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).

We discuss the existence of positive solutions of some singularity perturbed elliptic equations on convex domains with nonlinearity changing sign. In particular, we obtain solutions with both a boundary layer and a sharp interior peak.

Using the fibrering method, we prove the existence of multiple positive solutions of quasilinear problems of second order. The main part of our differential operator is p-Laplacian and we consider solutions both in the bounded domain Ω⊂ℝN and in the whole of ℝN. We also prove nonexistence results.

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

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.

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.

We discuss the stability and instability of steady-state solutions for a hydrodynamic model of semiconductors. We study the case where the doping profile is close to a positive constant and depends on the special variable x. We shall show that a given steady-state solution is asymptotically stable or unstable, depending on whether or not the density of the initial data satisfies P = 0, where P is defined in (3.12).

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 discuss existence and lowersemicontinuity for clusters of materials minimising an energy given by a collection of norms φij on the interfaces between regions Ri and Rj. Following Ambrosio and Braides, we exhibit a problem for which the triangle inequality holds but existence fails, and we state a new sufficient condition for lowersemicontinuity, which may be necessary.

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 hyperbolic system of conservation laws that govern the motion of a homogeneous isotropic, nonlinearly elastic body is shown to have a discontinuous solution for a class of stored-energy functions of slow growth. This solution is admissible by the usual entropy criterion and is in fact preferred by the entropy-rate criterion over the smooth equilibrium solution to the same problem. The existence of such a dissipative solution shows that the equilibrium solution is dynamically unstable. This instability cannot be ascertained by linearisation.

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.

Non-negative solutions of the initial boundary value problem for a degenerate parabolic equation are investigated. It is shown that solutions blow up regionally in finite tine. The size of blow-up sets is determined for radially symmetric cases.