This paper is concerned with the existence of minimizers for functionals having a double-well integrand with affine boundary conditions. Such functionals are related to the so-called Kohn–Strang functional, which arises in optimal shape design problems in electrostatics or elasticity. They are known to be not quasiconvex, and therefore existence of minimizers is, in general, guaranteed only for their quasiconvex envelopes. We generalize the previous results of Allaire and Francfort, and give necessary and sufficient conditions on the affine boundary conditions for existence of minimizers. Our method relies on the computation of the quasiconvexification of these functionals by using homogenization theory. We also prove by a general argument that their rank-one convexifications coincide with their quasiconvexifications.

Two relative compactness results for two-scale convergence in homogenization, due to G. Nguetseng, were recently extended to the multi-scale case by G. Allaire and M. Briane. Whereas their extension of Nguetseng's first result, which is in L2, is straightforward, their extension of his second result, which takes place in the Sobolev space H1, is quite complicated, even though it follows Nguetseng by using the fact that the image of H1 under the gradient mapping is the orthogonal complement of the set of divergence-free functions. Here a much simpler proof is provided by deriving the H1-type result from combining the first extension result with the fact that the above-mentioned image space is also the space of all rotation-free fields. Moreover, this approach reveals that the two results can be seen as corollaries of a fundamental relative compactness result for Young measures.

Presented herein is a new method for analysing the long-time behaviour of solutions of nonlinear, dispersive, dissipative wave equations. The method is applied to the generalized Korteweg–de Vries equation posed on the entire real axis, with a homogeneous dissipative mechanism included. Solutions of such equations that commence with finite energy decay to zero as time becomes unboundedly large. In circumstances to be spelled out presently, we establish the existence of a universal asymptotic structure that governs the final stages of decay of solutions. The method entails a splitting of Fourier modes into long and short wavelengths which permits the exploitation of the Hamiltonian structure of the equation obtained by ignoring dissipation. We also develop a helpful enhancement of Schwartz's inequality. This approach applies particularly well to cases where the damping increases in strength sublinearly with wavenumber. Thus the present theory complements earlier work using centre-manifold and group-renormalization ideas to tackle the situation wherein the nonlinearity is quasilinear with regard to the dissipative mechanism.

We outline a completely probabilistic study of travelling-wave solutions of the FKPP reaction-diffusion equation that are monotone and connect 0 to 1. The necessary asymptotics of such travelling-waves are proved using martingale and Brownian motion techniques. Recalling the connection between the FKPP equation and branching Brownian motion through the work of McKean and Neveu, we show how the necessary asymptotics and results about branching Brownian motion combine to give the existence and uniqueness of travelling waves of all speeds greater than or equal to the critical speed.

We deal with the large-time behaviour of scalar hyperbolic conservation laws with source terms

which are often called hyperbolic balance laws. Fan and Hale have proved existence of a global attractor for this equation with x ∈ S1. consists of spatially homogeneous equilibria, a large number of rotating waves and of heteroclinic orbits between these objects. In this paper, we solve the connection problem and show which equilibria and rotating waves are connected by a heteroclinic orbit. Apart from existence results, our approach via generalized characteristics also gives geometric information about the heteroclinic solutions, e.g. about the shock curves and their strength.

We study the Jacobian determinants J = det(∂fi/∂xj) of mappings f: Ω ⊂ ℝn → ℝn in a Sobolev–Orlicz space W1,Φ (Ω,ℝn). Their natural generalizations are the wedge products of differential forms. These products turn out to be in the Hardy–Orlicz spaces ℌp (Ω). Other nonlinear quantities involving the Jacobian, such as J log |J|, are also studied. In general, the Jacobians may change sign and in this sense our results generalize the existing ones concerning positive Jacobians.

We study the linear stability of equilibrium points of a semilinear phase-field model, giving criteria for stability and instability. In the one-dimensional case, we study the distribution of equilibria and also prove the existence of metastable solutions that evolve very slowly in time.

This paper studies geometric properties, in particular extreme points and rays, of various generalizations of Young measures. Applications of the knowledge of extreme points are illustrated on existence results for optimal control problems and on various convergence results for Young measures by using the Choquet theory.

A vector equation which gives the velocity of a vortex filament embedded in an inviscid incompressible flow is considered. It comprises terms representing effects from the localized self-induction and from the external flow. The initial value problem is proved to have at least a solution for a suitable external flow term.

The time-dependent Ginzburg-Landau equations of superconductivity in three spatial dimensions are investigated in this paper. We establish the existence of global weak solutions for this model with any Lp (p ≧ 3) initial data. This work generalizes the results of Wang and Zhan.

Developing an approach of Repnitskii and Volkov, we focus on properties of semigroups of order-preserving mappings on finite chains; in particular, we show that the class of all these semigroups has no finite quasi-identity basis (although it has an infinite one).

In this note, we present some Liouville type theorems about the non-negative solutions to some indefinite elliptic equations.