In this article we study the behaviour of a heterogeneous thin film whose microstructure oscillates on a scale that is comparable to that of the thickness of the domain. The argument is based on a three-dimensional–two-dimensional reduction through a Γ-convergence analysis, techniques of two-scale convergence and a decoupling procedure between the oscillating variable and the in-plane variable.

The scalar equation with variable delay r(t) ≥ 0 is investigated, where t−r(t) is increasing and xg(x) > 0 (x ≠ 0) in a neighbourhood of x = 0. We find conditions for r, a and g so that for a given continuous initial function ψ a mapping P for (1) can be defined on a complete metric space Cψ and in which P has a unique fixed point. The end result is not only conditions for the existence and uniqueness of solutions of (1) but also for the stability of the zero solution. We also find conditions ensuring that the zero solution is asymptotically stable by changing to an exponentially weighted metric on a closed subset of Cψ. Finally, we parlay the methods for (1) into results for

Let (M, g) and (N, h) be compact Riemannian manifolds of dimensions m and n, respectively. For p-homogeneous convex functions f(s, t) on [0,∞) × [0, ∞), we study the validity and non-validity of the first-order optimal Sobolev inequality on H1, p(M × N) where and Kf = Kf (m, n, p) is the best constant of the homogeneous Sobolev inequality on D1, p (Rm+n), The proof of the non-validity relies on the knowledge of extremal functions associated with the Sobolev inequality above. In order to obtain such extremals we use mass transportation and convex analysis results. Since variational arguments do not work for general functions f, we investigate the validity in a uniform sense on f and argue with suitable approximations of f which are also essential in the non-validity. Homogeneous Sobolev inequalities on product manifolds are connected to elliptic problems involving a general class of operators.

In this paper, the second-order nonlinear elliptic system with α, γ < 1 and β ≥ 1, is considered in RN, N ≥ 3. Under suitable hypotheses on functions fi, gi, hi (i = 1, 2) and P, it is shown that this system possesses an entire positive solution , 0 < θ < 1, such that both u and v are bounded below and above by constant multiples of |x|2−N for all |x| ≥ 1.

We prove a semi-continuity theorem for an integral functional made up by a polyconvex energy and a surface term. Our result extends a well-known result by Ball to the BV framework.

We prove the compactness of the Whittaker sublocus of the moduli space of Riemann surfaces (complex algebraic curves). This is the subset of points representing hyperelliptic curves that satisfy Whittaker's conjecture on the uniformization of hyperelliptic curves via the monodromy of Fuchsian differential equations. In the last part of the paper we devote our attention to the statement made by R. A. Rankin more than 40 years ago, to the effect that the conjecture ‘has not been proved for any algebraic equation containing irremovable arbitrary constants’. We combine our compactness result with other facts about Teichmüller theory to show that, in the most natural interpretations of this statement we can think of, this result is, in fact, impossible.

Let D be a bounded domain in the complex plane whose boundary consists of m ≥ 2 pairwise disjoint simple closed curves and let A(bD) be the algebra of all continuous functions on bD which extend holomorphically through D. We show that a continuous function Φ on bD belongs to A(bD) if for each g ∈ A (bD) the harmonic extension of Re(gΦ) to D has a single-valued conjugate.

Under the assumption that the Radon measure μ on Rd satisfies only some growth condition, the authors prove that, for the maximal singular integral operator associated with a singular integral whose kernel only satisfies a standard size condition and the Hörmander condition, its boundedness in Lebesgue spaces Lp(μ) for any p ∈ (1, ∞) is equivalent to its boundedness from L1(μ) into weak L1(μ). As an application, the authors verify that if the truncated singular integral operators are bounded on L2(μ) uniformly, then the associated maximal singular integral operator is also bounded on Lp(μ) for any p ∈ (1, ∞).

For a class of nonlinear parabolic equations and inequalities, we establish conditions guaranteeing that any solution tends to 0 as t → ∞.

The full nonlinear dynamic von Kárm´n system depending on a small parameter ε > 0 is considered. We study the asymptotic behaviour of the total energy associated with the model for large t and ε → 0. Introducing appropriate boundary feedback, we show that the total energy of a solution of the corresponding damped model decays exponentially as t → +∞, uniformly with respect to the parameter ε > 0. As ε → 0, we obtain a damped plate model for which the energy also tends to zero exponentially. The limit system can be viewed as new variant of the so-called Timoshenko model. It consists of a second-order hyperbolic equation for transversal vibrations of the plate coupled with a first-order ordinary differential equation whose solution appears as coefficient of the plate model and takes into account (when ε → 0) the contribution of the tangential components.

In this paper we investigate the critical Fujita exponent for the initial-value problem of the degenerate and singular nonlinear parabolic equation with a non-negative initial value, where p > m ≥ 1 and 0 ≤ λ1 ≤ λ2 < p(λ1 + 1) − 1. We prove that, for m < p ≤ pc = m + (2 + λ2)/(n + λ1), every non-trivial solution blows up in finite time, while, for p > pc, there exist both global and non-global solutions to the pro

The Lyapunov functional is constructed for a quasilinear parabolic system which models chemotaxis and takes into account a volume-filling effect. For some typical case it is proved that the ω-limit set of any trajectory consists of regular stationary solutions. Some lower and upper bounds on the stationary solutions are found. For a given range of parameters there are stationary solutions which are inhomogeneous in space.