We study a semilinear elliptic problem on ℝN where a potential is not a bounded function but can also be an infinite measure. We analyse the lack of compactness of the problem, obtaining a structure theorem for Palais-Smale sequences. This result allows us to obtain different kinds of existence theorems by a variational approach.

The paper is concerned with equations of the form x' = A(t)x +f(t, x), where A is a continuous matrix function defined on ℝ, f is a continuous vector-valued function of (t, x) with f(t, 0) = 0. It is proved that if x' = A(t)x has an exponential trichotomy, A is bounded and f satisfies the Lipschitz condition with coefficient sufficiently small, then these equations are topologically equivalent to the systems of equations of the form , where B, g satisfy the same conditions as A, f.

In this paper we establish bounds constraining the effective conductivity tensor of composites made of an arbitrary number n of possibly anisotropic phases in prescribed volume fractions. The bounds are valid in any spatial dimension d≧2. The bounds have a very simple and concise form and include those previously obtained by Hashin and Shtrikman, Murat and Tartar, Lurie and Cherkaev, Kohn and Milton, Avellaneda, Cherkaev, Lurie and Milton and Dell'Antonio and Nesi.

Multiple integrals with polyconvex integrands are studied on the class of all sense-preserving diffeomorphisms from W1,p(Ω, Rn) where Ω is an open subset of Rn. They are proved to be sequentially weakly lower semicontinuous if 1 < p = n –1. An example is presented showing that a similar result is not valid if p <n –1.

The problem

is considered, where X is a normed space, F: X →] –∞, + ∞] is a (possibly non-convex) functional and L ∈ X'. We look for the values of γy for which the infimum above is attained. Applications to nonconvex functionals denned on measures and on the BV space are given.

Let Cf, Pf, Qf and Rf be respectively the convex, polyconvex, quasi-convex and rank-one-convex envelopes of a given function f. If fp: RNxM→R and fq(ξ) behaves as |ξ|q at infinity q ∈ (1, ∞), we show that . This is the case for (Pfp)p provided that q ≠1,…, min (N, M), otherwise . In the last part of this work, we show that f(ξ) = g(|ξ|) does not imply in general Pf = Qf.

Hyperasymptotic expansions were recently introduced by Berry and Howls, and yield refined information by expanding remainders in asymptotic expansions. In a recent paper of Olde Daalhuis, a method was given for obtaining hyperasymptotic expansions of integrals that represent the confluent hypergeometric U-function. This paper gives an extension of that method to neighbourhoods of the so-called Stokes lines. At each level, the remainder is exponentially small compared with the previous remainders. Two numerical illustrations confirm these exponential improvements.

This paper, although self-contained, is a continuation of the work done in [8], where the motion of a viscous, incompressible fluid is considered in conjunction with the rotation of a rigid body which is immersed in the fluid. The resulting mathematical model is a Navier-Stokes problem with dynamic boundary conditions. In [8] a unique L2,3 solution is constructed under certain regularity assumptions on the initial states. In this paper we consider the Navier-Stokes problem with dynamic boundary conditions in the Lebesgue spaces Lr,3 (3<r<∞) and prove the existence of a unique solution, local in time, without imposing any regularity conditions on the initial states.

This paper is devoted to the study of the asymptotic behaviour of radial solutions to an elliptic equation in ℝn. The equation is derived from the blow-up problem in the non-linear Schrödinger equation.