We study a mean-field model of superconducting vortices in one and two dimensions. The existence of a weak solution and a steady-state solution of the model are proved. A special case of the steady-state problem is shown to be of the form of a free boundary problem. The solutions of this free boundary problem are investigated. It is also shown that the weak solution of the one-dimensional model is unique and satisfies an entropy inequality.

We study the interior spike solutions to a steady state problem of the shadow system of the Gierer–Meinhardt system arising from biological pattern formation. We first show that at a non-degenerate peak point the interior spike solution is locally unique, and then we establish the spectrum estimates of the associated linearized operator. We also prove that the corresponding solution to the shadow system is unstable. Furthermore, the metastability of such solutions is analysed.

As a paradigm for non-interpenetrating crack models, the Poisson equation in a nonsmooth domain in R2 is considered. The geometrical domain has a cut (a crack) of variable length. At the crack faces, inequality type boundary conditions are prescribed. The behaviour of the energy functional is analysed with respect to the crack length changes. In particular, the derivative of the energy functional with respect to the crack length is obtained. The associated Griffith formula is derived, and properties of the solution are investigated. It is shown that the Rice–Cherepanov integral defined for the solutions of the unilateral problem defined in the nonsmooth domain is path-independent. Finally, a non-negative measure characterising interaction forces between the crack faces is constructed.

The thermal equilibrium state of a bipolar, isothermal quantum fluid confined to a bounded domain Ω⊂ℝd, d = 1, 2 or d = 3 is the minimizer of the total energy [Escr]ελ; [Escr]ελ involves the squares of the scaled Planck's constant ε and the scaled minimal Debye length λ. In applications one frequently has λ2[Lt]1. In these cases the zero-space-charge approximation is rigorously justified. As λ → 0, the particle densities converge to the minimizer of a limiting quantum zero-space-charge functional exactly in those cases where the doping profile satisfies some compatibility conditions. Under natural additional assumptions on the internal energies one gets an differential-algebraic system for the limiting (λ = 0) particle densities, namely the quantum zero-space-charge model. The analysis of the subsequent limit ε → 0 exhibits the importance of quantum gaps. The semiclassical zero-space-charge model is, for small ε, a reasonable approximation of the quantum model if and only if the quantum gap vanishes. The simultaneous limit ε = λ → 0 is analyzed.