We define the family $\mathscr F$ of certain type of carpets, and calculate the fractal dimensions of fibres $F_x$ for all $F \in \mathscr F$ and for almost all $x$.

We answer a question posed by Ilmanen on the integrality of varifolds which appear as the singular perturbation limit of the Allen-Cahn equation. We show that the density of the limit measure is integer multiple of the surface constant almost everywhere at almost all time. This shows that limit measures obtained via the Allen- Chan equation and those via Brakke’s construction share the same integrality property as well as being weak solutions for the mean curvature flow equation.

We deal with reaction-diffusion equations of bistable type in an inhomogeneous medium. When the reaction term is balanced in the sense that a bulk potential energy attains the same global minimum at the two stable equilibria for each spatial point, we derive a free-boundary problem whose solutions determine equilibirum interfaces. We show that a non-degenerate solution of the free-boundary problem gives rise to an equilibrium internal layer solution of the reaction-diffusion equation, and moreover, the stability property of the latter is obtained from a linearization of the free boundary problem.

The aim of this paper is to give a simple proof of results by González- Koskela concerning the radial growth of $C\sp 2$ functions satisfying Bloch type condition. Our results also give generalizations of their results.