Abstract
In this paper, we prove that solutions of the anisotropic
Allen-Cahn equation in double-obstacle form with kinetic term
\begin{displaymath}
\varepsilon \beta(\nabla \varphi) \partial_t \varphi -
\varepsilon \nabla A'(\nabla \varphi) -
\frac{1}{\varepsilon} \varphi
= \frac{\pi}{4} u \quad \mbox{ in } [|\varphi| < 1],
\end{displaymath}
where $\ A\ $ is a convex function, homogeneous of degree two,
and $\ \beta\ $ depends only on the direction of $\ \nabla \varphi\ $,
converge to an anisotropic mean-curvature flow
\begin{displaymath}
\beta(N) V_N = - \mbox{tr}(B(N) D^2 B(N) R) - B(N) u.
\end{displaymath}
Here $\ V_N \mbox{ and } R\ $ respectively denote the normal
velocity and
the second fundamental form of the interface, and $\ B := \sqrt{2A}\ $.
We prove this in the case when the above flow admits a smooth
solution, and we establish that the Hausdorff-distance
between the zero-level set of $\ \varphi\ $ and the interface
of the flow is of order $\ O(\varepsilon^2)\ $.
Lingua originale | English |
---|---|
pagine (da-a) | 1103-1118 |
Numero di pagine | 16 |
Rivista | Mathematical Models and Methods in Applied Sciences |
Stato di pubblicazione | Pubblicato - 1996 |
Keywords
- anisotropic allen-cahn equation
- anisotropic mean curvature flow
- double obstacle