Let be the one-sided maximal operator and let Ф be a convex non-decreasing function on (0, ∞), Ф(0) = 0. We present necessary and sufficient conditions on a couple of weight functions (σ, ϱ) such that the integral inqualities of weak type

and of extra-weak type

hold. Our proofs do not refer to the theory of Orlicz spaces.

We consider the following equation:

where d(x) = d(x, ∂Ω), θ > –2 and Ω is a half-space. The existence and non-existence of several kinds of positive solutions to this equation when , f(u) = up (p > 1) and Ω is a bounded smooth domain were studied by Bandle, Moroz and Reichel in 2008. Here, we study exact the behaviour of positive solutions to this equation as d(x) → 0+ and d(x) → ∞, respectively, and the symmetry of positive solutions when , Ω is a half-space and f(u) is a more general nonlinearity term than up . Under suitable conditions for f, we show that the equation has a unique positive solution W, which is a function of x 1 only, and W satisfies

We prove that the two-dimensional Brown–Ravenhall operator is bounded from below when the coupling constant is below a specified critical value—a property also referred to as stability. As a consequence, the operator is then self-adjoint. The proof is based on the strategy followed by Evans et al. and Lieb and Yau, with some relevant changes characteristic of the dimension. Our analysis also yields a sharp Kato inequality.

Several additional members of the series of normal saturated dicarboxylic acids have been investigated by X-rays, and the results are in agreement with the conclusions arrived at in an earlier paper. In the series of mono- and di-alkyl malonic acids, successive reflection planes are separated by the length of one molecule. The series of fatty acid amides closely resembles that of the fatty acids themselves. The cleavage planes are separated by the length of two molecules oriented in opposite directions.

My thanks are due to Mr N. K. Adam for the specimens of mono- and dioctadecyl malonic acid, and to Sir James Walker for his advice and helpful criticism.

I propose making a few brief observations on the relations in which we stand to the aborigines of the Andaman Islands, their place in the economy of this convict settlement, and their more noticeable habits and customs. In the consideration of the latter, I shall only state such information as I have obtained myself from conversation with, and personal observation on the manners of the natives. Detailed monographs on their ethnology, &c., have been written already by several observers, and are to be found in the “Transactions of the Anthropological Society,” &c.

We consider elliptic systems with discontinuous coefficients and prove that if the known term belongs to the Morrey space Lp,λ, then the highest-order derivatives of the local solution belong to the same space. We also obtain local Hölder continuity for lower-order derivatives.

We continue the research on reiteration results between interpolation methods associated to polygons and the real method. Applications are given to N-tuples of function spaces, of spaces of bounded linear operators and Banach algebras.

Abstract versions of the Cauchy problem for the Euler-Poisson-Darboux equation and the Dirichlet problem for the equation of generalized axially symmetric potential theory are related by an integral transformation. In certain special cases, this leads to abstract versions of (1) the Poisson formula for the solution of G.A.S.P.T. in a half-space, (2) pseudo-analytic functions in a half-space, and (3) a generalized Hilbert transform related to the work of Heywood, Kober, and Okikiolu. Some properties of this generalized Hilbert transform are studied including an inversion theorem.

We consider reaction–diffusion equations under nonlinear boundary conditions where the nonlinearities are asymptotically linear at infinity and depend on a parameter. We prove that, as the parameter crosses some critical values, a resonance-type phenomenon provides solutions that bifurcate from infinity. We characterize the bifurcated branches when they are sub- or supercritical. We obtain both Landesman–Lazer-type conditions that guarantee the existence of solutions in the resonant case and an anti-maximum principle.

The stationary Gross–Pitaevskii equation in one dimension is considered with a complex periodic potential satisfying the conditions of the 𝒫𝒯 (parity-time reversal) symmetry. Under rather general assumptions on the potentials, we prove bifurcations of 𝒫𝒯-symmetric nonlinear bound states from the end points of a real interval in the spectrum of the non-selfadjoint linear Schrödinger operator with a complex 𝒫𝒯-symmetric periodic potential. The nonlinear bound states are approximated by the effective amplitude equation, which bears the form of the cubic nonlinear Schrödinger equation. In addition, we provide sufficient conditions for the appearance of complex spectral bands when the complex 𝒫𝒯-symmetric potential has an asymptotically small imaginary part.

We establish that if Ω is an open square and if P is an arbitrary point of Ω, then the solutions of the wave equation with Dirichlet boundary condition utt − Δu = 0 in R × Ω, u = 0 on R × Γ can remain strictly positive during an arbitrarily large time. In fact stronger results are proved, considering several points of Ω simultaneously and prescribing different, more general, behaviour of the solution at these points.

