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

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.
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2013
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: `https://hdl.handle.net/10807/45109`
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