbert spaces and by M. Gonzales and V. M. Onieva [6] in Banach spaces. The aim of this paper is to study this problem for closed operators in operator ranges. The main results are a characterisation of the non-semi-Fredholm operators with respect to α-closed and α-compact operators as well as a generalisation of a result of M. Goldman [5]. We also give some applications of the theory developed to ordinary differential operators.
Perturbation theory for dual semigroups II. Time-dependent perturbations in the sun-reflexive case
Authors: Clément Ph. , Diekmann O. , Gyllenberg M. , Heijmans H. J. A. M. , Thieme H. R.
Published: November 2011

Synopsis

We consider time-dependent perturbations of generators of strongly continuous semigroups on a Banach space. The perturbations map the Banach space into a bigger space, which is the second dual of the original space in a specific semigroup sense. Using the theory of dual semigroups we show that the solutions of a generalised variation-of-constants formuladefine an evolutionary system. We investigate continuity and differentiability propertiesof this evolutionary system and its dual system and examine in what sense the perturbed generator and its adjoint generate these evolutionary systems. It is shown that the results apply naturally to retarded functional differential equations and age structured population dynamics.
Levinson's theorem and Titchmarsh-Weyl m(λ) theory for Dirac systems*
Authors: Hinton D. B. , Klaus M. , Shaw J. K.
Published: November 2011

Synopsis

A Levinson theorem is proved for a Dirac system with one singular endpoint. The number ofbound state is expressed in terms of the change in asymptotic phase of an appropriate solution and in terms of factors whose values depend on the presence of half-bound states. The behaviour of the asymptotic phase is used to determine the asymptotic behaviour of the Titchmarsh-Weyl m-function.
Regularity results for equilibria in a variational model for fracture
Authors: Acerbi Emilio , Fonseca Irene , Fusco Nicola
Published: November 2011

Synopsis

In recent years models describing interactions between fracture and damage have been proposed in which the relaxed energy of the material is given by a functional involving bulk and interfacial terms, of the form

where Ω is an open, bounded subset of ℝN, q ≧1, g ∈ L∞ (Ω ℝN), λ, β > 0, the bulk energy density F is quasiconvex, K⊂ℝN is closed, and the admissible deformation u:Ω→ ℝN is C1 in Ω\K One of the main issues has to do with regularity properties of the ‘crack site’ K for a minimising pair (K, u). In the scalar case, i.e. when uΩ→ ℝ, similar models were adopted to image segmentation problems, and the regularity of the ‘edge’ set K has been successfully resolved for a quite broad class of convex functions F with growth p > 1 at infinity. In turn, this regularity entails the existence of classical solutions. The methods thus used cannot be carried out to the vectorial case, except for a very restrictive class of integrands. In this paper we deal with a vector-valued case on the plane, obtaining regularity for minimisers of corresponding to polyconvex bulk energy densities of the form

where the convex function h grows linearly at infinity.
Stochastic systems governed by B-evolutions on Hilbert spaces
Authors: Ahmed N. U. , Kerbal S.
Published: November 2011

Synopsis

In this paper, we consider the question of existence of solutions and their regularity properties for a large class of stochastic evolution equations governed by B-evolutions involving two different Hilbert spaces. This allows dynamic boundary conditions together with noisy boundary data. They cover also stochastic boundary value problems. Our results are illustrated by two practical examples.
Positive solution curves of semipositone problems with concave nonlinearities
Authors: Castro Alfonso , Gadam Sudhasree , Shivaji R.
Published: November 2011

Synopsis

We consider the positive solutions to the semilinear equation:

where Ω denotes a smooth bounded region in ℝN(N> 1) and ℷ 0. Heref:[0, ∞)→ℝ is assumed to be monotonically increasing, concave and such thatf(0)<0 (semipositone). Assuming thatf′(∞)≡limt→∞f(t)> 0, we establish the stability and uniqueness of large positive solutions in terms of (f(t)/t)′ When Ω is a ball, we determine the exact number of positive solutions for each λ > 0. We also obtain the geometry of the branches of positive solutions completely and establish how they evolve. This work extends and complements that of [3, 7] wheref′(∞)≦0.
Stable summands ofU(n)
Authors: Crabb M. C. , Hubbuck J. R. , McCall J. A. W.
Published: November 2011

