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 $\epsilon$. We analyze some properties of the formal limit of solutions of this system as $\epsilon \to 0$, and show its connection with nonconvex mean curvature flow. Several numerical experiments substantiating the formal asymptotic analysis are presented.

Paolini, M., Pasquarelli, F., Bellettini, G., Nonconvex mean curvature flow as a formal singular limit of the nonlinear bidomain model, <<Quaderni del Seminario Matematico di Brescia>>, 2012; 2012 (10): 1-35 [http://hdl.handle.net/10807/31199]

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

#### Abstract

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 $\epsilon$. We analyze some properties of the formal limit of solutions of this system as $\epsilon \to 0$, and show its connection with nonconvex mean curvature flow. Several numerical experiments substantiating the formal asymptotic analysis are presented.
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Paolini, M., Pasquarelli, F., Bellettini, G., Nonconvex mean curvature flow as a formal singular limit of the nonlinear bidomain model, <<Quaderni del Seminario Matematico di Brescia>>, 2012; 2012 (10): 1-35 [http://hdl.handle.net/10807/31199]
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/10807/31199
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