Consider an anisotropic area functional, giving rise to a variational principle for the shape of crystal surfaces. Sometimes such a functional is regularised with an additional curvature term to avoid difficulties coming from a lack of convexity. We study the asymptotic behaviour of the resulting functional as the strength of the regularisation tends to 0. We consider two cases. The first corresponds to a cubic crystal structure. The expected shapes of the crystal surfaces are polyhedra with faces parallel to the coordinate planes, and for the regularised functionals, we discover a limiting energy depending on the lengths of the edges. In the second case, we have a uniaxial anisotropy. We calculate the limiting energy for surfaces of revolution and give a lower bound for topological spheres.
|Title of host publication||Differential Geometry and Continuum Mechanics|
|Editors||Gui-Qiang Chen, Michael Grinfeld, R. J. Knops|
|Publication status||Published - 2015|
|Name||Springer Proceedings in Mathematics & Statistics|
Moser, R. (2015). Singular perturbation problems involving curvature. In G-Q. Chen, M. Grinfeld, & R. J. Knops (Eds.), Differential Geometry and Continuum Mechanics (pp. 49-75). (Springer Proceedings in Mathematics & Statistics; Vol. 137). Springer.