We study the homogenization and localization of high-frequency waves in a locally periodic media with period ε. We consider initial data that are localized Bloch-wave packets, i.e. that are the product of a fast oscillating Bloch wave at a given frequency ξ and of a smooth envelope function whose support is concentrated at a point x with length scale . We assume that (ξ, x) is a stationary point in the phase space of the Hamiltonian λ(ξ, x), i.e. of the corresponding Bloch eigenvalue. Upon rescaling at size we prove that the solution of the wave equation is approximately the sum of two terms with opposite phases which are the product of the oscillating Bloch wave and of two limit envelope functions which are the solution of two Schrödinger type equations with quadratic potential. Furthermore, if the full Hessian of the Hamiltonian λ(ξ, x) is positive definite, then localization takes place in the sense that the spectrum of each homogenized Schrödinger equation is made of a countable sequence of finite multiplicity eigenvalues with exponentially decaying eigenfunctions.

We generalize the ‘sup+C inf’ inequality obtained by Shafrir to the solutions of

with Ω ⊂ ℝ2 open and bounded, α ∈ (0,1) and V any measurable function which satisfies 0 < a ≤ V ≤ b < + ∞.

We consider the spatially inhomogeneous and anisotropic reaction–diffusion equation ut = m(x)−1 div[m(x)ap(x,∇u)] + ε−2f(u), involving a small parameter ε > 0 and a bistable nonlinear term whose stable equilibria are 0 and 1. We use a Finsler metric related to the anisotropic diffusion term and work in relative geometry. We prove a weak comparison principle and perform an analysis of both the generation and the motion of interfaces. More precisely, we show that, within the time-scale of order ε2|ln ε|, the unique weak solution uε develops a steep transition layer that separates the regions {uε ≈ 0} and {uε | 1}. Then, on a much slower time-scale, the layer starts to propagate. Consequently, as ε → 0, the solution uε converges almost everywhere (a.e.) to 0 in Ω−t and 1 in Ω+t , where Ω−t and Ω+t are sub-domains of Ω separated by an interface Гt, whose motion is driven by its anisotropic mean curvature. We also prove that the thickness of the transition layer is of order ε.

We extend a result from Phillips by showing that one-homogeneous solutions of certain elliptic systems in divergence form either do not exist or must be affine. The result is novel in two ways. Firstly, the system is allowed to depend (in a sufficiently smooth way) on the spatial variable x. Secondly, Phillips's original result is shown to apply to W one-homogeneous solutions, from which his treatment of Lipschitz solutions follows as a special case. A singular one-homogeneous solution to an elliptic system violating the hypotheses of the main theorem is constructed using a variational method.

A version of the Darboux transformation is explored for Sturm-Liouville problems with eigenvalue-dependent boundary conditions, from differential-equation and operator-theoretic viewpoints. Some of the literature on Darboux's transformation is related in a historical introduction.

Given a complete m-primary ideal J in a local regular two-dimensional ring (R,m), we describe every adjacent complete ideal above J as the integral closure of some ideal (f, g) for suitable f, g associated to J. We also provide a geometrical procedure that gives its base points, thus determining its equisingularity class. We decompose the set IJ of these adjacent ideals in terms of the Rees valuations of J. As a consequence, we obtain a geometrical characterization of the finiteness of IJ.

We describe the numerical invariants and the curvature ellipse attached to the second fundamental form of a spacelike surface in a four-dimensional Minkowski space. We then study the configuration of the V-principal curvature lines on a spacelike surface when the normal field V is lightlike (the lightcone configuration). We end with some observations on the mean directionally curved lines and on the asymptotic lines on spacelike surfaces.

For a measure μ on ℝn (or on a doubling metric measure space) and a Young function Φ, we define two versions of Orlicz–Poincaré inequalities as generalizations of the usual p-Poincaré inequality. It is shown that, on ℝ, one of them is equivalent to the boundedness of the Hardy–Littlewood maximal operator from LΦ(ℝ,μ) to LΦ(ℝ,μ), while the other is equivalent to a generalization of the Muckenhoupt Ap-condition. While one direction in these equivalences is valid only on ℝ, the other holds in the general setting of doubling metric measure spaces. We also characterize both Orlicz–Poincaré inequalities on metric measure spaces by means of pointwise inequalities involving maximal functions of the gradient.

We look for standing waves for nonlinear Schrödinger equations

with cylindrically symmetric potentials g vanishing at infinity and non-increasing, and a C1 nonlinear term satisfying weak assumptions. In particular, we show the existence of standing waves with non-vanishing angular momentum with prescribed L2 norm. The solutions are obtained via a minimization argument, and the proof is given for an abstract functional which presents a lack of compactness. As a specific case, we prove the existence of standing waves with non-vanishing angular momentum for the nonlinear hydrogen atom equation.

We apply the theory of planar dynamical systems to carry out a qualitative analysis for the planar dynamical system corresponding to the fluidized-bed modelling equation. We obtain the global phase portraits of this system under various parameter conditions and the existence conditions of bounded travelling-wave solutions of this equation. According to the discussion on relationships between the behaviours of bounded travelling-wave solutions and the dissipation coefficients ε and δ, we find a critical value λ0 for arbitrary travelling-wave velocity υ. This equation has a unique damped oscillatory solution as ∥ε + δυ∥ < λ0 and ∥ε + δυ∥ ≠ 0, while it has a unique monotone kink profile solitary-wave solution as ∥ε + δυ∥ > λ0. By means of the undetermined coefficients method, we obtain the exact bell profile solitary-wave solution and monotone kink profile solitary-wave solution. Meanwhile, we obtain the approximate damped oscillatory solution. We point out the positions of these solutions in the global phase portraits. Finally, based on integral equations that reflect the relationships between the approximate damped oscillatory solutions and the implicit exact damped oscillatory solutions, error estimates for the approximate damped oscillatory solutions are presented.

