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.

It is shown that the Cesàro averaging operators Cα, Re α > −1, introduced by Stempak, are bounded on the Dirichlet space Da if and only if a > 0, while the associated operators Aα are bounded on Da if and only if −1 < a < 2. This extends results of Galanopoulos, who considered the particular case α = 0 for 0 ≤ a ≤ 1.

We consider the Cauchy problem for a class of scalar conservation laws with flux having a single inflection point. We prove existence of global weak solutions satisfying a single entropy inequality together with a kinetic relation, in a class of bounded variation functions. The kinetic relation is obtained by the travelling-wave criterion for a regularization consisting of balanced diffusive and dispersive terms. The result is applied to the one-dimensional Buckley-Leverett equation.

We study a class of initial-boundary-value problems for which an auxiliary condition of the form is prescribed. We determine bounds on an energy expression by means of differential inequalities and derive pointwise bounds for the solution and its gradient by use of a parabolic maximum principle.

We study the Cauchy problem for the spatially homogeneous Landau equation for Fermi–Dirac particles, in the case of hard and Maxwellian potentials. We establish existence and uniqueness of a weak solution for a large class of initial data.

We describe surjective linear maps preserving commutativity from (symmetric elements of) any algebra (with involution) onto (symmetric elements of) a prime algebra (with involution) not satisfying polynomial identities of low degree. Bijective commutativity preservers on skew elements of centrally closed prime algebras with involution of the first kind are also investigated.

The aim of this paper is to study the asymptotic behaviour of the solutions of the linearized elasticity system, posed on thin reticulated structures involving several small parameters. We show that this behaviour depends on the relative size of the parameters. In each case, we obtain a limit system where the microstructure and macrostructure appear simultaneously. From it, we get a suitable approximation in L2 of the displacements and the linearized strain tensor.

In this paper we prove new existence results for non-autonomous systems of first order ordinary differential equations under weak conditions on the nonlinear part. Discontinuities with respect to the unknown are allowed to occur over general classes of time-dependent sets which are assumed to satisfy a kind of inverse viability condition.

We study a boundary-value problem for a particular semilinear elliptic equation on Rn (n ≥ 2), whose solutions represent generalized stream functions for steady axisymmetric ideal fluid flows. Solutions are shown to exist that generalize those already known in dimensions 2 and 3. An isoperimetric characterization is given for our solutions, which represent generalized spherical vortex-rings. As a corollary, a sharp Sobolev-type inequality is obtained.

We give an example of an indefinite weight Sturm-Liouville problem whose eigenfunctions form a Riesz basis under Dirichlet boundary conditions but not under anti-periodic boundary conditions.

In this paper we obtain, for a semilinear elliptic problem in RN, families of solutions bifurcating from the bottom of the spectrum of −Δ. The problem is variational in nature and we apply a nonlinear reduction method that allows us to search for solutions as critical points of suitable functionals defined on finite-dimensional manifolds.

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 prove the existence of a non-trivial solution for the nonlinear elliptic problem −Δu + V(x)u = a(x)g(u) in RN, where g is superlinear near zero and near infinity, a(x) changes sign and V ∈ C(RN) is positive at infinity. For g odd, we prove the existence of an infinite number of solutions.

Given an infinite-dimensional vector space V, we consider the semigroup GS (m, n) consisting of all injective linear α: V → V for which codim ran α = n, where dim V = m ≥ n ≥ ℵ0. This is a linear version of the well-known Baer–Levi semigroup BL (p, q) defined on an infinite set X, where |X| = p ≥ q ≥ ℵ0. We show that, although the basic properties of GS (m, n) are the same as those of BL (p, q), the two semigroups are never isomorphic. We also determine all left ideals of GS (m, n) and some of its maximal subsemigroups; in this, we follow previous work on BL (p, q) by Sutov and Sullivan as well as Levi and Wood.

Let V be an infinite-dimensional locally convex complex space, X a closed subset of P(V) defined by finitely many continuous homogeneous equations and E a holomorphic vector bundle on X with finite rank. Here we show that E is holomorphically trivial if it is topologically trivial and spanned by its global sections and in a few other cases.

We study a generalization of the Svarc genus of a fibre map. For an arbitrary collection ɛ of spaces and a map f : X → Y, we define a numerical invariant, the ɛ-sectional category of f, in terms of open covers of Y. We obtain several basic properties of ɛ-sectional category, including those dealing with homotopy domination and homotopy pushouts. We then give three simple axioms which characterize the ɛ-sectional category. In the final section, we obtain inequalities for the ɛ-sectional category of a composition and inequalities relating the ɛ-sectional category to the Fadell–Husseini category of a map and the Clapp–Puppe category of a map.

This paper studies additive properties of the generalized Drazin inverse (g-Drazin inverse) in a Banach algebra and finds an explicit expression for the g-Drazin inverse of the sum a + b in terms of a and b and their g-Drazin inverses under fairly mild conditions on a and b.

We prove the uniqueness of the very singular solution to when 1 < p < (N + 2)/(N + 1), thus completing the previous result by Qi and Wang, restricted to self-similar solutions.

Let Ω ⊂ R2 denote a bounded Lipschitz domain and consider some portion Γ0 of ∂Ω representing the austenite–twinned-martensite interface which is not assumed to be a straight segment. We prove that for an elastic energy density ϖ: R2 → [0 ∞) such that ϖ(0, ±1) = 0. Here, W(Ω) consists of all functions u from the Sobolev class W1, ∞(Ω) such that |uy| = 1 almost everywhere on Ω together with u = 0 on Γ0. We will first show that, for Γ0 having a vertical tangent, one cannot always expect a finite surface energy, i.e. in the above problem, the condition in general cannot be included. This generalizes a result of [12] where Γ0is a vertical straight line. Property (*) is established by constructing some minimizing sequences vanishing on the whole boundary ∂Ω, that is, one can even take Γ0 = ∂Ω. We also show that the existence or non-existence of minimizers depends on the shape of the austenite–twinned-martensite interface Γ0.