A class of three-dimensional crack problems is considered, of which a prototype example is provided by a half-space containing a long internal crack, located in a plane perpendicular to the boundary. By means of an asymptotic procedure, the original three-dimensional problem is split up into a sequence of two-dimensional formulations. Results of its numerical implementation are in good agreement with results of more computer-intensive finite-element calculations.

In Part I of this paper (Terrill & Byatt-Smith, 1993) the problem of the flow between an obstacle in the form of a wedge and a porous flexible tow was modelled using lubrication theory. In the region close to the wedge where bending is important the following equation

was derived for the non-dimensional gap width, H(X), between the tow and the wedge.

We study radially symmetric stationary points of the functional

where u denotes the density of a fluid confined to a container Ω, W(u) is the course-grain free energy and ε accounts for surface energy. Under the further assumption of small energy, that is

for small ε, we prove existence of precisely two solutions for the corresponding Euler-Lagrange equation. Each of these solutions is monotone in the radial direction and converges as ε→0 to one of two possible radially symmetric single interface minimizers of E0. Our main tool is the method of matched asymptotic expansions from which we construct exact solutions.

In the Calculus of Variations, several notions of convexity have emerged, corresponding to different properties of the functionals to be minimized. The relations between these various notions are not yet fully understood. In this context, we present a numerical study of quasiconvexity for some functions of the type f(ξ) = g(|ξ|2, det ξ), where Ξ is a 2×2-matrix. The corresponding global optimization problems are solved using a simulated annealing-like algorithm. The computations strongly indicate that the considered functions are quasiconvex if and only if they are rank-one convex. The relation to Morrey's conjecture, various applications and implementation problems are discussed.

We consider the question of the existence/non-existence of solutions for the non-local nonlinear elliptic system which models a thermistor driven by a current source. Specifically, we show that for small input current there exists a solution, while this will not in general be the case for a sufficiently large current. A feature of our estimates is that the conditions for non-existence are determined by local criteria on the domain and the coefficients. Our basic tools for existence involve truncation, L2.μ estimates and fixed point arguments. Non-existence is obtained by averaging procedures and an application of Barta's Inequality.