Singular perturbation problems involving curvature

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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.
Original languageEnglish
Title of host publicationDifferential Geometry and Continuum Mechanics
EditorsGui-Qiang Chen, Michael Grinfeld, R. J. Knops
ISBN (Print)978-3-319-18572-9
Publication statusPublished - 2015

Publication series

NameSpringer Proceedings in Mathematics & Statistics


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