an:05365922
Zbl 1185.53076
Dirr, N.; Karali, G.; Yip, N. K.
Pulsating wave for mean curvature flow in inhomogeneous medium
EN
Eur. J. Appl. Math. 19, No. 6, 661-699 (2008).
0956-7925 1469-4425
2008
j
53C44 76D05 35Q30
mean curvature flow; inhomogeneous medians; pulsating wave
In the present study the authors deal with the mean curvature flow of a hypersurface in a periodic inhomogeneous medium. More precisely, they consider the evolution \(\{\Gamma(t)\subset \mathbb{R}^{n+1}\mid t\geq 0\}\) of an \(n\)-dimensional surface with its motion law given by
\[
V_N(p)=H(p)+\delta f(p),\;p\in\Gamma(t),\tag{1}
\]
where \(V_N\) and \(H\) are the normal velocity and mean curvature of \(\Gamma (t)\), and \(\delta\) is a positive number which measures the strength of the spatial inhomogeneity, represented by \(f:\mathbb{R}^{n+1}\to \mathbb{R}\). Under rather weak assumptions on the data of (1), the authors are able to show for any direction \(\vec\nu\) the existence of a unique speed \(c_\nu\) and a number \(D<\infty\) such that the solution of (1) starting from a plane with normal \(\vec\nu\) stays as a graph over the same plane for all times, and moreover, this graph lies within a distance \(D\) from a plane which has normal \(\vec\nu\) and moves with normal velocity \(c_\nu\). Furthermore, if \(c_\nu\neq 0\), the authors show that pulsating waves exist.
Messoud A. Efendiev (Berlin)