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

Paolini, M., Bellettini, G., Pasquarelli, F., Nonconvex mean curvature flow as a formal singular limit of the nonlinear bidomain model, <<ADVANCES IN DIFFERENTIAL EQUATIONS>>, 2013; 18 (Settembre): 895-934 [http://hdl.handle.net/10807/45109]

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

Paolini, Maurizio;Bellettini, Giovanni;Pasquarelli, Franco
2013

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 ϵ. We analyze some properties of the formal limit of solutions of this system as ϵ→0+, and show its connection with nonconvex mean curvature flow. Several numerical experiments substantiating the formal asymptotic analysis are presented.
Inglese
Paolini, M., Bellettini, G., Pasquarelli, F., Nonconvex mean curvature flow as a formal singular limit of the nonlinear bidomain model, <<ADVANCES IN DIFFERENTIAL EQUATIONS>>, 2013; 18 (Settembre): 895-934 [http://hdl.handle.net/10807/45109]
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/10807/45109
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