If a Sturm—Liouville problem is given in an open interval of the real line, then regular boundary value problems can be considered on compact sub-intervals. For these regular problems, all with necessarily discrete spectra, the eigenvalues depend on both the end-points of the compact intervals, and upon the choice of the real separated boundary conditions at these end-points. These eigenvalues are not, in general, continuous functionsof the end-points and boundary conditions. The paper shows the surprising form of these discontinuities. The results have applications to the approximations of singular Sturm—Liouville problems by regular problems, and to the theoretical aspects of the Sleign2 Computer program.

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.

Asymptotic formulae for the Titchmarsh—Weyl m-coefficient on rays in the complex λ-plane for the equation − y ″ + qy = λy whenthe potential is limit circle and non-oscillatory at x = 0 are obtained under assumptions slightly more general than xq(x) ∈ L1(0,c). The behaviour of q at the right end-point is arbitrary and may fall in either the limit-point or limit-circle case. A method of regularization of the equation is given that can be made to depend either on a solution of the equation for λ = 0 or more directly on an approximation to the solution in terms of q. This enables equivalent definitions of the m-coefficient to be given for the singular Sturm—Liouville problem associated with a singular limit-circle boundary condition, and its associatedregular Sturm—Liouville problem. As a consequence, it becomes possible to apply asymptotic results obtained by Atkinson for the regular problem in order to give asymptoticresults for the singular problem. Potentials of the form q(x) = C/xj, 1 ≤ j < 2, are included. In the case j = 1, an independent calculation of the limit-point m-coefficient over the range (0,∞), relying on Whittaker functions, verifies the main result.

A stability theorem is proved for non-monotone waves in a reaction–diffusion system modelling three competing species, where few stability results currently exist for systems with more than two species. Asymptotic stability with respect to the nonlinear equations is established by showing that the spectrum of the linearized operator has no unstable eigenvalues and that the zero eigenvalue associated with translation invariance is simple. The result is obtained in a singular regime where strong pattern formation occurs and solutions to the linear equations can be separated into particular solutions related to the fast–slow structure of the underlying wave and its singular limit. In this system both the fast and slow waves can contribute an instability and the global characterization of these solutions must address certain difficulties not present in lower-dimensional systems. The topological index of Alexander, Gardner and Jones is used to count the eigenvalues of the linear operator.

Two relative compactness results for two-scale convergence in homogenization, due to G. Nguetseng, were recently extended to the multi-scale case by G. Allaire and M. Briane. Whereas their extension of Nguetseng's first result, which is in L2, is straightforward, their extension of his second result, which takes place in the Sobolev space H1, is quite complicated, even though it follows Nguetseng by using the fact that the image of H1 under the gradient mapping is the orthogonal complement of the set of divergence-free functions. Here a much simpler proof is provided by deriving the H1-type result from combining the first extension result with the fact that the above-mentioned image space is also the space of all rotation-free fields. Moreover, this approach reveals that the two results can be seen as corollaries of a fundamental relative compactness result for Young measures.

This paper studies geometric properties, in particular extreme points and rays, of various generalizations of Young measures. Applications of the knowledge of extreme points are illustrated on existence results for optimal control problems and on various convergence results for Young measures by using the Choquet theory.

This paper deals with the theory of point interactions for the one-dimensional Schrödinger equation. The familiar example of the δ-potential V(x) = gδ(x−x0), for which the transfer matrix across the singularity (point transfer matrix) is given by

is extended to cover cases in which the transfer matrix M(z) is dependent on the (complex) spectral parameter z, and which can be obtained as limits of transfer matrices across finite intervals for sequences of approximating potentials Vn.

The case of point transfer matrices polynomially dependent on z is treated in detail, with a complete characterization of such matrices and a proof of their factorization as products of point transfer matrices linearly dependent on z.

The theory presented here has applications to the study of point interactions in quantum mechanics, and provides new classes of point interactions which can be obtained as limiting cases of regular potentials.

Let m: [ 0, ∞) → [ 0, ∞) be an increasing continuous function with m(t) = 0 if and only if t = 0, m(t) → ∞ as t → ∞ and Ω C ℝN a bounded domain. In this note we show that for every r > 0 there exists a function ur solving the minimization problem

where Moreover, the function ur is a weak solution to the corresponding Euler–Lagrange equation

for some λ > 0. We emphasize that no Δ2-condition is needed for M or M; so the associated functionals are not continuously differentiable, in general.

Symmetric groupoids which classify the monomorphism contexts of objects in arbitrary categories are studied, along with their connections to symmetric inverse monoids and symmetric inverse algebras. Particular attention is given to symmetric groupoids of objects in free categories and to the inverse algebras induced from them by 0-closure. These graph algebras generalize both the class of polycyclic semigroups and the class of combinatorial ω-semigroups with adjoined zeros. Since all such algebras are E*-unitary, an analogue of McAlister's theory of.E-unitary inverse monoids is developed for a special class of E*-unitary inverse monoids and then illustrated on the class of graph algebras.

