Convergence to the grim reaper for a curvature flow with unbounded boundary slopes

• Published in: Calculus of variations and partial differential equations 2021-08, Vol.60 (4), Article 159
• Main Authors: Lou, Bendong, Wang, Xiaoliu, Yuan, Lixia
• Format: Article
• Language: English
• Subjects: Analysis; Calculus of Variations and Optimal Control; Optimization; Control; Convergence; Curvature; Estimates; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Slopes; Systems Theory; Theoretical; Topology
• ISSN: 0944-2669; 1432-0835
• DOI: 10.1007/s00526-021-01991-x

We consider a curvature flow V = H in the band domain Ω : = [ - 1 , 1 ] × R , where, for a graphic curve Γ t , V denotes its normal velocity and H denotes its curvature. If Γ t contacts the two boundaries ∂ ± Ω of Ω with constant slopes, in 1993, Altschular and Wu (Math Ann 295:761–765, 1993) proved that Γ t converges to a grim reaper contacting ∂ ± Ω with the same prescribed slopes. In this paper we consider the case where Γ t contacts ∂ ± Ω with slopes equaling to ± 1 times of its height. When the curve moves to infinity, the global gradient estimate is impossible due to the unbounded boundary slopes. We first consider a special symmetric curve and derive its uniform interior gradient estimates by using the zero number argument, and then use these estimates to present uniform interior gradient estimates for general non-symmetric curves, which lead to the convergence of the curve in C loc 2 , 1 ( ( - 1 , 1 ) × R ) topology to the grim reaper with span ( - 1 , 1 ) .