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

Paolini, Maurizio;Pasquarelli, Franco;Bellettini, Giovanni
2012

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.
Inglese
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]
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/10807/31199
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus ND
  • ???jsp.display-item.citation.isi??? 4
social impact