Let Hn be a monotone sequence of non-negative self-adjoint operators or relations in a Hilbert space. Then there exists a self-adjoint relation H∞ such that Hn converges to H∞ in the strong resolvent sense. This result and related limit results are explored in detail and new simple proofs are presented. The corresponding statements for monotone sequences of semi-bounded closed forms are established as immediate consequences. Applications and examples, illustrating the general results, include sequences of multiplication operators, Sturm–Liouville operators with increasing potentials, forms associated with Kreĭn–Feller differential operators, singular perturbations of non-negative self-adjoint operators and the characterization of the Friedrichs and Kreĭn–von Neumann extensions of a non-negative operator or relation.

We study the weak stability index of an immersion ϕ: M → Sn+1 (1) ⊂ Rn+2 of an n-dimensional compact Riemannian manifold. We prove that the weak stability index of a compact hypersurface M with constant scalar curvature in Sn+1 (1), which is not totally umbilical, is greater than or equal to n + 2 if the mean curvature H1 and H3 are constant, and that the equality holds if and only if M is . As an application, we show that the weak stability index of an n-dimensional compact hypersurface with constant scalar curvature in Sn+1 (1), which is neither totally umbilical nor a Clifford hypersurface, is greater than or equal to 2n + 4 if the mean curvature H1 and H3 are constant.

We study the Bhatnagar–Gross–Krook (BGK) approximation to first-order scalar conservation laws with a flux which is discontinuous in the space variable. We show that the Cauchy problem for the BGK approximation is well posed and that, as the relaxation parameter tends to 0, it converges to the (entropy) solution of the limit problem.

We study the asymptotic behaviour of solutions to a quasilinear equation with high-contrast coefficients. The energy formulation of the problem leads to work with variable exponent Lebesgue spaces Lpε (·) in a domain Ω with a complex microstructure depending on a small parameter ε. Assuming only that the functions pε converge uniformly to a limit function p0 and that p0 satisfy certain logarithmic uniform continuity conditions, we rigorously derive the corresponding homogenized problem which is completely described in terms of local energy characteristics of the original domain. In the framework of our method we do not have to specify the geometrical structure Ω. We illustrate our result with periodical examples, extending, in particular, the classical extension results to variable exponent Sobolev spaces.

We study the thermodynamic consistency of phase-field models, which include gradient terms of the density ρ in the free-energy functional such as the van der Waals–Cahn–Hilliard model. It is well known that the entropy inequality admits gradient and higher-order gradient terms of ρ in the free energy only if either the energy flux or the entropy flux is represented by a non-classical form. We identify a non-classical entropy flux, which is not restricted to isothermal processes, so that gradient contributions are possible.

We then investigate equilibrium conditions for the van der Waals–Cahn–Hilliard phase-field model in the sharp interface limit. For a single substance thermodynamics provides two jump conditions at the sharp interface, namely the continuity of the Gibbs free energies of the adjacent phases and the discontinuity of the corresponding pressures, which is balanced by the mean curvature. We show that these conditions can be also extracted from the van der Waals–Cahn–Hilliard phase-field model in the sharp interface limit. To this end we prove an asymptotic expansion of the density up to the first order. The results are based on local energy estimates and uniform convergence results for the density.

Using variational methods we study the non-existence and multiplicity of non-negative solutions for a class of quasilinear elliptic equations of p(x)-Laplacian type with nonlinear boundary conditions of the form

where Ω; is a bounded domain with smooth boundary, n is the outer unit normal to ∂Ω and λ is a parameter. Furthermore, we want to emphasize that g : ∂Ω × [0,∞)→ ℝ is a continuous function that may or may not satisfy the Ambrosetti–Rabinowitz-type condition.

We consider the existence of positive solutions for the boundary-value problem

(q(t)ϕ(u′))′ + λf(t,u) = 0, r < t < R,

au(r) − bϕ−1 (q(r))u′(r) = 0, cu(R) + dϕ−1(q(R))u′(R) = 0,

where ϕ(u′) = |u′|p−2u′, p > 1, λ > 0, f is p-superlinear or p-sublinear at ∞ and is allowed to become −∞ at u = 0. Our results unify and extend many known results in the literature.

The Hermite-Hadamard inequality not only is a consequence of convexity but also characterizes it: if a continuous function satisfies either its left-hand side or its right-hand side on each compact subinterval of the domain, then it is necessarily convex. The aim of this paper is to prove analogous statements for the higher-order extensions of the Hermite-Hadamard inequality. The main tools of the proofs are smoothing by convolution and the support properties of higher-order monotone functions.

We study the existence, multiplicity and non-existence of positive solutions for the singular two-point boundary-value problems

where λ is a non-negative real parameter and f ∈ C((0, 1) × [0,∞), (0,∞)). Here, f(t, u) may be singular at t = 0 and/or 1. To obtain the main results we use the global continuation theorem and fixed-point index theory.

In a recent article Aldaz proved that the weak L1 bounds for the centred maximal operator associated to finite radial measures cannot be taken independently with respect to the dimension. We show that the same result holds for the Lp bounds of such measures with decreasing densities, at least for small p near to one. We also give some concrete examples, including the Gaussian measure, where better estimates with respect to the general case are obtained.