We establish the existence of isolated local minimisers to the problem of partitioning certain two-dimensional domains into three subdomains having least interfacial area. The solution we exhibit has the special property that the three boundaries of the minimising partition meet at a common point or “triple junction”. The configuration represents a likely candidate for a stable equilibrium in the dynamical problem of two-dimensional motion by curvature and also leads to the existence of local minimisers possessing triple junction structure to the energy associated with the vector Ginzburg–Landau and Cahn–Hilliard evolutions.

In this paper, we study all the stationary solutions of the form u(r)einθ to the complex-valued Ginzburg–Landau equation on the complex plane: here (r, θ) are the polar coordinates, and n is any real number. In particular, we show that there exists a unique solution which approaches to a nonzero constant as r → ∞.

Nonstandard analysis is used, in this paper, to give a construction of a Wiener -process Wt, t ∈ [0, ∞). From this, a hyperfinite representation of stochastic integrals for operatorvalued processes with respect to Wt is derived, and existence theorems in the spirit of Keisler are proved for (infinite-dimensional) stochastic differential equations of Itô's type one and a certain kind of Itô's type two, via regularity of hyperfinite stochastic difference equations.

We consider an emulsion of two Stokes fluids, one of which is periodically distributed in the form of small spherical bubbles. The effects of surface tension on the bubble boundaries are modelled mathematically, as in the work of G. I. Taylor, by a jump only in the normal component of the traction. For a given volume fraction of bubbles, we consider the two-scale convergence, and in the fine phase limit we find that the bulk flow is described by an anisotropic Stokes fluid. The effective viscosity tensor is consistent with the bulk stress formula obtained by Batchelor [2].

In this paper we characterise the linear transformations of an infinite-dimensional vector space that can be written as the product of nilpotent transformations. This and a linear version of Malcev's congruence on transformation semigroups are then used to construct a new class of congruence-free semigroups.

We consider an integrable case of the Henon-Heiles system and use an isomorphism with the two-gap KdV-flow to construct families of real elliptic trajectories which are associated with two-gap elliptic solitons of the KdV equation. Some of these solutions exhibit blow-up in finite time.

We study the existence of changing sign solutions of an elliptic semilinear boundary value problem, which arises as a limiting equation of the two species Lotka–Volterra competing equations system. Using variational methods and a result of D'Aujourd'hui, we find conditions which are both sufficient and necessary for this existence problem.

We consider the following problem:

where is continuous on RN and h(x)≢0. By using Ekeland's variational principle and the Mountain Pass Theorem without (PS) conditions, through a careful inspection of the energy balance for the approximated solutions, we show that the probelm (*) has at least two solutions for some λ* > 0 and λ ∈ (0, λ*). In particular, if p = 2, in a different way we prove that problem (*) with λ ≡ 1 and h(x) ≧ 0 has at least two positive solutions as

For a finite semilattice S, is is proved that if every noninvertible endomorphism is a product of idempotents, then S is a chain; the converse was proved, independently, by A. Ya. Aĭzenštat and J. M. Howie. For a finite pseudocomplemented semilattice S, with pseudocomplementation regarded as a unary operation, it is proved that all noninvertible endomorphisms are products of idempotents if and only if S is Boolean or a chain.

The problem of describing one-relator pro-p-groups of cohomological dimension two (along the lines of Lyndon's description of discrete one-relator groups of cohomological dimension two) is still open. The known method of passing by means of a suitable p-filtration to a graded Lie algebra is not applicable to the family of one-relator pro-p-groups presented in this article, since the relators cannot be separated from the p-th powers in the free pro-p-group. In terms of the p-filtrations, the relators come arbitrarily close to a p-th power, yet the groups they define have cohomological dimension two.

Conditions for the existence of a centre in two-dimensional systems are considered along the lines of Darboux. We show how these methods can be used in the search for maximal numbers of bifurcating limit cycles. We also extend the method to include more degenerate cases such as are encountered in less generic systems. These lead to new classes of integrals. In particular, the Kukles system is considered, and new centre conditions for this system are obtained.

We consider a family of dispersive equations whose simplest representative would be a Benjamin–Bona–Mahony equation with a Burger's type dissipation. The effect of possible unevenness of the bottom surface is considered and our main result gives decay rates of the solutions in Lβ(ℝ) spaces, 2 ≦ β ≦ + ∞.

In this paper we look for closed geodesies on a noncomplete Riemannian manifold ℳ. We prove that if ℳ has convex boundary, then there exists at least one closed nonconstant geodesic on it.

In this paper we prove the uniqueness and existence of harmonic maps of finite energy from a complete, noncompact Riemannian manifold (M, g) with Sobolev constant S2(M) > 0 and Ricci curvature Ric (M) ≧ 0 outside some compact subset, into a complete manifold of nonpositive curvature or a regular ball. In particular, we prove the uniqueness and existence of bounded harmonic functions on (M, g).