Synopsis

The special unitary groupSU(n)has the stable homotopy type of a wedge ofn− 1 finite complexes. The ‘first’ of these complexes is ΣℂPn–1, which is well known to be indecomposable at the prime 2 whethernis finite or infinite. We show that the ‘second’ finite complex is again indecomposable at the prime 2 whennis finite, but splits into a wedge of two pieces whennis infinite.
Singular boundary value problems for P-Laplacian-like equations
Authors: Hai D. D. , Oppenheimer Seth F.
Published: November 2011

Synopsis

We consider the existence of positive solutions to a class of singular nonlinear boundary value problems for P-Laplacian-like equations. Our approach is based on the Schauder Fixed-Point Theorem.
The viscous damping prevents propagation of singularities in the system of viscoelasticity
Authors: Rybka Piotr
Published: November 2011

Synopsis

We show that the linear viscous damping Δut, is so strong that it altogether prevents propagation of singularities of the gradient of solutions to the system of viscoelasticity. Moreover, no creation or annihilation of singularities is possible in finite time.
Cauchy problem for the Ginzburg-Landau equation for the superconductivity model
Authors: Guo Boling , Yuan Guangwei
Published: November 2011

In this paper, the existence and uniqueness of the global smooth solution are proved for an evolutionary Ginzburg–Landau model for superconductivity under the Coulomb and Lorentz gauge.
Pattern formation of the stationary Cahn-Hilliard model
Authors: Kielhöfer Hansjörg
Published: November 2011

We investigate critical points of the free energy Eε(u) of the Cahn–Hilliard model over the unit square under the constraint of a mean value ü. We show that for any fixed value ü in the so-called spinodal region and to any mode of an infinite class, there are critical points of Eε(u) having the characteristic symmetries of that mode provided ε > 0 is small enough. As ε tends to zero, these critical points have singular limits forming characteristic patterns for each mode. Furthermore, any singular limit is a stable critical point of E0(u)). Our method consists of a global bifurcation analysis of critical points of the energy Eε(u) where the bifurcation parameter is the mean value ü.
First-order corrections to the homogenised eigenvalues of a periodic composite medium. A convergence proof
Authors: Moskow Shari , Vogelius Michael
Published: November 2011

Let λε be a Dirichlet eigenvalue of the ‘periodically, rapidly oscillating’ elliptic operator –∇·(a(x/ε)∇) and let ∇ be a corresponding (simple) eigenvalue of the homogenised operator –∇·(A∇). We characterise the possible limit points of the ratio (λε–λ)/ε as ε→0. Our characterisation is quite explicit when the underlying domain is a (planar) convex, classical polygon with sides of rational or infinite slopes. In particular, in this case it implies that there is often a continuum of such limit points.
A generic property of the resonance set of an elliptic operator with respect to the domain
Authors: Pistoia Angela
Published: November 2011

The set of points (α, β) ∈ℝ2 for which the problem − δu = αu+ − βu− in Ω, u = 0 on ∂Ω(u+ = max {u, o} and u− = min {−u, 0}) has nontrivial solutions is important for the study of certain nonlinear problems. It is shown that for ‘most’ bounded domains Ω in ℝn, such a set is locally the union of a finite number of curves.
On the essential self-adjointness of Schrödinger operators with locally integrable potentials
Authors: Kalf H. , Rofe-Beketov F. S.
Published: November 2011

Results by Simader, Brézis and Cycon of the genre ‘locally essentially self-adjoint implies globally essentially self-adjoint’ are generalised to Schrödinger operators that are not necessarily bounded from below.
Optimal bounds and blow up phenomena for parabolic problems in narrowing domains
Authors: Andreucci Daniele , Tedeev Anatoli F.
Published: November 2011

We consider degenerate parabolic problems in domains with noncompact boundary and infinite volume, in any spatial dimension. The equation is of doubly nonlinear type. On the boundary we prescribe a homogeneous Neumann condition. The spatial domain is narrowing at infinity. We prove uniform convergence to 0 of solutions as time approaches ∞. To this end, due to the geometry of the domain, the requirement that the initial datum have finite mass is not enough, and we have to stipulate the further assumption that a certain moment of the initial datum (connected with the geometry of the domain) is finite. We prove optimal asymptotic estimates of the solution. Moreover, we apply our method to the investigation of blow-up problems in narrowing domains, obtaining a sharp condition, in integral form, for the existence of solutions defined for all positive times.
Multi-peak periodic semiclassical states for a class of nonlinear Schrödinger equations
Authors: Cingolani Silvia , Nolasco Margherita
Published: November 2011

