an:05047621
Zbl 1107.35069
Caselles, Vicent; Chambolle, Antonin
Anisotropic curvature-driven flow of convex sets
EN
Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 65, No. 8, 1547-1577 (2006).
0362-546X
2006
j
35K65 58J35 53C44
anisotropic mean curvature flow; preservation of convexity; extinction
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 [\textit{F. Almgren, J. E. Taylor} and \textit{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 \(\partial E\) [\textit{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.
Christine Guenther (Forest Grove)
0783.35002; 1061.35147