Advances in Differential Equations

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

Giovanni Bellettini, Maurizio Paolini, and Franco Pasquarelli

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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.

Article information

Source
Adv. Differential Equations Volume 18, Number 9/10 (2013), 895-934.

Dates
First available in Project Euclid: 2 July 2013

Permanent link to this document
https://projecteuclid.org/euclid.ade/1372777763

Mathematical Reviews number (MathSciNet)
MR3100055

Zentralblatt MATH identifier
1272.53054

Subjects
Primary: 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.) 35K40: Second-order parabolic systems 47J06: Nonlinear ill-posed problems [See also 35R25, 47A52, 65F22, 65J20, 65L08, 65M30, 65R30]

Citation

Bellettini, Giovanni; Paolini, Maurizio; Pasquarelli, Franco. Nonconvex mean curvature flow as a formal singular limit of the nonlinear bidomain model. Adv. Differential Equations 18 (2013), no. 9/10, 895--934. https://projecteuclid.org/euclid.ade/1372777763.


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