For a class of nonlinear Schrodinger equations, we prove the existence of semiclassical stationary states with possibly infinitely many concentration points. As h → 0, these states concentrate near critical points of the potential. Furthermore, for periodic potential, these states can be constructed to satisfy periodic boundary conditions.
On invariant subspaces for nonlinear finite-difference operators
Authors: Galaktionov Victor A.
Published: November 2011

We study linear subspaces invariant under discrete operators corresponding to finitedifference approximations of differential operators with polynomial nonlinearities. In several cases, we establish a certain structural stability of invariant subspaces and sets of nonlinear differential operators of reaction–diffusion type with respect to their spatial discretisation. The corresponding lower-dimensional reductions of the finite-difference solutions on the invariant subspaces are constructed.
Divergence-preserving geodesic transformations
Authors: García-Río E. , Vanhecke L.
Published: November 2011

We discuss divergence- and volume-preserving geodesic transformations with respect to submanifolds and in particular, with respect to hypersurfaces. We use these transformations to derive characterisations of special classes of hypersurfaces such as isoparametric hypersurfaces and Hopf hypersurfaces with constant principal curvatures. Furthermore, we consider divergence-preserving geodesic transformations with respect to geodesic spheres.
On a singular nonlinear semilinear elliptic problem
Authors: Shi Junping , Yao Miaoxin
Published: November 2011

We consider the singular boundary value problem

We study the existence, uniqueness, regularity and the dependency on parameters of the positive solutions under various assumptions.
On lower semicontinuity of a defect energy obtained by a singular limit of the Ginzburg–Landau type energy for gradient fields
Authors: Aviles Patricio , Giga Yoshikazu
Published: November 2011

A defect energy Jβ, which measures jump discontinuities of a unit length gradient field, is studied. The number β indicates the power of the jumps of the gradient fields that appear in the density of Jβ. It is shown that Jβ for β = 3 is lower semicontinuous (on the space of unit gradient fields belonging to BV) in L1-convergence of gradient fields. A similar result holds for the modified energy , which measures only a particular type of defect. The result turns out to be very subtle, since with β > 3 is not lower semicontinuous, as is shown in this paper. The key idea behind semicontinuity is a duality representation for J3 and . The duality representation is also important for obtaining a lower bound by using J3+ for the relaxation limit of the Ginzburg–Landau type energy for gradient fields. The lower bound obtained here agrees with the conjectured value of the relaxation limit.
Inequalities and monotonicity properties for zeros of Hermite functions
Authors: Elbert Árpád , Muldoon Martin E.
Published: November 2011

We study the variation of the zeros of the Hermite function Hλ(t) with respect to the positive real variable λ. We show that, for each non-negative integer n, Hλ(t) has exactly n + 1 real zeros when n < λ ≤ n + 1, and that each zero increases from – ∞ to ∞ as λ increases. We establish a formula for the derivative of a zero with respect to the parameter λ; this derivative is a completely monotonic function of λ. By-products include some results on the regular sign behaviour of differences of zeros of Hermite polynomials as well as a proof of some inequalities, related to work of W. K. Hayman and E. L. Ortiz for the largest zero of Hλ(t). Similar results on zeros of certain confluent hypergeometric functions are given too. These specialize to results on the first, second, etc., positive zeros of Hermite polynomials.
Examples of degenerate symmetric differential operators with infinite deficiency indices in L2(ℝm)
Authors: Orochko Yurii B.
Published: November 2011

For an unbounded self-adjoint operator A in a separable Hilbert space ℌ and scalar real-valued functions a(t), q(t), r(t), t ∊ ℝ, consider the differential expression

acting on ℌ-valued functions f(t), t ∊ ℝ, and degenerating at t = 0. Let Sp denotethe corresponding minimal symmetric operator in the Hilbert space (ℝ) of ℌ-valued functions f(t) with ℌ-norm ∥f(t)∥ square integrable on the line. The infiniteness of the deficiency indices of Sp, 1/2 < p < 3/2, is proved under natural restrictions on a(t), r(t), q(t). The conditions implying their equality to 0 for p ≥ 3/2 are given. In the case of a self-adjoint differential operator A acting in ℌ = L2(ℝn), the first of these results implies examples of symmetric degenerate differential operators with infinite deficiency indices in L2(ℝm), m = n + 1.
Regularity of almost extremal solutions of Monge–Ampère equations
Authors: Urbas John I. E.
Published: November 2011

