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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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