We give an alternative self-contained proof of the homogenization theorem for periodic multi-parameter integrals that was established by the authors. The proof in that paper relies on the so-called compactness method for Γ-convergence, while the one presented here is by direct verification: the candidate to be the limit homogenized functional is first exhibited and the definition of Γ-convergence is then verified. This is done by an extension of bounded gradient sequences using the Acerbi et al. extension theorem from connected sets, and by the adaptation of some localization and blow-up techniques developed by Fonseca and Müller, together with De Giorgi's slicing method.

We consider scalar hyperbolic conservation laws with non-convex flux and vanishing, nonlinear and possibly singular, diffusion and dispersion terms. The diffusion has the form R(u, ux)x and we cover, for instance, the singular diffusion (|ux|pux)x, where p ≥ 0 is arbitrary. We investigate the existence, uniqueness and various properties of classical and non-classical travelling waves and of the kinetic function. The latter serves to characterize non-classical shock waves, via an additional algebraic constrain called a kinetic relation. We discover that p = ⅓ is a somewhat unexpected critical value. For p ≤ ⅓, we obtain properties that are qualitatively similar to those we established earlier for regular and linear diffusion. However, for p > ⅓, the behaviour of the kinetic function is very different, as, for instance, non-classical shocks can have arbitrary small strength. The behaviour of the kinetic function near the origin is carefully investigated and depends on whether p < ½, p = ½ or p > ½. In particular, in the special case of the cubic flux-function and for the regularization (|ux|pux)x with p = 0, ½ or 1, the kinetic function can be computed explicitly. When p = ½, the kinetic function is simply a linear function of its argument.

An integral representation of a relaxed functional arising in the derivation of a nonlinear membrane model accounting for the density of bending moments is obtained in terms of a special class of parametrized probability measures (bending Young measures). A complete characterization of this class is given.

We study the linear stability of a vortex sheet in a limit case that corresponds to a transition between a weakly stable regime and a violently unstable regime. We prove an energy estimate that reflects the high degeneracy of the uniform Kreiss–Lopatinskii condition.

In this paper we study the existence of radially symmetric solitary waves for nonlinear Klein–Gordon equations and nonlinear Schrödinger equations coupled with Maxwell equations. The method relies on a variational approach and the solutions are obtained as mountain-pass critical points for the associated energy functional.

We discuss some existence theorems for partial differential inclusions, subject to Dirichlet boundary conditions, of the form where Φ is a quasi-affine function and so, in particular, for Φ(Du) = det Du.

We then apply it to minimization problems of the form

The structure of positive boundary blow-up solutions to semilinear problems of the form −Δu = λf(u) in Ω, u = ∞ on ∂Ω, Ω ⊂ RN a bounded smooth domain, is studied for a class of nonlinearities f ∈ C1 ([0, ∞)\{z2}) satisfying f (0) = f(z1) = f (z2) = 0 with 0 < z1 < z2, f < 0 in (0, z1)∪(z2, ∞), f > 0 in (z1, z2). Two positive boundary-layer solutions and infinitely many positive spike-layer solutions are obtained for λ sufficiently large.

We study the Cauchy problem for the one-dimensional equations of a viscous heat-conducting gas in the Lagrangian mass coordinates with the initial data in the Lebesgue spaces. We prove the existence, the uniqueness and the Lipschitz continuous dependence on the initial data of global weak solutions.

We introduce a new non-classical Riemann solver for scalar conservation laws with concave–convex flux-function. This solver is based on both a kinetic relation, which determines the propagation speed of (under-compressive) non-classical shock waves, and a nucleation criterion, which makes a choice between a classical Riemann solution and a non-classical one. We establish the existence of (non-classical entropy) solutions of the Cauchy problem and discuss several examples of wave interactions. We also show the existence of a class of solutions, called splitting–merging solutions, which are made of two large shocks and small bounded-variation perturbations. The nucleation solvers, as we call them, are applied to (and actually motivated by) the theory of thin-film flows; they help explain numerical results observed for such flows.

Given a smooth function 𝔏 of a real or complex variable and taking its values in the class of Fredholm operators of index zero in a Banach space, there are some available definitions in the literature of algebraic multiplicity of the family 𝔏 at a point x0 of the parameter at which the operator 𝔏(x0) becomes non-invertible. The purpose of this paper is to suggest an axiomatic for the multiplicity and to prove that the algebraic multiplicity is uniquely determined by a few of its properties, independently of its construction.

In this paper, a host-vector model is considered for a disease without immunity in which the current density of infectious vectors is related to the number of infectious hosts at earlier times. Spatial spread in a region is modelled in the partial integro-differential equation by a diffusion term. For the general model, we first study the stability of the steady states using the contracting-convex-sets technique. When the spatial variable is one dimensional and the delay kernel assumes some special form, we establish the existence of travelling wave solutions by using the linear chain trick and the geometric singular perturbation method.

Consider the second-order discrete system where f ∈ C (R × Rm, Rm), f(t + M, Z) = f(t, Z) for any (t, Z) ∈ R × Rm and M is a positive integer. By making use of critical-point theory, the existence of M-periodic solutions of (*) is obtained.