Nonlinear Diﬀer. Equ. Appl.
2017 Springer International Publishing AG
Nonlinear Diﬀerential Equations
and Applications NoDEA
On the rate of convergence of the p-curve
Jean C. Cortissoz, Andr´es Galindo and Alexander Murcia
Abstract. In this paper we give rates of convergence for the p-curve short-
ening ﬂow for p ≥ 1 an integer, which improves on the known estimates
and which are probably sharp.
Mathematics Subject Classiﬁcation. 53C44, 35K55, 58J35.
Keywords. p-curve shortening ﬂow, Convergence rate, Blow-up.
Let us ﬁrst introduce the main character of this story, the p-curve shortening
ﬂow, with p a positive integer. So, we let
x : S
× [0,T) −→ R
be a family of smooth convex embeddings of S
, the unit circle, into R
say that x satisﬁes the p-curve shortening ﬂow, p ≥ 1, if x satisﬁes
where k is the curvature of the embedding and N is the normal vector pointing
outwards the region bounded by x (·,t).
The p-curve shortening ﬂow is just a natural generalisation of the well
known and well studied curve shortening ﬂow. A solution to (1) starting from
an embedded convex simple curve will contract, via embedded convex curves,
towards a round point in ﬁnite time: this means that if we start with a sim-
ple convex curve, via the p-curve shortening, after a convenient normalisa-
tion, which includes a time reparametrisation, the embedded curves converge
smoothly to a circle (see ). It is also known that this convergence is expo-
nential in the following sense (here
k denotes the curvature of the embedded
curves after normalisation)