We prove the existence of a solution to a free boundary value problem arising in elastohydrodynamic lubrication. Previously, the existence was proved only under severe restrictions on the range of the physical parameters. We remove those restrictions here.

An eigenvalue problem is considered for a multi-structure consisting of a three-dimensional finite solid connected to an arbitrary smooth elastic thin shell of revolution. Two-sided estimates are obtained for the first six eigenfrequencies of the multi-structure. Explicit asymptotic formulae are given.

The linearized Ginzburg–Landau equations in both semi-infinite strips and rectangles are transformed into equivalent one-dimensional integral equations. Then, the properties of the integral equations are utilized to prove that the onset field for a semi-infinite strip is isolated. We solve the integral equations numerically to obtain the onset field for both rectangles and semi-infinite strips. A formal asymptotic expansion of the onset field in the long rectangle limit is also obtained. Using this formal expansion, we show that the onset field converges in this limit faster than any finite exponential rate, and as a byproduct, that the onset mode in a semi-infinite strip must be asymptotically symmetric.

Let $G=GL_n(F)$, where $F$ is a $p$-adic field of characteristic zero and residual characteristic $p$. Assuming that $p>2n$, we compare germs of characters of irreducible admissible representations of $G$ with germs of characters of unipotent representations of direct products of general linear groups over finite extensions of $F$. We show that the character of an irreducible admissible representation has an $s$-asymptotic germ expansion, for some semisimple $s$ in the Lie algebra of $G$. Furthermore, this expansion matches with the $0$-asymptotic expansion (that is, the local character expansion) of the character of a unipotent representation of the centralizer of $s$ in $G$.

AMS 2000 Mathematics subject classification: Primary 22E50; 22E35

Starting from an age-structured model, we derive a partial differential equation satisfied by the total number of mature adult members of a population, on an infinite one-dimensional domain. The formulation involves a distribution of possible ages of maturation and uses a probability density function on which ecologically realistic assumptions are made. It is found that the existence and value of a positive equilibrium solution depends on the mean maturation delay. When no positive equilibrium exists, we prove global attractivity of the zero solution. For a particular ecologically reasonable choice of the distribution function, we show that travelling fronts exist connecting the zero equilibrium with the positive one provided the mean maturation delay is sufficiently small, and the dependence of the front's propagation speed on the mean delay is discussed.

Let Ω be an open bounded subset of Rn and f a continuous function on Ω̄ satisfying f(x) > 0 for all x ∈ Ω̄. We consider the maximization problem for the integral ∫Ωf(x)u(x)dx over all Lipschitz continuous functions u subject to the Dirichlet boundary condition u = 0 on ∂Ω and to the gradient constraint of the form H(Du(x)) ≤ 1, and prove that the supremum is ‘achieved’ by the viscosity solution of Ĥ(Du(x)) = 1 in Ω and u = 0 on ∂Ω, where Ĥ denotes the convex envelope of H. This result is applied to an asymptotic problem, as p → ∞, for quasi-minimizers of the integralAn asymptotic problem as k → ∞ for infis also considered, where the infimum is taken all over and the set K is given by {ξ | H(ξ) ≤ 1}.

Let be a complete and minimal set in a Hilbert space X, which is not an unconditional basis. Then there exists an operator that is densely defined on X, does not have a good spectral behaviour and all the vectors {ei} are its eigenvectors.

An estimate on the number of distinct relative periodic orbits around a stable relative equilibrium in a Hamiltonian system with continuous symmetry is given. This result constitutes a generalization to the Hamiltonian symmetric framework of a classical result by Weinstein and Moser on the existence of periodic orbits in the energy levels surrounding a stable equilibrium. The estimate obtained is very precise in the sense that it provides a lower bound for the number of relative periodic orbits at each prescribed energy and momentum values neighbouring the stable relative equilibrium in question and with any prefixed (spatio-temporal) isotropy subgroup. Moreover, it is easily computable in particular examples. It is interesting to see how, in our result, the existence of non-trivial relative periodic orbits requires (generic) conditions on the higher-order terms of the Taylor expansion of the Hamiltonian function, in contrast with the purely quadratic requirements of the Weinstein–Moser theorem, which emphasizes the highly nonlinear character of the relatively periodic dynamical objects.

We address the problem of parallelizability and stable parallelizability of a family of manifolds that are obtained as quotients of circle actions on complex Stiefel manifolds. We settle the question in all cases but one, and obtain in the remaining case a partial result.