We consider two boundary value problems (of Neumann or related type) associated with the equation in Ω. The existence of a solution was previously established assuming that p < N/(N −s2). (N dimension of Ω.) We prove that this exponent is critical for these problems, at least in the radially symmetric case when Ω is a ball. This is understood in the sense that the existence result does not hold when p ≧ N/(N − 2).

The interior initial-boundary value problem for the wave equation in m ≧ 1 space dimension is considered for vanishing boundary values. Certain regularity, dependent on m, is required for the solution and additionalboundary conditions, the number of which being also dependent on m, are imposed on the given right hand side. Emphazising the case m = 3, the Rothe method is applied after the problem has been rewritten as a hyperbolic first order evolution problem for m + 1 unknown functions. The sequence of discrete solutions obtained is shown to be discretely convergent to the continuous solution in the sense of uniform convergence if the solution of the continuous problem is assumed to exist. A priori estimates are derived both for the discrete solutions and the continuous solution.

The interpretation of optical interference experiments at low light intensities and with independent light sources is discussed. A new experiment on interference between modulated light beams is described briefly, and its significance is indicated. There are two appendices, one a fuller discussion of the conditions under which interference and intensity correlation may be produced, the other a comment on one aspect of Jánossy and Nagy's discussion (1956) of the feasibility of the alternating slit interference experiment.

Assume that the local compliance tensor of an elastic composite in two space dimensions is equal to a rank-one tensor plus a null-Lagrangian (there is only one symmetric one in two dimensions). The purpose of this paper is to prove that the effective compliance tensor has the same representation: rank-one plus the null-Lagrangian. This statement generalises the wellknown result of Hill that a composite of isotropic phases with a common shear modulus is necessarily elastically isotropic and shares the same shear modulus. It also generalises the surprising discovery of Avellaneda et al. that under a certain condition on the pure crystal moduli the shear modulus of an isotropic polycrystal is uniquely determined. The present paper sheds light on this effect by placing it in a more general framework and using some elliptic PDE theory rather than the translation method. Our results allow us to calculate the polycrystalline G-closure of the special class of crystals under consideration. Our analysis is contrasted with a two-dimensional model problem for shape-memory polycrystals. We show that the two problems can be thought of as ‘elastic percolation’ problems, one elliptic, one hyperbolic.

This paper is concerned with the following nonlinear Schrödinger system in ${\mathbb R}^3$

$$\left\{ {\beging{matrix}{ {-\Delta u + (1 + \alpha P(x))u = \mu u^3 + \beta uv^2,} \hfill & {x\in {\open R}^3,} \hfill \cr {-\Delta v + (1 + \alpha Q(x))u = \nu v^3 + \beta u^2v,} \hfill & {x\in {\open R}^3,} \hfill \cr {u,v > 0,} \hfill & {x\in {\open R}^3,} \hfill \cr } } \right.$$

$\beta \in {\mathbb R}$

$\mu ,\nu $

We prove that, for any positive integer k > 1, there exists a suitable range of α such that the above problem has a non-radial positive solution with exactly k maximum points which tend to infinity as $\alpha \to +\infty $ (or $0^+$). Moreover, we also construct prescribed number of sign-changing solutions.

Conditions are given which ensure that a weighted 2nth order ordinary differential equation on a half-line satisfy the strong limit-point and Dirichlet conditions. Perturbation terms are permitted which either satisfy certain pointwise bounds or integral type bounds. Not all of the coefficients of the equation are required to be non-negative.

A Rees matrix semigroup over an inverse semigroup contains a greatest regular subsemigroup. The regular semigroups obtained in this manner are abstractly characterized here. The greatest completely simple homomorphic image and the idempotent generated part of such semigroups are investigated. Rectangular bands of semilattices of groups and some special cases are characterized in several ways.

In this paper, we consider the equation

where A is a linear operator, N = ψ′ with ψ ∈ C1(E, R), and E is an Hilbert space. We suppose that N has a derivative at infinity N′(∞) and that 0 belongs to the spectrum of A–N′(∞). We prove an abstract theorem for multiplicity of solutions for the above equation. We then apply this theorem to the study of periodic solutions of Hamiltonian systems and of semilinear wave equations when the period is prescribed.

This paper deals with the following non-linear equation with a fractional Laplacian operator and almost critical exponents:

\[ (-\Delta)^{s} u=K(|y'|,y'')u^{({N+2s})/(N-2s)\pm\epsilon},\quad u > 0,\quad u\in D^{1,s}(\mathbb{R}^{N}), \]

$\epsilon ^{-(({N-2s-2})/({(N-2s)^2})}$