1. Introduction

Motion by mean curvature of an embedded smooth hypersurface without boundary has been the subject of several recent papers, because of its geometric interest and of its application to different areas, see for instance the pioneering book [Brakke 1978], or the papers [Allen and Cahn 1979], [Huisken 1984], [Osher and Sethian 1988], [Evans and Spruck 1991], [Almgren et al. 1993]. A smooth boundary ∂E of an open set E = E(0) ⊂ Rn flows by mean curvature if there exists a time-dependent family (∂E(t))t∈[0,T ] of smooth boundaries satisfying the following property: the normal velocity of any point x ∈ ∂E(t) is equal to the sum of the principal curvatures of ∂E(t) at x. One can show that, at each time t, such an evolution process reduces the area of ∂E(t) as fast as possible. Mean curvature flow has therefore a variational character, since it can be interpreted as the gradient flow associated with the area functional ∂E →Hn-1 (∂E), where Hn-1 indicates the (n−1)-dimensional Hausdorff measure in Rn.

In several physical processes (for instance in certain models of dendritic growth and crystal growth, see [Cahn et al. 1992], or in statistical physics (see for example [Spohn 1993]) it turns out, however, that the evolution of the surface is not simply by mean curvature, but is an anisotropic evolution. From the energy point of view, this means that the functional of which we are taking the gradient flow is not the area of ∂E anymore, but is a weighted area, which can be derived by looking at ℝn as a normed space.