We consider a family of multi-phase Stefan problems for a certain one-dimensional model of cell-to-cell adhesion and diffusion, which takes the form of a non-linear forward–backward parabolic equation. In each material phase the cell density stays either high or low, and phases are connected by jumps across an ‘unstable’ interval. We develop an existence theory for such problems which allows for the annihilation of phases and the subsequent continuation of solutions. Stability results for the long-time behaviour of solutions are also obtained, and, where necessary, the analysis is complemented by numerical simulations.

Proceeding from a recent construction (Fraenkel 2007, Arch. Rational Mech. Anal. 183, 187–214), this paper contains three contributions to the theory of the extreme gravity wave on water (or “wave of greatest height”) initiated by Stokes in 1880. The first is a solution, of the extreme form of the integral equation due to Nekrasov, that consists of an explicit function plus a (rigorously) estimated remainder; the remainder vanishes at the crest and at the trough of the wave and in between has a maximum value less than 0.0026 times the maximum value of the explicit part. The second contribution establishes local uniqueness for a variant of the Nekrasov equation; we prove that this equation has only one solution between upper and lower bounding functions that are moderately far apart. The third contribution consists of a table and graph of strict upper and lower bounds for the interface between air and water in the physical plane.

In the preceding paper by Fraenkel & Harwin, the existence theorem leaves a gap in the description of the extreme wave that is constructed there. The form of the wave in a neighbourhood of the crest is not described accurately enough. This weakness is repaired in the present paper.

We study equilibrium liquid crystal configurations in three-dimensional geometries, within the continuum Landau-de Gennes theory. We obtain explicit bounds for the equilibrium scalar order parameters in terms of the temperature and material-dependent constants. We explicitly quantify the temperature regimes where the Landau-de Gennes predictions match and the temperature regimes where the Landau-de Gennes predictions do not match the probabilistic second-moment definition of the Q-tensor order parameter. The regime of agreement may be interpreted as the regime of validity of the Landau-de Gennes theory since the Landau-de Gennes theory predicts large values of the equilibrium scalar order parameters – larger than unity, in the low-temperature regime. We discuss a modified Landau-de Gennes energy functional which yields physically realistic values of the equilibrium scalar order parameters in all temperature regimes.