Synopsis

We show that for a large class of Monge-Ampère equations, generalised solutions on a uniformly convex domain Ω⊂ℝn are classical solutions on any pre-assigned subdomain Ω′⋐Ω, provided the solution is almost extremal in a suitable sense. Alternatively, classical regularity holds on subdomains of Ω which are sufficiently distant from ∂Ω. We also show that classical regularity may fail to hold near ∂Ω in the nonextremal case. The main example of the class of equations considered is the equation of prescribed Gauss curvature.
The existence of a positive solution of semilinear elliptic equations with limiting Sobolev exponent
Authors: Wang Shixiao
Published: November 2011

Synopsis

Our paper concerns the existence of a positive solution for the equation:

A new condition, which guarantees the existence of a solution of the above equation, has been established. It has also given some sharp information in the cases where: (1) a(x) = λ = const. and Ω is a “thin” domain; (2) Ω is a ball and a(x) is a radially symmetrical function.
Homogeneous polynomial splines
Authors: Goodman T. N. T. , Lee S. L.
Published: November 2011

Synopsis

We construct functions which are piecewise homogeneous polynomials in the positive octant in three dimensions. These give a rich and elegant theory which combines properties of polynomial box splines see [6] and the references therein) with the explicit representation of simple exponential box splines [11], while enjoying complete symmetry in the three variables. By a linear transformation followed by a projection on suitable planes, one obtains piecewise polynomial functions of two variables on a mesh formed by three pencils of lines. The vertices of these pencils may be finite or one or two may be infinite, i.e. the corresponding pencils may comprise parallel lines. As a limiting case, all three vertices become infinite and one recovers polynomial box splines on a three-direction mesh.
Strongly elliptic boundary integral equations for electromagnetic transmission problems
Authors: Costabel Martin , Stephan Ernst P.
Published: November 2011

Synopsis

We study a boundary integral equation method for transmission problems for strongly elliptic differential operators, which yields a strongly elliptic system of pseudodifferential operators and which therefore can be used for numerical computations with Galerkin's procedure. The method is shown to work for the vector Helmholtz equation in ℝ3 with electromagnetic transmission conditions. We propose a slightly modified system of boundary values in order for the corresponding bilinear form to be coercive over H1. We analyse the boundary integral equations using the calculus of pseudodifferential operators. Here the concept of the principal symbol is used to derive existence and regularity results for the solution.
A combinatorial property of finite full transformation semigroups
Authors: Howie John M. , Robertson Edmund F. , Schein Boris M.
Published: November 2011

Synopsis

Let E be the set of idempotents in the semigroup Singn of singular self-maps of N = {1, …, n}. Let α ∊ Singn. Then α ∊ E2 if and only if for every x in im α the set xα−1 either contains x or contains an element of (im α)′.

Write rank α for |im α| and fix α for |{x ∊ N: xa = x}|. Define (x, xα, xα2) to be an admissible α-triple if x ∊ (im α)′, xα3 ≠ xα2. Let comp α (the complexity of α) be the maximum number of disjoint admissible α-triples. Then α ∊ E3 if and only if
On the domains of minimal and maximal operators for regularisable singular differential expressions
Authors: Race D.
Published: November 2011

Synopsis

Certain classical differential expressions which are singular at a finite end-point (or at an interior point) can be represented as regular, scalar quasi-differential expressions, the best-known examples being the Boyd Equation and Laplace Tidal Wave Equation. We show here that in all such cases the domains of the minimal and maximal operators in the appropriate weighted Hilbert space, for the regularised expression, coincide with the corresponding domains for the expression in its original, singular form.

This is contrasted with a known property of the corresponding expression domains. Whereas for an expressionM, the operator domains contain only functions y for which bothyandMylie in the appropriate Hilbert space, the expression domain comprises a much larger set of functions with no such restrictions beyond those necessary forMyto exist as a function. In the second-order case, the expression domain of the regularisation of a singular expression is known to be a strict subset of the original expression domain, contrasting with the results proved here for the operator domains.