Calculus of Variations and Geometric Measure Theory

G. Bellettini - M. Paolini - F. Pasquarelli

Nonconvex mean curvature flow as a formal singular limit of the nonlinear bidomain model

created by belletti on 17 Feb 2012


Submitted Paper

Inserted: 17 feb 2012
Last Updated: 17 feb 2012

Year: 2012


In this paper we study the nonconvex anisotropic mean curvature flow of a hypersurface. This corresponds to an anisotropic mean curvature flow where the anisotropy has a nonconvex Frank diagram. The geometric evolution law is therefore forward-backward parabolic in character, hence ill-posed in general. We study a particular regularization of this geometric evolution, obtained with a nonlinear version of the so-called bidomain model. This is described by a degenerate system of two uniformly parabolic equations of reaction-diffusion type, scaled with a positive parameter $\eps$. We analyze some properties of the formal limit of solutions of this system as $\eps \to 0^+$, and show its connection with nonconvex mean curvature flow. Several numerical experiments substantiating the formal asymptotic analysis are presented.