Application of Andrews and Green-Osher inequalities to nonlocal flow of convex plane curves

A curve shortening flow rule for closed embedded plane curves with a prescribed rate of change in enclosed area

“…Apart from the area-preserving flow rule already mentioned, other relevant generalizations include a signed-area-preserving flow [19], a lengthpreserving flow [20,21] and the gradient flow of the isoperimetric ratio L 2 /4π A [22]. Non-local flows like (1.2) with k replaced by 1/k have also received attention recently [23][24][25][26]. While we will not be treating flows of hypersurfaces in higher dimensions [27,28], it is worth noting that nonlocal generalizations of mean curvature flow are also of interest [9,29,30].…”

“…For a general flow rule, ( 1.3 ) leads to the result which implies that is the steepest descent gradient flow of L 2 −2(2 π − β ) A . Thus, we see that ( 1.2 ), ( 1.4 ) can be thought of as the gradient flow of L 2 −2(2 π − β ) A after a simple rescaling in time, where the rescaled time satisfies [ 23 ]. In this way, we again see that ( 1.2 ), ( 1.4 ) is a generalization of both standard curve shortening flow ( 1.1 ) (which is the gradient flow of L 2 after the same time rescaling) and area-preserving flow (which is the gradient flow of the isoperimetric difference L 2 −4 πA after the rescaling).…”

On a planar area-preserving curvature flow

“…As a variation of the popular curve shortening flow [6,7,2,4,18,19,3], the nonlocal curvature flow, arising in many application fields [17], such as phase transitions, image processing, etc., has received much attention in recent years. The relevant research is focused on so-called area-preserving flow [5,13], perimeterpreserving flow [16,14], and other analogues [8,11,12,15]. The relations between these nonlocal flows are nicely summarized by Lin and Tsai [10].…”

Higher order Wirtinger-type inequalities and sharp bounds for the isoperimetric deficit

“…The first inequality in (1.7) is also a consequence of our second order Wirtinger-type inequality (1.2), and is equivalent to the Lin-Tsai inequality [18,Lemma 1.7]. The second inequality in (1.7) is a consequence of our third order Wirtinger-type inequality (1.3), and can be regarded as a lower bound of the isoperimetric deficit.…”