On length-preserving and area-preserving nonlocal flow of convex closed plane curves

Abstract

For any α > 0, we study k α -type length-preserving and area-preserving nonlocal flow of convex closed plane curves and show that these two types of flow evolve such curves into round circles in C ∞ -norm. Other relevant k α -type nonlocal flow is also discussed when α ≥ 1.

Global dynamics of a parabolic type equation arising from the curvature flow

“…Note that γ 0 is axis-symmetric and has 2mπ total curvature. Then the problem (1.1) could be used to describe the motion of γ 0 driven by a lengthpreserving curvature flow, see related studies in [15,17,19] and references therein. In particular, when a = π, γ 0 is a simple closed convex curve and the flow converges to a round circle as time goes to infinity according to the result in [17].…”

Evolution of non-simple closed curves in the area-preserving curvature flow

“…Since it is unknown whether the Bonnesen inequality holds for a general non-simple closed curve, we are compelled to use a different method in this paper, that is, an energy method. For other types of planar non-local flow, the reader is referred to [14,15,19,20,22]. In the higher dimensional case, people also consider non-local flows.…”

Evolving convex curves to constant-width ones by a perimeter-preserving flow

“…Lin and Tsai [20] summarised the previews length-preserving or area-preserving flows as the so called κ α -type and 1 κ α -type nonlocal flows, where α > 0 is a constant. Later, Wang and Tsai [28] proved that κ α -type length-preserving or areapreserving flows for convex curves exist globally and drive the evolving curve into circles as t → +∞. The α-homogeneity of F plays an essential role in Wang-Tsai's research.…”

“…), there are no extra convexity condition on the function F . Apart from the case of F = κ α (α > 0) studied by Wang-Tsai [28], there are lots of other examples satisfying above conditions (i)-(iii), such as F (κ) = ln(1 + κ), e κ , 2κ + sin κ, κ 2 ln κ + κ and so on. Another remark is that the parabolic property F ′ > 0 and the positivity condition F > 0 imply the limit lim u→0 + F ′ (u) • u = 0 if this limit exists (see Lemma 3.1).…”