We show that in an infinite straight pipe of arbitrary (sufficiently smooth) cross-section, a generalized non-Newtonian liquid admits one and only one fully developed time-periodic flow (Womersley flow) when either the flow rate (problem 1) or the axial pressure gradient (problem 2) is prescribed in analogous time-periodic fashion. In addition, we show that the relevant solution depends continuously upon the data in appropriate norms. As is well known from the Newtonian counterpart of the problem, the latter is pivotal for the analysis of flow in a general unbounded pipe system with cylindrical outlets (Leray's problem). It is also worth remarking that problem 1 possesses an intrinsic interest from both mathematical and physical viewpoints, in that it constitutes a (nonlinear) inverse problem with a significant bearing on several applications, including blood flow modelling in large arteries.

Using a regular Borel measure μ ⩾ 0 we derive a proper subspace of the commonly used Sobolev space D1(ℝN) when N ⩾ 3. The space resembles the standard Sobolev space H1(Ω) when Ω is a bounded region with a compact Lipschitz boundary ∂Ω. An equivalence characterization and an example are provided that guarantee that is compactly embedded into L1(RN). In addition, as an application we prove an existence result of positive solutions to an elliptic equation in ℝN that involves the Laplace operator with the critical Sobolev nonlinearity, or with a general nonlinear term that has a subcritical and superlinear growth. We also briefly discuss the compact embedding of to Lp(ℝN) when N ⩾ 2 and 2 ⩽ p ⩽ N.

We investigate two kinds of Fricke families, those consisting of Fricke functions and those consisting of Siegel functions. In terms of their special values we then generate ray class fields of imaginary quadratic fields over the Hilbert class fields, which are related to the Lang–Schertz conjecture.

We reconcile two classical models of edge dislocations in solids. The first, from the early 1900s, models isolated edge dislocations as line singularities in locally Euclidean manifolds. The second, from the 1950s, models continuously distributed edge dislocations as smooth manifolds endowed with non-symmetric affine connections (equivalently, endowed with torsion fields). In both models, the solid is modelled as a Weitzenböck manifold. We prove, using a weak notion of convergence, that the second model can be obtained rigorously as a homogenization limit of the first model as the density of singular edge dislocation tends to infinity.

By giving some new treatments we can improve a classical result of Walfisz (1963) on the asymptotic formula of Euler's function.

We study classical solutions of the Cauchy problem for a class of non-Lipschitz semilinear parabolic partial differential equations in one spatial dimension with sufficiently smooth initial data. When the nonlinearity is Lipschitz continuous, results concerning existence, uniqueness and continuous dependence on initial data are well established (see, for example, the texts of Friedman and Smoller and, in the context of the present paper, see also Meyer), as are the associated results concerning Hadamard well-posedness. We consider the situations when the nonlinearity is Hölder continuous and when the nonlinearity is upper Lipschitz continuous. Finally, we consider the situation when the nonlinearity is both Hölder continuous and upper Lipschitz continuous. In each case we focus upon the question of existence, uniqueness and continuous dependence on initial data, and thus upon aspects of Hadamard well-posedness.

The aim of this paper is to provide a comprehensive study of some linear non-local diffusion problems in metric measure spaces. These include, for example, open subsets in ℝN, graphs, manifolds, multi-structures and some fractal sets. For this, we study regularity, compactness, positivity and the spectrum of the stationary non-local operator. We then study the solutions of linear evolution non-local diffusion problems, with emphasis on similarities and differences with the standard heat equation in smooth domains. In particular, we prove weak and strong maximum principles and describe the asymptotic behaviour using spectral methods.

We characterize, in the context of rearrangement invariant spaces, the optimal range space for a class of monotone operators related to the Hardy operator. The connection between the optimal range and the optimal domain for these operators is carefully analysed.