Application of Andrews and Green-Osher inequalities to nonlocal flow of convex plane curves
In this nice paper, the authors study a class of non-local flows on convex planar curves which keeps either the length of the curves, or their enclosed areas, constant. The normal speed is, pointwise, a difference of a function of the curvature at that point and a function of time (usually determined by the curves' length and/or area). A generalized Hölder inequality due to Andrews and a Green-Osher inequality (proved by Green and Osher via a flow) are exploited to show the convergence of these flows to circles. The key ingredient is to study the asymptotics of the isoperimetric difference of the evolving curves, \(L^2 (t) - 4 \pi A (t)\), where the above inequalities are used to show that it decays exponentially. Many examples of this class of flows are given and further applications of these inequalities are discussed.
scientific article; zbMATH DE number 6136967
scientific article