Let f : G → H be a fixed homomorphism and p′ : G * H → G and p″ : G * H → H the two projections of the free product. Then a co-action relative to f is a homomorphism s : G → G * H such that p′s = id and p″s = f. We study this notion and investigate the following questions. What restrictions does s place on the structure of the group G? What form does s take in special cases? When does s induce a co-multiplication on H? What is the relation between associativity of s and associativity of the induced co-multiplication m on H? What are the properties of the operation of Hom(H, B) on Hom(G, B) induced by s : G → G * H? In addition, we give several diverse examples of co-actions in the last section.

Let L : dom L ⊂ L2(Ω) → L2(Ω) be a self-adjoint operator, Ω being open and bounded in RN. We give a description of the Fučík spectrum of L away from the essential spectrum. Let λ be a point in the discrete spectrum of L; provided that some non-degeneracy conditions are satisfied, we prove that the Fučík spectrum consists locally of a finite number of curves crossing at (λ, λ). Each of these curves can be associated to a critical point of the function H : x ↦ 〈|x|,x〉L2 restricted to the unit sphere in ker(L – λI). The corresponding critical values determine the slopes of these curves. We also give global results describing the Fučík spectrum, and existence results for semilinear equations, by performing degree computations between the Fučík curves.

We consider the maximal operator defined on the real line by which is related to the Cesàro convergence of the singular integrals. We characterize the weights w for which Mα is of weak type, strong type and restricted weak type (p, p) with respect to the measure w(x) dx.

Mathematical models of phase transitions in solids lead to the variational problem, minimize ∫Ω W (Du) dx, where W has a multi-well structure, i.e. W = 0 on a multi-well set K and W > 0 otherwise. We study this problem in two dimensions in the case of equal determinant, i.e. for K = SO(2)U1 ∪ … ∪SO(2)Uk or K = O(2)U1 ∪ … ∪ O(2)Uk for U1, … , Uk ∈ M2×2 with det Ui = δ in three dimensions when the matrices Ui are essentially two-dimensional and also for K = SO(3)Û1 ∪ … ∪ SO(3)Ûk for U1, … , Uk ∈ M3×3 with , which arises in the study of thin films. Here, Ûi denotes the (3×2) matrix formed with the first two columns of Ui. We characterize generalized convex hulls, including the quasiconvex hull, of these sets, prove existence of minimizers and identify conditions for the uniqueness of the minimizing Young measure. Finally, we use the characterization of the quasiconvex hull to propose ‘approximate relaxed energies’, quasiconvex functions which vanish on the quasiconvex hull of K and grow quadratically away from it.

We study a predator–prey system with Holling–Tanner interaction terms. We show that if the saturation rate m is large, spatially inhomogeneous steady-state solutions arise, contrasting sharply with the small-m case, where no such solution could exist. Furthermore, for large m, we give sharp estimates on the ranges of other parameters where spatially inhomogeneous solutions can exist. We also determine the asymptotic behaviour of the spatially inhomogeneous solutions as m → ∞, and an interesting relation between this population model and free boundary problems is revealed.

Keller's phase-interchange relation, which computes the overall conductivity of a two-dimensional checkerboard with alternating conductivity, is revisited in the context of nonlinear incompressible elasticity. A general phase-interchange relation is obtained in a monotone setting through H-convergence and in a convex setting through variational methods. Several Keller-like applications are presented.

We consider here the radial Stefan problem with Gibbs–Thomson law, which is a classical model describing growth or melting of a spherical crystal in a surrounding liquid. We shall specialize to the cases of two and three space dimensions and discuss the asymptotic behaviour of a melting crystal near its dissolution time t* > 0. We prove here that, when the interface shrinks monotonically, the asymptotics near t = t* is of the form Here, R(t) denotes the radius of the crystal, σ is a surface tension parameter and u(r, t) represents the field temperature. An important point to be noticed is that (*) exhibits no dependence on the space dimension N, in sharp contrast with results known for the case σ = 0.

In this paper we prove the existence and analyticity of solitary waves of generalized Kadomtsev–Petviashvili equations satisfying a set of conditions on linear and nonlinear terms which determine their behaviour at infinity and around 0.

In this paper we prove a certain regularity property of configurations of immiscible fluids, filling a bounded container Ω and locally minimizing the interface energy ∑i<jcij‖Sij‖, where Sij represents the interface between fluid i and fluid j, ‖·‖ stands for area or more general area-type functional, and cij is a positive coefficient. More precisely, we show that, under strict triangularity of the cij, no infiltrations of other fluids are allowed between two main ones. A remarkable consequence of this fact is the almost-everywhere regularity of the interfaces. Our analysis is performed in general dimension n ≥ 2 and with volume constraints on fluids.

The global solution, stability and evolution of regularities are studied for the initial-boundary-value problem of a viscous compressible gas flow. With global a priori estimates and careful analysis, the existence of global classical solutions in Hölder space is established.

In the first part, we study several exterior boundary-value problems covering three types of semilinear equations: elliptic, parabolic and hyperbolic. By a unified approach, we show that these problems share a common critical behaviour. In the second part we prove a blow-up result for an inhomogeneous porous medium equation with the critical exponent, which was left open in a previous paper.