This paper is concerned with the existence of topological multivortex solutions in (2 + 1) self-dual gauge theories such as the classical abelian Higgs model or the Chern–Simons Higgs gauge theories. A general form of topological multivortex equations is presented with the inclusion of antivortices. Two kinds of solutions are considered; topological vortex solutions and topological vortex–antivortex solutions. We construct these solutions by super and subsolution methods, and derive the exponential decay of solutions at infinity and the quantized integral formula. As an application, we prove the existence of a topological multivortex solutions in a generalized Chern–Simons Higgs theory.

We study the Cauchy problem for the nonlinear Schrödinger equation with dissipation where L is a linear pseudodifferential operator with dissipative symbol ReL(ξ) ≥ C1|ξ|2/(1 + ξ2) and |L′(ξ)| ≤ C2(|ξ|+ |ξ|n) for all ξ ∈ R. Here, C1, C2 > 0, n ≥ 1. Moreover, we assume that L(ξ) = αξ2 + O(|ξ|2+γ) for all |ξ| < 1, where γ > 0, Re α > 0, Im α ≥ 0. When L(ξ) = αξ2, equation (A) is the nonlinear Schrödinger equation with dissipation ut − αuxx + i|u|2u = 0. Our purpose is to prove that solutions of (A) satisfy the time decay estimate under the conditions that u0 ∈ Hn,0 ∩ H0,1 have the mean value and the norm ‖u0‖Hn,0 + ‖u0‖H0,1 = ε is sufficiently small, where σ = 1 if Im α > 0 and σ = 2 if Im α = 0, and Therefore, equation (A) is considered as a critical case for the large-time asymptotic behaviour because the solutions of the Cauchy problem for the equation ut − αuxx + i|u|p−1u = 0, with p > 3 have the same time decay estimate ‖u‖L∞ = O(t−½) as that of solutions to the linear equation. On the other hand, note that solutions of the Cauchy problem (A) have an additional logarithmic time decay. Our strategy of the proof of the large-time asymptotics of solutions is to translate (A) to another nonlinear equation in which the mean value of the nonlinearity is zero for all time.

This paper is concerned with the existence of maximizers for a certain non-convex energy functional, relative to a class of rearrangements of a given function. Physically, the solution represents a uniform shear flow containing a bounded vortex anomaly, in R2. The prescribed data are the rearrangement class of the vorticity field.

Let D be a domain in Rn with bounded complement and let n ≠ 2. For the initial-boundary value problemwe prove that there are no non-trivial global (non-negative) solutions if 0 < n (p − 1) ≤ 2 and there exist both global non-trivial and non-global solutions if n (p − 1) > 2.

This paper provides a topological and ergodic analysis of random linear Hamiltonian systems. We consider a class of Hamiltonian equations presenting absolutely continuous dynamics and prove the existence of the radial limits of the Weyl M-functions in the L1-topology. The proof is based on previous ergodic relations obtained for the Floquet coefficient. The second part of the paper is devoted to the qualitative description of disconjugate linear Hamiltonian equations. We show that the principal solutions at ±∞ define singular ergodic measures, and determine an invariant region in the Lagrange bundle which concentrates the essential dynamical information. We apply this theory to the study of the n-dimensional Schrödinger equation at the first point of the spectrum.

We are concerned with positive solutions decaying at infinity for a class of semilinear elliptic equations in all of RN having superlinear subcritical nonlinearity. The corresponding variational problem lacks compactness because of the unboundedness of the domain and, in particular, it cannot be solved by minimization methods. However, we prove the existence of a positive solution, corresponding to a higher critical value of the related functional, under a suitable fast decay condition on the coefficient of the linear term. Moreover, we analyse the behaviour of the solution as this coefficient goes to infinity and show that the solution tends to split as the sum of two positive functions sliding to infinity in opposite directions. Finally, we use this property to prove the existence of at least 2k − 1 distinct positive solutions, when this coefficient splits as the sum of k bumps sufficiently far apart.

We construct non-trivial laminates distributed continuously along a smooth closed ‘evolute’ curve by approximating an auxiliary (‘involute’) curve by a polygon with rank-1 sides. We give an explicit example in which the evolute has no rank-1 connections. Using these techniques, given a finite set of matrices B1,…,Br and any other matrix A, we show how to construct a laminate with positive mass at each Bi, any proportion less than 1 of its support on the Bi, and centre of mass at A.

The asymptotic behaviour of solutions of second-order quasilinear elliptic partial differential equations defined on unbounded domains in Rn contained in strips (when n = 2) or slabs (when n > 2) is investigated when such solutions satisfy Dirichlet boundary conditions and the Dirichlet boundary data have appropriate asymptotic behaviour at infinity. We prove Phragmèn–Lindelöf theorems for large classes of elliptic operators, including uniformly elliptic operators and operators with well-defined genre, establish exponential decay estimates for uniformly elliptic operators when the Dirichlet boundary data vanish outside a compact set, establish the uniqueness of solutions, and give examples of solutions for non-uniformly elliptic operators which decay but do not decay exponentially. Our principal theorems are proven using special barrier functions; these barriers are constructed by considering an operator associated to our original operator.

