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Anisotropic curvature-driven flow of convex sets. (English) Zbl 1107.35069
The authors study the anisotropic mean curvature flow of convex sets $$E\subset {\mathbb R}^n$$. The sets move along the gradient flow of the surface energy functional $P_\phi (E) = \int_{\partial E} \phi^0(\nu^E)\,dH^{n-1},$ where $$\phi^0$$ is surface tension, $$\phi^0(\psi)=\sup_{\phi(\eta)\leq 1}\eta \psi$$, and $$\nu^E$$ is the normal vector. For smooth anisotropy, they show that in any dimension $$E(t)$$ remains convex up to extinction. If the anisotropy is crystalline, they “build a convex evolution which satisfies an equation which is a weak form of the crystalline curvature motion equation”.
As they describe in the introduction, they employ the variational approach developed in [F. Almgren, J. E. Taylor and L. Wang, SIAM J. Control Optim. 31, 387–438 (1993; Zbl 0783.35002)]. Discretizing in time, the authors there construct a surface at time $$t + h$$ from that at time $$t$$ via a solution $$T_h E$$ of
$\min_F P_\phi(F) + \frac{1}{h}\int_{F\Delta E} d(x,\partial E)\,dx,$ where $$d(x,\partial E)$$ is the distance from $$x$$ to $$\partial E$$, and $$F\Delta E$$ is the symmetric difference of $$F$$ and $$E$$. They study the convergence to a limit flow.
One of the authors of the paper under review found a way to construct $$T_h E$$ by defining it as a level set $$\{x:u(x)<0\},$$ where $$u$$ minimizes $\int_\Omega \phi^0(Du)+\frac{1}{2h} \int_\Omega (u(x)-d_E(x))^2 \,dx.\tag{1}$ Here $$\Omega$$ is an open subset of $${\mathbb R}^n$$, and $$d_E$$ is signed distance to $$\partial E$$ [A. Chambolle, Interfaces Free Bound. 6, 195–218 (2004; Zbl 1061.35147)].
In the current paper the authors proceed similarly, but let $$u$$ satisfy the Euler equation of (1) over $${\mathbb R}^n$$. They prove that when $$E$$ is convex, then $$u$$ and hence $$T_h E$$ is convex. This implies in the smooth case that the convexity of $$E(t)$$ is preserved in any dimension.

##### MSC:
 35K65 Degenerate parabolic equations 58J35 Heat and other parabolic equation methods for PDEs on manifolds 53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
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##### References:
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