偏微分方程式セミナー: Oscillation of curvature and preserving local convexity for the curve diffusion flow

開催日時
2013年   8月 20日 15時 00分 ~ 2013年   8月 20日 16時 00分
場所
北海道大学理学部4号館4-501室
講演者
Glen Wheeler 氏 (ウーロンゴン大学)
 
In 1998, Giga and Ito provided the first rigorous proof that the
curve diffusion flow can drive a simple, closed, strictly embedded
planar curve to a self-intersection in finite time. They further
showed in 1999 that strict convexity is also lost in finite
time. Since this discovery work on the curve diffusion flow has
been intense, with research efforts focused on attempting to
find sharp conditions which guarantee global existence, regularity,
and convergence. This has proven to be a formidable
task, with many results in this direction. Elliott-Garcke showed
that graphs over a circle with small W^{3,2} \cap C^1 norm
converge to a circle, and Dziuk-Kuwert-Schaetzle (as well as
Chou) provided a finite-time blowup criterion. In this talk I will
explain my recent application of the normalised oscillation of
curvature to the curve diffusion flow, and how it has allowed
me to prove an improved global existence and exponential
convergence result as well as providing the first quantisation
of the prevalence of non-convexity along the flow. Precisely,
one can prove a sharp estimate for the Lebesgue measure of
the time during which a solution of curve diffusion flow in a
given class remains non-convex.

関連項目

研究集会・セミナー・集中講義の一覧へ