In this paper, we study the non-existence of solutions for the following (model) problem in a bounded open subset Ω of RN:with Dirichlet boundary conditions, where p > 1, q > 1 and μ is a bounded Radon measure. We prove that if λ is a measure which is concentrated on a set of zero r capacity (p < r ≤ N), and if q > r (p − 1)/(r − p), then there is no solution to the above problem, in the sense that if one approximates the measure λ with a sequence of regular functions fn, and if un is the sequence of solutions of the corresponding problems, then un converges to zero.

We also study the non-existence of solutions for the bilateral obstacle problem with datum a measure λ concentrated on a set of zero p capacity,with u in for every υ in K, finding again that the only solution obtained by approximation is u = 0.

In this paper we prove, on one hand, extrapolation from infinity for the one-sided classes , and on the other hand, the BMO-boundedness for one-sided singular integrals. We also provide several applications of our results.

We study a Stefan problem for coarsening of many small particles of solid phase in undercooled liquid phase which includes surface tension and kinetic undercooling effects. A mean-field model is derived by homogenization of the free boundary problem. It extends the classical theory of Lifshitz et al. by taking into account effects due to kinetic undercooling.

Sufficient conditions are obtained for a ring R, faithfully graded by a bisimple inverse semigroup S, to be (a) prime and (b) right primitive, these conditions being on the subring RG consisting of all elements of R with support contained in G, a maximal subgroup of S. Earlier results on semigroup rings arise as special cases.

If A and B are C*-algebras and X is an operator A, B-bimodule, then points of X can be separated from closed A, B-absolutely convex subsets of X by completely bounded A, B-bimodule homomorphisms from X into B(K), where K is a Hilbert space and the A, B-bimodule structure on B(K) is induced by a pair of representations π : A → B(K) and σ : B → B(K). If A and B are von Neumann algebras and X is a normal (not necessarily dual) operator A, B-bimodule, those A, B-absolutely convex subsets of X are characterized which can be separated from points of X as above, but with the additional requirement that the two representations π and σ are normal. This requires a new topology on X, which has appeared also in connection with some other questions concerning operator modules.

This paper is devoted to a study of local linear independence of refinable vectors of functions. A vector of functions is said to be refinable if it satisfies the vector refinement equation where a is a finitely supported sequence of r × r matrices called the refinement mask. A complete characterization for the local linear independence of the shifts of ϕ1,…,ϕr is given strictly in terms of the mask. Several examples are provided to illustrate the general theory. This investigation is important for construction of wavelets on bounded domains and nonlinear approximation by wavelets.

It is known that the condition ‘either ∂L (F) ≠ Ø or there exist υ1,…,υq ∈ Rnsuch thatF ∈ int co {υ1,…,υq} characterizes solvability of the problemwith f(·) = 〈F,·〉.

We extend this result to the case of lower semicontinuous integrands L : Rn → R.

We also show that validity of this condition for all F ∈ Rn is both a necessary and sufficient requirement for solvability of all minimization problems with sufficiently regular Ω and f. Moreover, the assumptions on Ω and f can be completely dropped if L has sufficiently fast growth at infinity.

An example is given of a quasiconvex f : M2×3 → R such that the transposed function f̃ : M3×2 → R given by f̃(F) = f(FT) is not quasiconvex. For f̃ one can take Sverák's quartic polynomial that is rank-one convex but not quasiconvex. The proof is closely related to the observation that the map v ↦ v1v2v3 is weakly continuous from L3(R3; R3) into distributions provided that A(Dv) = (∂2v1, ∂3v1, ∂1v2, ∂3v2, ∂1v3, ∂2v3) is compact in W−1,3(R3; R6).

We study the stability of periodic solutions of the scalar delay differential equation where f(t) is a periodic forcing term and δ,p∈R. We study stability in the first approximation showing that the non-smooth equation (*) can be linearized along some ‘non-singular’ periodic solutions. Then the corresponding variational equation together with the Krasnosel'skij index are used to prove the existence of multiple periodic solutions to (*). Finally, we apply a generalization of Halanay's inequality to establish conditions for global attractivity in equations with maxima.

We study some global geometric properties of a static Lorentzian manifold Λ embedded in a differentiable manifold M, with possibly non-smooth boundary ∂Λ. We prove a variational principle for geodesics in static manifolds, and using this principle we establish the existence of geodesics that do not touch ∂Λ and that join two fixed points of Λ. The results are obtained under a suitable completeness assumption for Λ that generalizes the property of global hyperbolicity, and a weak convexity assumption on ∂Λ. Moreover, under a non-triviality assumption on the topology of Λ, we also get a multiplicity result for geodesics in Λ joining two fixed points.

The initial-boundary value problem for the non-homogeneous Navier-Stokes equations including the slipping on the solid boundary is considered. The unique solvability is established locally in time for the three-dimensional problem and globally in time for the two-dimensional problem without so-called smallness restrictions.

A reduction method is used to prove the existence and uniqueness of strong solutions to stochastic Kolmogorov–Petrovskii–Piskunov (KPP) equations, where the initial condition may be anticipating. The asymptotic behaviour of the solution for large time and space and the random travelling waves are then studied under two different basic assumptions.