In this paper, the existence and uniqueness of the global smooth solution are proved for an evolutionary Ginzburg–Landau model for superconductivity under the Coulomb and Lorentz gauge.

We consider the one-dimensional, nonlocal, evolution equation derived by De Masi et al. (1995) for Ising systems with Glauber dynamics, Kac potentials and magnetic field. We prove the existence of travelling fronts, their uniqueness modulo translations among the monotone profiles and their linear stability for all the admissible values of the magnetic field for which the underlying spin system exhibits a stable and metastable phase.

The existence of periodic solutions is studied for certain singularly perturbed differential inclusions. Applications are given to dry friction problems.

We study the asymptotic behaviour, for a sequence of varying open sets Ωn, of the solutions un of nonlinear Dirichlet problems for a monotone Leray–Lions operator. The method is based on the comparison between the gradient of un and the corrector for the p-Laplacian corresponding to the same geometry as the monotone operator. The representation of the limit problem and a corrector result are obtained.

We prove the existence of an arbitrarily large number of periodic solutions for a class of nonlinear differential equations generalising the dynamics of a forced pendulum with small length.

Homogenisation of the first-order Hamilton-Jacobi equations when H is periodic in the second variable, leads to an effective Hamiltonian H satisfying: uε converges, as ε → 0, to the solution u of ut + H(Du) = 0. In our first paper, we assumed that H is convex and we derived a variational formula giving H. In this second paper, we consider eikonal equations, i.e. H(p, x) = ½|p|2 – V(x). Using our variational formula, we compute explicitly the effective Hamiltonian in several cases, and we study precisely the lack of strict convexity for H (‘flat part’ around the origin).

In this paper we study the Poisson geometry of the second Hamiltonian structure for the periodic N Toda lattice, around a certain family of singularities. We show that their singular leaves are not isolated and that the regular codimension of the leaves at points of this kind is always equal to three. This result is based on a rather unexpected result about a certain Toeplitz matrix.

We describe all pairs of semigroups S and radicals ρ, such that ρ is invariant in S-graded rings. This generalises several known results due to Amitsur and Sands.

Let λε be a Dirichlet eigenvalue of the ‘periodically, rapidly oscillating’ elliptic operator –∇·(a(x/ε)∇) and let ∇ be a corresponding (simple) eigenvalue of the homogenised operator –∇·(A∇). We characterise the possible limit points of the ratio (λε–λ)/ε as ε→0. Our characterisation is quite explicit when the underlying domain is a (planar) convex, classical polygon with sides of rational or infinite slopes. In particular, in this case it implies that there is often a continuum of such limit points.

Let there be given a non-negative, quasiconvex function F satisfying the growth condition

for some p ∈]1, ∞[. For an open and bounded set Ω⊂ℝm, we show that if

then the variational integral

is lower semicontinuous on sequences of W1, p functions converging weakly in W1, q. In the proof, we make use of an extension operator to fix the boundary values. This idea is due to Meyers [26] and Maly [22], and the main contribution here is contained in Lemma 4.1, where a more efficient extension operator than the one in [22] (and in [14]) is used. The properties of this extension operator are in a certain sense best possible.

We study the existence and uniqueness of non-negative solutions of the nonlinear parabolic equation

posed in Q = RN × (0, ∞) with general initial data u(x, 0) = u0(x) ≧ 0. We find optimal exponential growth conditions for existence of solutions. Similar conditions apply for uniqueness, but the growth rate is different. Such conditions strongly depart from the linear case m = 1, ut = Δu – u, and also from the purely diffusive case ut = Δum.

This paper gives an explicit infinitesimal (to all orders) description of the period map associated to a smooth projective hypersurface, as well as related objects such as the full Hodge filtration on the middle cohomology, the local moduli space and the Gauss–Manin connection and its iterates.

In this paper, we prove the global existence and uniqueness of solutions to the Cauchy problem of a hyperbolic system, which probably contains so-called δ-waves.

We define the rational de Rham cohomology associated with the generalised confluent hypergeometric functions. Purity of the cohomology is proved and an explicit ℂ-basis of the nontrivial cohomology is computed.

In this paper we prove the existence and uniqueness of a renormalised solution of the nonlinear problem

where the data f and u0 belong to L1(Ω × (0, T)) and L1 (Ω), and where the function a:(0, T) × Ω × ℝN → ℝN is monotone (but not necessarily strictly monotone) and defines a bounded coercive continuous operator from the space into its dual space. The renormalised solution is an element of C0 ([ 0, T] L1 (Ω)) such that its truncates TK(u) belong to with

this solution satisfies the equation formally obtained by using in the equation the test function S(u)φ, where φ belongs to and where S belongs to C∞(ℝ) with

A nonlocal variational problem modelling phase transitions is studied in the framework of Young measures. The existence of global minimisers among functions with internal layers on an infinite tube is proved by combining a weak convergence result for Young measures and the principle of concentration-compactness. The regularity of such global minimisers is discussed, and the nonlocal variational problem is also considered on asymptotic tubes.

Positive definite temperature functions u(x, t) in ℝn+1 = {(x, t)| x ∈ ℝn,t > 0} are characterised by

where μ is a positive measure satisfying that for every ℰ > 0,

is finite. A transform is introduced to give an isomorphism between the class ofall positive definite temperature functions and the class of all possible temperature functions in Then correspondence given by generalises the Bochner–Schwartz Theorem for the Schwartz distributions and extends Widder's correspondence characterising some subclass of the positive temperature functions by the Fourier-Stieltjes transform.

This paper presents a study of linear operators associated with the linearisation of general semilinear strongly damped wave equations around stationary solutions. The structure of the spectrum of such operators is considered in detail, with an emphasis on stability questions. Necessary and sufficient conditions for the stability of the trivial solution of the linear equation are given, together with conditions for this solution to become unstable. In the latter case, the mechanisms which are responsible for the change of stability are analysed. These results are then applied to obtain stability and instability conditions for the semilinear problem. In particular, a condition is given which ensures that the dimensions of the centre and unstable manifolds of a stationary solution are the same as when that solution is considered as a stationary solution of an associated parabolic problem.