We study the effects of a small symmetry breaking perturbation on a system of differential equations at a coupled Hopf bifurcation with O(2) symmetry, where the perturbation breaks the continuous rotation symmetry, but retains a reflection (Z2) symmetry. It is shown that for a large range of parameter values, the invariant manifolds of the unperturbed bifurcation persist and that for some values of normal form coefficients there are secondary bifurcations of nonsymmetric periodic standing wave solutions.

We use the modern tools of the duality principles and the calculus of variations to formulate, analyse and solve a class of plasticity problems involving second order partial derivatives. The Serrin-type integrals can most appropriately facilitate the existence statements for the extrema from either side of the duality relation in a larger class of BV functions, and interpret the solutions with possible discontinuities on sets of measure zero. The exact solutions of a beam and numerical solutions of a circular plate are presented to demonstrate the theoretical conclusions.

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.

In this paper we establish Perron and Krein–Rutman-like theorems for an operator mapping a cone into the interior of the cone, by considering the discrete dynamical system for the induced operator on the projective space (= sphere). Existence of a positive eigenvector reduces to showing that the ω-limit set of the induced operator consists of a single equilibrium. A special feature of our approach is that the convexity of the cone is needed only for establishing the non-emptiness of the w-limit set. This allows us in finite dimensions to establish an abstract Perron Theorem for non-convex cones.

Two theorems on the spectrum of operator pencils generated by the three-dimensional system of anisotropic elasticity in the complement of a plane angle are proved.

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.

Asymptotic estimates are established for nontrivial positive radial eigenfunctions of the nonlinear eigenvalue problem −Δu = λ(up − uq) in the unit ball B in ℝN (N > 2) with Neumann boundary conditions, as the supremum norm tends to infinity. Here p is the critical Sobolev exponent (N + 2)/(N − 2) and 0 < q < p − 1 = 4/(N − 2).

The variational eigenvalue problem for a real quadratic form b with respect to a positive definite form a on a vector space V may be represented by the triple (b, a, V). Methods of intermediate problems provide approximations from below to the lower eigenvalues of (b, a, V) using monotone increasing sequences of such triples. It is shown that every such approximation method is canonically equivalent to Weinstein's method. General convergence theorems are proved for such methods. These results generalise known convergence results for increasing sequences of quadratic forms. The results are applied to some specific approximation methods and are illustrated using a differential eigenvalue problem.

We prove local Lipschitz regularity for minimisers of integral functionals of the form J(u) = ∫Ω{f(Du(x)) + g(x, u(x))} dx, where the integrand f is not convex but satisfies some asymptotic convexity assumption.

We study solutions in ℝn of the nonlinear Schrödinger equation iut + Δu = λ |u|γu, where γ is the fixed power 4/n. For this particular power, these solutions satisfy the “pseudo-conformal” conservation law, and the set of solutions is invariant under a related transformation. This transformation gives a correspondence between global and non-global solutions (if λ < 0), and therefore allows us to deduce properties of global solutions from properties of non-global solutions, and vice versa. In particular, we show that a global solution is stable if and only if it decays at the same rate as a solution to the linear problem (with λ = 0). Also, we obtain an explicit formula for the inverse of the wave operator; and we give a sufficient condition (if λ < 0) that the blow up time of a non-global solution is a continuous function on the set of initial values with (for example) negative energy.

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.

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.

Let Gk be the gauge group of Pk, the principal SU(2) bundle over S4 with c2(Pk) = k. In this paper we show that Gk ≃ Gk. if and only if (12, k) = (12, k′) where (12, k) is the GCD of 12 and k.

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.

We consider the behaviour at x = ±∞ of solutions to reaction-diffusion equations modelling laminar flames in a premixed reactive gas. We show that if the initial data have limits at ±∞, then the solutions satisfy ODEs at ±∞ for all positive time. We then analyse the qualitative behaviour of solutions to the ODEs. Our applications include extensions of previous results on questions of flame propagation versus extinction, and a new decay result: if the initial temperature is above ignition temperature at one end of the domani and if the initial concentration vanishes at the other, then we show that the concentration decays^to zero uniformly as the time variable goes to infinity.

Let X be a uniformly convex Banach space, and N(X) the normal structure coefficient of X. In this paper it is proved that N(X) can be calculated by considering only sets whose points are equidistant from their Chebyshev centre. This result is applied to prove that N(LP(Ω)) = min {21−1/p, 21/p}, Ω being a σ-finite measure space. The computation of N(Lp) lets us also calculate some other coefficients related to the normal structure.

Consider the semilinear evolution equation

(P) u + Au = f(t,u)

where A is a sectorial operator on a Banach space and f is ω-periodic in t. Using a time-discrete Conley index developed in a previous paper [6], we prove a few existence results on bounded solutions of (P) defined for all t ∊ R. More specific results are given for time-periodic scalar parabolic equations.

A gp-toolkit consisting of computer implementations of various group theory methods, in particular a Tietze transformation program, was designed. Special cases of a conjecture were solved by the gp-toolkit. Examination of the method used by the gp-toolkit to deduce relations showed that a general approach had been employed. We present a proof verifying that the conjecture is true which is a straightforward generalisation of the method discovered by the gp-toolkit.

This paper presents a systematic account of ℤ/2-equivariant KO-theoretic methods in the study of r-fields (that is, r linearly independent cross-sections, r ≧ 1) on a real vector bundle. Applications of the theory to the problem of immersing complex and quaternionic projective spaces in Euclidean space are given in [13].

Stability and asymptotic stability of the null solution of the differential-difference equation (E)x′(t) = f(x(t), x(t − r)), f: RNxRN → RN, f(0, 0) = 0, are studied by means of an extension of the Liapunov–Razumikhin method. Let V: RN → R be a differentiate map, let C = C(+ −r, 0=, RN), and let x(t, ψ) denote the solution of (E) with initial condition ψ in C at t = 0. For t ≧ 0 let xt(ψ) be defined by xt,(ψ)(θ) = x(t + θ, ψ), −r ≦θ ≦0. Let V′ (ψ) be the variation of V along the solution x(t, ψ). We say that V is dichotomic with respect to (E) if there exist T ≧0 and Ω, a neighbourhood of the origin in C, such that if ψ is in the closure of the set where V′ (xT(ψ)) >; 0, then V(x(T, ψ)) ≦ V(x(s, ψ)) for some s, −r ≦ sT. It is proved that if V is positive definite, continuously differentiable, and dichotomic, then the null solution of (E) is stable. A concept of strict dichotomic map is introduced and used to prove asymptotic stability. A number of examples are given to illustrate the applications of the method.