Anisotropic mean curvature flow in higher codimension

Anisotropic mean curvature flow has been widely studied, the motivation coming from its wide range of applications and its intrinsic geometric appeal. Until now research concentrated on the study of motion of hypersurfaces, both from an analytical and a numerical point of view.

I am interested in motion by anisotropic mean curvature flow in higher codimension, a topic that has received much less attention. More precisely, I consider motion of parametrized curves and two-dimensional surfaces in R^n, for arbitrarily big n.

The anisotropic mean curvature flow is defined as the L^2-gradient flow for the anisotropic area functional. The latter is described by means of an anisotropy function, a map which, roughly speaking, assigns a positive weight to each tangent space and behaves like a norm. The anisotropy function expresses the idea that directions in spaces counts differently, i.e. the space is anisotropic.

For example if

parametrizes a closed curve Γ, then the anisotropic length functional is given by

where

denotes the anisotropy function. The isotropic case is recovered by taking Φ to be the usual Euclidean norm.

Below you can find some graphical examples. My discretization schemes use piecewise linear finite elements.

My research has been partially supported by the Deutsche Forschungsgemeinschaft via DFG-Forschergruppe "Nonlinear Partial Differential Equations: Theoretical and Numerical Analysis". The graphics were created with GRAPE.

References:

	P. Pozzi Anisotropic curve shortening flow in higher codimension Math. Methods Appl. Sci. 30 (2007), no. 11, 1243-1281
	P. Pozzi Anisotropic mean curvature flow for two dimensional surfaces in higher codimension: a numerical scheme Preprint Fakultät für Mathematik und Physik, Universität Freiburg, Nr. 12-07 (2007) To appear in Interfaces and Free Bound.

Here you see the evolution of a curve under an anisotropy whose unit ball is an ellipsoid strongly elungated along the x-Axis.

Here you see the evolution of an ellipsoid under a crystalline anisotropy.

Autor: Paola Pozzi : Letzte Änderung: 18.6.2008