We study the asymptotic behaviour, for a sequence of varying open sets Ωn, of the solutions un of nonlinear Dirichlet problems for a monotone Leray–Lions operator. The method is based on the comparison between the gradient of un and the corrector for the p-Laplacian corresponding to the same geometry as the monotone operator. The representation of the limit problem and a corrector result are obtained.

Techniques from the theory of partial differential equations are employed to prove the uniform convergence of the eigenfunction expansion associated with a two-parameter system of ordinary differential equations of the second order.

In this paper, we study the decomposition of the Nehari manifold by exploiting the combination of concave and convex nonlinearities. The result is subsequently used, in conjunction with the Ljusternik–Schnirelmann category and variational methods, to prove the existence and multiplicity of positive solutions for an indefinite elliptic problem involving a critical Sobolev exponent.

We study the semiclassical limit of the least energy solutions to the nonlinear Dirac equationfor x ∈ ℝ3. We prove that the equation has least energy solutions for all ħ > 0 small, and, in addition, that the solutions converge in a certain sense to the least energy solution of the associated limit problem as ħ → 0.

In this paper we give a sufficient condition which guarantees that the zero solution of a population growth equation is uniformly stable.

We characterize those holomorphic symbols ϕ: D → D for which the induced composition operator Cϕ: Bω → Bμ (respectively, Bω,0 → Bμ,0) is bounded or compact, where D is the unit disc in the complex plane C, ω is a normal function on [0, 1) and μ is a non-negative function on [0, 1) with μ(tn) > 0 for some sequence satisfying limn→∞tn= 1.

We consider the ergodic (or additive eigenvalue) problem for the Neumann-type boundary-value problem for Hamilton–Jacobi equations and the corresponding discounted problems. Denoting by uλ the solution of the discounted problem with discount factor λ > 0, we establish the convergence of the whole family to a solution of the ergodic problem as λ → 0, and give a representation formula for the limit function via the Mather measures and Peierls function. As an interesting by-product, we introduce Mather measures associated with Hamilton–Jacobi equations with the Neumann-type boundary conditions. These results are variants of the main results in a recent paper by Davini et al., who study the same convergence problem on smooth compact manifolds without boundary.

1. This paper is the continuation of a former one (Erdélyi, 1939), and deals with the integration of the system of two partial linear differential equations of the second order

The former paper will be referred to as I; all the notations of I will be retained without further explanation.

6. The expected score of an individual on a test consisting of a large number of items is assumed to be given by a formula involving the ability of the individual and also two quantities constant for the test. An expression is then derived for the covariance between two tests measuring different abilities. It appears that if a factorial analysis is performed on a set of tests of unequal difficulty, using the matrix of variances and covariances, a spurious factor will tend to be introduced depending mainly on the differences in difficulty. The effect of this is removed by transforming the variances and covariances to a new set of coefficients. A numerical example of the process is given.

In conclusion I should like to thank the Carnegie Trust for the Universities of Scotland for a grant to cover the cost of the setting and printing of mathematical formulæ in a paper previously published in the Society's Proceedings (LXI, A, 1943, 273–287).

Let In be the symmetric inverse semigroup on Xn = {1,…, n}, let Sln be the subsemigroup of strictly partial one-to-one self-maps of Xn and let = { α ∊ SIn: x} ≦ x = U = ∅= be the semigroup of all partial one-to-one decreasing maps including the empty or zero map of Xn. In this paper it is shown that is an (irregular, for n ≧ 2) type A semigroup with n D*-classes and D* = I*. Further, it is shown that is generated by the n(n + l)/2 quasi-idempotents in

We study a Stefan problem for coarsening of many small particles of solid phase in undercooled liquid phase which includes surface tension and kinetic undercooling effects. A mean-field model is derived by homogenization of the free boundary problem. It extends the classical theory of Lifshitz et al. by taking into account effects due to kinetic undercooling.

This paper considers autonomous parabolic equations which have a homoclinic orbit to an isolated equilibrium point. We study these systems under autonomous perturbations. Firstly we prove that the perturbation under which the homoclinicorbit persists forms a submanifold of codimension one. Then, if a perturbation of this manifold is considered, we prove that a unique stable periodic orbit arises from the homoclinic orbit under certain conditions for the eigenvalues of thesaddle point.

In this paper we analyse the methodology of the theory of differential inclusions. First, we emphasize that any sequence of piecewise affine functions with successive elements obtained by perturbations of preceding functions in the sets of their affinity converges strongly, together with the gradients. This gives a simple algorithm with which to construct sequences of approximate solutions that converge to exact solutions (neither the specific choice suggested by ‘the method of convex integration for Lipschitz functions’ nor Baire category methodology is required). We then suggest a functional that is defined in the set of admissible functions and measures maximal oscillations produced by sequences of admissible functions weakly convergent to a given function. This functional can be used to prove that the set of stable solutions is dense in the weak topology in the closure of the set of admissible functions either via the Baire category lemma or via a specific choice of strictly convergent sequences.

We explain how the above-mentioned methods of finding solutions to differential inclusions are connected to earlier results on weak–strong convergence, i.e. to results on stability, in the calculus of variations and in differential inclusions.

We also include information on developments in the subject in the three years after the results of this work were obtained.

V. B. Stanojevic suggested in her recent paper that it would be of interest to prove a corresponding L1-convergence theorem for Fourier series with complex O-regularly varying quasimonotonc coefficients. The present paper will discuss this question and establish L1-convergence and. furthermore. L1-approximation theorems for complex-valued integrable functions.

We obtain an integral representation of an energy for structured deformations of continua in the space of functions of bounded variation, as a first step to the study of asymptotic models for thin defective crystalline structures, where phenomena as slips, vacancies and dislocations prevent the effectiveness of classical theories.

In this paper, we prove that for a system of ordinary differential equations of class Cr+1,1, r≧0 and two arbitrary Cr+1, 1 local centre manifolds of a given equilibrium point, the equations when restricted to the centre manifolds are Cr conjugate. The same result is proved for similinear parabolic equations. The method is based on the geometric theory of invariant foliations for centre-stable and centre-unstable manifolds.

It is proved that elliptic boundary-value problems have a global smoothing property in Lebesgue spaces, provided the underlying space of weak solutions admits a Sobolev-type inequality. The results apply to all standard boundary conditions, and a wide range of non-smooth domains, even if the classical estimates fail. The dependence on the data is explicit. In particular, this provides good control over the domain dependence, which is important for applications involving varying domains.

We give a complete solution to the G-closure problem for mixtures of two well-ordered possibly anisotropic conductors. Both the G-closure with fixed volume fractions and the full G-closure are computed. The conductivity tensors are considered in a fixed frame and no rotations are allowed.

We prove that viscosity solutions to fully nonlinear elliptic equations with degeneracy of double phase type are locally C1, γ-regular.

Connections between (positive) mean ergodic operators acting in Banach lattices and properties of the underlying lattice itself are well understood (see the works of Emel'yanov, Wolff and Zaharopol). For Fréchet lattices (or more general locally convex solid Riesz spaces) there is virtually no information available. For a Fréchet lattice E, it is shown here that (amongst other things) every power-bounded linear operator on E is mean ergodic if and only if E is reflexive if and only if E is Dedekind σ-complete and every positive power-bounded operator on E is mean ergodic if and only if every positive power-bounded operator in the strong dual E′β (no longer a Fréchet lattice) is mean ergodic. An important technique is to develop criteria that detect when E admits a (positively) complemented lattice copy of c0, l1 or l∞.