A method of Jensen is extended to show that the second derivatives of the solutions of various linear obstacle problems are bounded under weaker regularity hypotheses on the dataof the problem than were allowed by Jensen. They are, in fact, weak enough that the linear results imply the boundedness of the second derivatives for quasilinear problems as well. Comparisons are made with previously known results, some of which are proved by similar methods. Both Dirichlet and oblique derivative boundary conditions are considered. Corresponding results for parabolic obstacle problems are proved.

In this paper we consider a one-dimensional Stefan problem with a source term. Under the assumption that the initial profile is monotone, we obtain continuity of the free boundaries between the solid, the mushy and the liquid region.

We consider a mathematical model for the motion of a marked monomer in a system of reacting polymers at equilibrium. A well-posed integro-differential initial value problem for the probability of finding the marked monomer in a molecule of a given length is formulated. We prove exponential convergence of the probability to a unique equilibrium distribution. A quite complete spectral analysis is carried out for a self adjoint operator, which is a perturbation of a multiplication operator by an integral operator and is related to the generator of the time evolution.

The number of occurrences of the Steinberg representation St of GL(n, p) as a composition factor in the symmetric algebra Fp[x1, … xn] has been determined by several authors. We extend this result to the representations of GL(n, p) which are the closest neighbours of St in the SL(n, p) weight diagram. The method is to play off duality for GL(n, p)-modules against connectivity for M(n, p)-modules. The result is equivalent to determining the cohomology groups of the corresponding indecomposable stable summands of the localisation of an n-fold product of complex projective spaces at the prime p.

The concept of a transition system is extended to a parametrised family of differential equations

where x ∊ ℝn and λ ∊ Λ = [0, l]m, an m-cube. Furthermore, algebraic formulae for comparing connection matrices at the various parameter values are obtained. Finally, several applications of these techniques are indicated.

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.

The Hopf algebra structure of H*(ΩG, F2) and the action of the dual Steenrod algebra are completely and explicitly determined when G isone of the connected, simply connected, exceptional, simple Lie groups. The approach is homological, using connected coverings and spectral sequences.

Suppose f: M →N is a continuous map from a Riemannian manifold (M, d) into a manifold N. The main result of this paper is to give some conditions under which f identifies a pair of cut points. This result leads to generalisations of the classical Borsuk-Ulam theorem. As a consequence some topological properties of locally symmetric spaces are discovered.

We consider automorphic actions on von Neumann algebras of a locally compact group E given as a topological extension 0 → A → E → G → 0, where A is compact abelian and second countable. Motivated by the wish to describe and classify ergodic actions of E when G is finite, we classify (up to conjugacy) first the ergodic actions of locally compact groups on finite-dimensional factors and then compact abelian actions with the property that the fixed-point algebra is of type I with atomic centre. We then handle the case of ergodic actions of E with the property that the action is already ergodic when restricted to A, and then, as a generalisation, the case of (not necessarily ergodic) actions of E with the property that the restriction to A is an action with abelian atomic fixed-point algebra. Both these cases are handled for general locally compact-countable G. Finally, we combine the obtained results to classify the ergodic actions of E when G is finite, provided that either the extension is central and Hom (G, T) = 0, or G is abelian and either cyclic or of an order not divisible by a square.

Let then, for certain weight functions u and v and indices p,q, it is shown that ∥Tαf∥q, u≦C∥grad f ∥p, v'q > n / α holds. For α=l,p=q and u = v ≡ l this reduces to a result of M. Weiss. In addition we establish n-dimensional weighted Hardy—Littlewood type inequalities ofthe form for large classes of weights.

We discuss the radially, symmetric solutions and the symmetry breaking of the equation Δu + 2δe −u = 0 in D and u + b(∂u/∂n) = 0 on ∂D, where D is the unit disk in ℝ2, δ >0 and b is a constant. We prove that for any b < 0, there exists > 0 such that there are exactly two radially symmetric solutions for δ ∊ (0, ), one for δ = and none for δ > δ*b. For , where m is a positive integer, there are (b), k = 1, …, m, such that the equation has symmetry breaking at δ*k (b) on the lower branch of radially symmetric solutions.