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.

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

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

Some special cases of the result are also discussed.

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

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

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

The exact blow-up rate of u is obtained. Uniqueness, regularity, decay at infinity of the function , as well a s a representation formula for the case m = 1, are also proved.

In commutative Banach algebras with factorisation, the existence of an identity (bounded approximate identity) modulo a topologically nilpotent radical implies the existence of a global identity (bounded approximate identity), respectively.

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.

In this paper we prove a theorem concerning existence and nonexistence of minimising harmonic maps in each connected component using the framework of Brezis, Coron and Soyeur.

A class of models of single-species dynamics with diffusion within and between patches is considered. It is shown that under our prescribed conditions, there is a unique, positive, globally asymptotically stable steady state.

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

As an example for the dissipative case, we find that the Ginzburg–Landau equation is the modulation equation for the Swift–Hohenberg problem. Moreover, the method also enables us to handle hyperbolic problems: the nonlinear Schrodinger equation is shown to describe the modulation of wave packets in the Sine–Gordon equation.

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.

We give here an abelian kernel (central) group extension sequence for calculating, for a non-simply-connected space X, the group of pointed self-homotopy-equivalence classes . This group extension sequence gives in terms of , where Xn is the nth stage of a Postnikov decomposition, and, in particular, determines up to extension for non-simplyconnected spaces X having at most two non-trivial homotopy groups in dimensions 1 and n. We give a simple geometric proof that the sequence splits in the case where is the generalised Eilenberg–McLane space corresponding to the action ϕ: π1 → aut πn, and give some information about the class of the extension in the general case.

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.

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|).

In this paper, an Lp version of the “div-curl lemma” is generalised in a very general framework. Another form of the Lp -theorem of compensated compactness is also exploited.