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Pulsating wave for mean curvature flow in inhomogeneous medium. (English) Zbl 1185.53076
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.

##### MSC:
 53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) 76D05 Navier-Stokes equations for incompressible viscous fluids 35Q30 Navier-Stokes equations
##### Keywords:
mean curvature flow; inhomogeneous medians; pulsating wave
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##### References:
 [1] DOI: 10.1016/j.anihpc.2004.10.009 · Zbl 1135.35092 · doi:10.1016/j.anihpc.2004.10.009 [2] DOI: 10.1007/s000300050029 · Zbl 0887.35070 · doi:10.1007/s000300050029 [3] Huisken, J. Diff. Geom. 20 pp 237– (1984) [4] Friedman, Pacific J. Math. 8 pp 201– (1958) · Zbl 0103.06403 · doi:10.2140/pjm.1958.8.201 [5] Evans, J. Geom. Anal. 2 pp 121– (1992) · Zbl 0768.53003 · doi:10.1007/BF02921385 [6] DOI: 10.1007/978-3-7643-7719-9_13 · doi:10.1007/978-3-7643-7719-9_13 [7] Evans, J. Differential Geom. 33 pp 635– (1991) [8] DOI: 10.1090/S0273-0979-1992-00266-5 · Zbl 0755.35015 · doi:10.1090/S0273-0979-1992-00266-5 [9] DOI: 10.1007/BF01232278 · Zbl 0707.53008 · doi:10.1007/BF01232278 [10] Chen, J. Diff. Geom. 33 pp 749– (1991) [11] DOI: 10.2307/1971452 · Zbl 0696.53036 · doi:10.2307/1971452 [12] Dirr, Interfaces Free Bound. 8 pp 79– (2006) [13] Cahn, SIAM J. Appl. Math. 59 pp 455– (1999) [14] Dirr, Interfaces Free Bound. 8 pp 47– (2006) [15] DOI: 10.1002/cpa.10008 · Zbl 1036.49040 · doi:10.1002/cpa.10008 [16] Bhattacharya, Interfaces Free Bound. 6 pp 151– (2004) [17] Bhattacharya, Proc. Royal. Soc. Edin. 133A pp 773– (2003) [18] Allen, Acta Metall. 27 pp 1084– (1979) · doi:10.1016/0001-6160(79)90196-2 [19] DOI: 10.1002/cpa.20046 · Zbl 1065.49011 · doi:10.1002/cpa.20046 [20] Phillips, Crystals, Defects and Microstructures (2001) · doi:10.1017/CBO9780511606236 [21] Korevaar, Proc. Sympos. Pure Math. Part 2, Amer. Math. Soc. 45 pp 81– (1986) · doi:10.1090/pspum/045.2/843597
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