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Convergence to the Grim Reaper for a Curvature Flow with Unbounded Boundary Slopes
Authors
Bendong Lou
Xiaoliu Wang
Lixia Yuan
Publication date
14 January 2020
Publisher
View
on
arXiv
Abstract
We consider a curvature flow
V
=
H
V=H
V
=
H
in the band domain
Ξ©
:
=
[
β
1
,
1
]
Γ
R
\Omega :=[-1,1]\times \R
Ξ©
:=
[
β
1
,
1
]
Γ
R
, where, for a graphic curve
Ξ
t
\Gamma_t
Ξ
t
β
,
V
V
V
denotes its normal velocity and
H
H
H
denotes its curvature. If
Ξ
t
\Gamma_t
Ξ
t
β
contacts the two boundaries
β
Β±
Ξ©
\partial_\pm \Omega
β
Β±
β
Ξ©
of
Ξ©
\Omega
Ξ©
with constant slopes, in 1993, Altschular and Wu \cite{AW1} proved that
Ξ
t
\Gamma_t
Ξ
t
β
converges to a {\it grim reaper} contacting
β
Β±
Ξ©
\partial_\pm \Omega
β
Β±
β
Ξ©
with the same prescribed slopes. In this paper we consider the case where
Ξ
t
\Gamma_t
Ξ
t
β
contacts
β
Β±
Ξ©
\partial_\pm \Omega
β
Β±
β
Ξ©
with slopes equaling to
Β±
1
\pm 1
Β±
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
l
o
c
2
,
1
(
(
β
1
,
1
)
Γ
R
)
C^{2,1}_{loc} ((-1,1)\times \R)
C
l
oc
2
,
1
β
((
β
1
,
1
)
Γ
R
)
topology to the {\it grim reaper} with span
(
β
1
,
1
)
(-1,1)
(
β
1
,
1
)
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oai:arXiv.org:1907.11535
Last time updated on 12/10/2020