The Riemann initial value problem is studied for scalar conservation laws whose fluxes have a single inflection point. For a regularization consisting of balanced diffusive and dispersive terms, the travelling wave criterion is used to select admissible shocks. In some cases, the Riemann problem solution contains an undercompressive shock. The analysis is illustrated by exploring parameter space for the Buckley–Leverett flux. The boundary of the set of parameters for which there is a physical solution of the Riemann problem for all data is computed. Within the region of acceptable parameters, the solution hasseveral different forms, depending on the initial data; the different forms are illustrated by numerical computations. Qualitatively similar behaviour is observed in Lax–Wendroff approximations of solutions of the Buckley–Leverett equation with no dissipation or dispersion.

Previous fixed-point index calculation results (see [4] and [7]) exploited in the study of population systems on bounded domains are no longer applicable to systems on the whole ℝn, due mainly to the lack of compactness. In this paper, we develop fixed-point index calculation techniques for non-compact operators so that they are applicable to systems on the whole ℝn. We illustrate the use of our fixed-point index calculation results by a simple representative model, namely, the Lotka–Volterra N-species periodic competition system on the whole ℝn.

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

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

In this paper, we consider the existence of multiple solutions of biharmonic equations boundary value problem

where Ω is a bounded smooth domain in ℝN, N ≥ 5; λ ∈ ℝ1 is a given constant; p = 2N/(N − 4) is the critical Sobolev exponent for the embedding ; Δ2 = ΔΔ denotes iterated N-dimensional Laplacian; f(x) is a given function. Some results on the existence and non-existence of multiple solutions for the above problem have been obtained by Ekeland's variational principle and the mountain-pass lemma under some assumptions on f(x) and N.

This paper is concerned with integral domains R, for which the factor group SL2(R)/U2(R) has a non-trivial, free quotient, where U2(R) is the subgroup of GL2(R) generated by the unipotent matrices. Recently, Krstić and McCool have proved that SL2(P[x])/U2(P[x]) has a free quotient of infinite rank, where P is a domain which is not a field. This extends earlier results of Grunewald, Mennicke and Vaserstein.

Any ring of the type P[x] has Krull dimension at least 2. The purpose of this paper is to show that result of Krstić and McCool extends to some domains of Krull dimension 1, in particular to certain Dedekind domains. This result, which represents a two-dimensional anomaly is the best possible in the following sense. It is well known that SL2(R) = U2(R), when R is a domain of Krull dimension zero, i.e. when R is a field. It is already known that for some arithmetic Dedekind domains A, the factor group SL2(A)/U2(A) has a free quotient of finite (and not infinite) rank.

Using the viscosity vanishing method, we obtain the existence of a smooth solution to the inhomogeneous Heisenberg chain equations in one dimension. The uniqueness of the solution to the viscosity equation is also given.

The main objects of study are the homotopically stratified metric spaces introduced by Quinn. Closed unions of strata are shown to be stratified forward tame. Stratified fibrations between spaces with stratifications are introduced. Paths that lie in a single stratum, except possibly at their initial points, form a space with a natural stratification, and the evaluation map from that space of paths is shown to be a stratified fibration. Applications to mapping cylinders and to the geometry of manifold stratified spaces are expected in future papers.

Let f be a rotationally invariant function defined on the set Lin+ of all tensors with positive determinant on a vector space of arbitrary dimension. Necessary and sufficient conditions are given for the rank 1 convexity of f in terms of its representation through the singular values. For the global rank 1 convexity on Lin+, the result is a generalization of a two-dimensional result of Aubert. Generally, the inequality on contains products of singular values of the type encountered in the definition of polyconvexity, but is weaker. It is also shown that the rank 1 convexity is equivalent to a restricted ordinary convexity when f is expressed in terms of signed invariants of the deformation.

We introduce the spectral heat function H associated with the Dirichlet-Laplace–Beltrami operator ΔM on a compact smooth Riemannian manifold M with a non-empty smooth boundary. We obtain two-term asymptotics for H without assuming any billiard conditions on M. As a corollary, we obtain estimates for the integral of the k′th Dirichlet eigenfunction of ΔM over M for k→∞.

We consider the Stokes problem with a small viscosity. When the viscosity goes to zero, the boundary-layer phenomenon can appear. In this case, the solution of the given perturbed Stokes equation cannot be properly approximated by the solution of its limiting equation ‘near’ the boundary Γ of the domain of study, say Ω To overcome this problem, we need to construct a corrector term in the neighbourhood of Γ Lions has studied this problem and has constructed a corrector for the case where Ω is a half space in ℝ2. The case where Ω is an open and bounded domain of ℝ2 or ℝ3, which remained unsolved, is the concern of this paper. The construction of the corrector to the perturbed Stokes equation depends heavily on the geometry of Ω In two dimensions, we construct the corrector in the form of a stream function, while in ℝ3 we construct it in the form of a potential vector. The corrector acts effectively in a neighbourhood of Γ that is the boundary layer. Using similar methods to those of Baranger and Tartar, we define the thickness of the boundary layer in a natural way. In addition, in this paper we study the behaviour of the corrected solution in some Hölder spaces.