For a bounded solution to the mean curvature equation in Rn, we shall obtain its Hölder estimates, with exponent α, which are in linear proportion to the number obtained by dividing its global oscillation by the αth power of the distance of a point to the boundary of the domain after a suitable rescaling.

Let B1 be a ball of radius R1 in RN with centre at the origin and let B0 be a smaller ball of radius R0 contained inside it. Let u be the solution of the problem −Δu = 1 in B1\B0 vanishing on the boundary. It is shown thatis minimal if and only if the balls are concentric. It is also shown that the first (Dirichlet) eigenvalue of the Laplacian in B1\B0 is maximal if and only if the balls are concentric. Generalizations are indicated.

For linear systems with a multi-symplectic structure, arising from the linearization of Hamiltonian partial differential equations about a solitary wave, the Evans function can be characterized as the determinant of a matrix, and each entry of this matrix is a restricted symplectic form. This variant of the Evans function is useful for a geometric analysis of the linear stability problem. But, in general, this matrix of two-forms may have branch points at isolated points, shrinking the natural region of analyticity. In this paper, a new construction of the symplectic Evans matrix is presented, which is based on individual vectors but is analytic at the branch points—indeed, maximally analytic. In fact, this result has greater generality than just the symplectic case; it solves the following open problem in the literature: can the Evans function be constructed in a maximally analytic way when individual vectors are used? Although the non-symplectic case will be discussed in passing, the paper will concentrate on the symplectic case, where there are geometric reasons for evaluating the Evans function on individual vectors. This result simplifies and generalizes the multi-symplectic framework for the stability analysis of solitary waves, and some of the implications are discussed.

A new characterization of the Dunford–Pettis property in terms of the extensions of multilinear operators to the biduals is obtained. For the first time, multilinear characterizations of the reciprocal Dunford–Pettis property and Pełczyński's property (V) are also found. Polynomial and holomorphic versions of these properties are given as well.

Let B be the unit ball in Rn, n ≥ 3. Let 0 < p < 1 < q ≤ (n + 2)/(n − 2). In 1994, Ambrosetti et al. found that the semilinear elliptic Dirichlet problemadmits at least two solutions for small λ > 0 and no solution for large λ. In this paper, we prove that there is a critical number Λ > 0 such that this problem has exactly two solutions for λ ∈ (0, Λ), exactly one solution for λ = Λ and no solution for λ > Λ.

In this paper, we consider the Cauchy problem of general symmetrizable hyperbolic systems in multi-dimensional space. When some components of the initial data have compact support, we give a sufficient condition on the non-existence of global C1 solutions. This non-existence theorem can be applied to some physical systems, such as Euler equations for compressible flow in multi-dimensional space. The blow-up phenomena here can come from the singularity developed at the interface, such as vacuum boundary, rather than the shock formation as studied in the previous works on strictly hyperbolic systems. Therefore, the systems considered here include those which are non-strictly hyperbolic.

We study the spectrum of regular and singular Sturm–Liouville problems with real-valued coefficients and a weight function that changes sign. The self-adjoint boundary conditions may be regular or singular, separated or coupled. Sufficient conditions are found for (i) the spectrum to be real and unbounded below as well as above and (ii) the essential spectrum to be empty. Also found is an upper bound for the number of non-real eigenvalues. These results are achieved by studying the interplay between the indefinite problems (with weight function which changes sign) and the corresponding definite problems. Our approach relies heavily on operator theory of Krein space.

We study the homogenization of ferromagnetic equations with periodic coefficients in space dimension 2. The obtained nonlinear homogenized law can only be written using a two-scale framework, which couples microscopic and macroscopic scales. It also involves corrector terms, at the microscopic scale, in the form of pseudo-differential operators. We prove the L2 two-scale strong convergence in the laminar case (one-dimensional periodicity).

We consider a general class of degenerate elliptic-parabolic problems associated with the equation b(υ)t = div a(υ, Dυ) + f. Existence of renormalized solutions is established for general L1 data. Uniqueness of renormalized solutions has already been shown in a previous work.

This special issue is devoted to papers from the meeting on Combinatorics, Probability and Computing, held at the Mathematisches Forschungsinstitut in Oberwolfach from the 23$^{\rm rd}$ to the 29$^{\rm th}$ of September 2001. As is typical of meetings at Oberwolfach, this was an exciting and stimulating conference; there was a large number of excellent talks, many of which provoked a great deal of interest and discussion among the participants.