On a Gradient Flow of Plane Curves Minimizing the Anisoperimetric Ratio
Abstract
We analyze a gradient flow of closed planar curves minimizing the anisoperimetric ratio. For such a flow the normal velocity is a function of the anisotropic curvature and it also depends on the total interfacial energy and enclosed area of the curve. In contrast to the gradient flow for the isoperimetric ratio, we show there exist initial curves for which the enclosed area is decreasing with respect to time. We also derive a mixed anisoperimetric inequality for the product of total interfacial energies corresponding to different anisotropy functions. Finally, we present several computational examples illustrating theoretical results.
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On a Gradient Flow of Plane Curves Minimizing the Anisoperimetric Ratio
Daniel Ševčovič and Shigetoshi Yazaki
^{0}^{0}footnotetext: Manuscript received April 27, 2013; revised June 6, 2013. This work was supported in part by the VEGA grant 1/0747/12 (DS) and GrantinAid for Scientific Research (C) 23540150 (SY)^{0}^{0}footnotetext: Prof. D. Ševčovič, PhD., is with the Department of Applied Mathematics and Statistics, Comenius University, 842 48 Bratislava, Slovak Republic. email: sevcovic@fmph.uniba.sk^{0}^{0}footnotetext: Prof. S. Yazaki, PhD., is with the Department of Mathematics, School of Science and Technology, Meiji University, Kanagawa 2148571, Japan. email: syazaki@meiji.ac.jp
Index Terms
Anisoperimetric ratio, gradient geometric flows, mixed anisoperimetric ratio inequality, tangential stabilization
I Introduction
T HE goal of this paper is to investigate a geometric flow of closed plane curves , minimizing the anisoperimetric ratio. We will show that the normal velocity for such a geometric flow is a function of the anisotropic curvature, the total interfacial energy and enclosed area of an evolved curve,
(1) 
where is the curvature and is a strictly positive coefficient depending on the tangent angle at a point . Here is a nonlocal part of the normal velocity depending on the entire shape of the curve and the term represents the anisotropic curvature. In typical situations, the nonlocal part is a function of the enclosed area and the interfacial energy , i.e. . As an example one can consider
where is the length of an evolved closed curve . It is well known that such a flow represents the area preserving geometric evolution of closed embedded plane curves investigated by Gage [8]. Among other geometric flows with nonlocal normal velocity we mention the curvature driven length preserving flow in which studied by Ma and Zhu [16] and the inverse curvature driven flow preserving the length studied by Pan and Yang [21]. The isoperimetric ratio gradient flow with has been proposed and investigated by Jiang and Zhu [14] for convex curves and by the authors in [23] for general closed Jordan curves evolving in the plane.
Recently, a classical nonlocal curvature flow preserving the enclosed area was reinvestigated by Xiao et al. in [25]. They proved uniform upper bound and lower bound on the curvature. Furthermore, Mao et al. [17] showed that such a nonlocal flow will decrease the perimeter of the evolving curve and make the curve more and more circular during the evolution process. Applying inequalities of Andrews and GreenOsher type, Lin and Tsai [15] showed that the evolving curves will converge to a round circle, provided that the curvature is apriori bounded. However, most of those fine results for area preserving flow still have to be extended to the case of a class of nonlocal flows minimizing the isoperimetric and/or anisoperimetric ratio.
The main goal of this paper is twofold. First we derive the normal velocity corresponding to the anisoperimetric ratio gradient flow. It turns out that where is the anisotropic curvature, i.e. has the form of (I). We derive and analyze several important properties of such a geometric flow. In contrast to the isoperimetric ratio gradient flow (c.f. Jiang and Zhu [14], [23]), we show that the anisoperimetric ratio gradient flow may initially increase the total length and, conversely, decrease the enclosed area of evolved curves. In order to verify such striking phenomena, an accurate numerical discretization scheme for fine approximation of the geometric flow has to be proposed. This is the second principal goal of the paper. We derive a numerical scheme based on the method of flowing finite volumes with combination of asymptotically uniform tangential redistribution of grid points. The idea of a uniform tangential redistribution has been proposed by How et al in [13] and further analyzed by Mikula and Ševčovič in [18]. The asymptotically uniform tangential redistribution has been analyzed in [20, 19]. The scheme is tested on the areadecrease and lengthincrease phenomena as well as on various other examples of evolution of initial curves having large variations in the curvature.
The paper is organized as follows. In the next section we recall the system of governing PDEs describing the evolution of all relevant geometric quantities. In section 3 we recall basic properties the anisotropic curvature and Wulff shape. We prove an important duality identity between total interfacial energies corresponding to different anisotropies. In section 4 we investigate a gradient flow for the anisoperimetric ratio. It turns out that the flow of plane minimizing the anisoperimetric ratio has the normal velocity locally depending on the anisotropic curvature and nonlocally depending on the total interfacial energy and the enclosed area of the evolved curve. Section 5 is devoted to the proof of a mixed anisoperimetric inequality for the product of two total interfacial energies corresponding to two anisotropy functions. In section 6 we investigate properties of the enclosed area for the anisoperimetric gradient flow. In contrast to a gradient flow for the isoperimetric ratio, we will show that there are initial convex curves for which the enclosed area is strictly decreasing. Finally, in section 7 we construct a counterexample to a comparison principle showing that there initial noninteresting curves such that they intersect each other immediately when evolved in the normal direction by the anisoperimetric ratio gradient flow. In section 8 we derive a numerical scheme for solving curvature driven flows with normal velocity depending on nolocal terms. The scheme is based on a flowing finite volume method combined with a precise scheme for approximation of nonlocal terms. We present several numerical examples illustrating theoretical results and interesting phenomena for the gradient flow for anisoperimetric ratio.
Ii System of governing equations and curvature adjusted tangential redistribution
In this section we recall description and basic properties of geometric evolution of a closed plane Jordan curve which can be parameterized by a smooth function such that and . We identify the interval with the quotient space by imposing periodic boundary conditions for at . We denote , and where is the Euclidean inner product between vectors and . The unit tangent vector is given by , where is the arclength parameter . The unit inward normal vector is defined in such a way that . Then the signed curvature in the direction is given by . Let be a tangent angle, i.e., and . From the Frenét formulae and we deduce that .
Geometric evolution problem can be formulated as follows: for a given initial curve , find a family of curve , starting from and evolving in the normal direction with the velocity . In this paper we follow the socalled direct approach in which evolution of the position vector is governed by the equation:
(2) 
Here is the tangential component of the velocity vector. Note that has no effect on the shape of evolving closed curves, and the shape is determined by the value of the normal velocity only. Therefore, one can take take when analyzing analytical properties of the geometric flow driven by (2). On the other hand, the impact of a suitable choice of a tangential velocity on construction of robust and stable numerical schemes has been pointed out by many authors (see [22, 23] and references therein).
In what follows, we shall assume that where is a strictly positive periodic smooth function of the tangent angle and is a nonlocal part of the normal velocity depending on the entire shape of the curve . According to [19] (see also [18, 20]) the system of PDEs governing evolution of plane curves evolving in the normal and tangential directions with velocities and reads as follows:
(3)  
(4)  
(5)  
(6) 
for and . Here is the socalled local length (c.f. [18]). A solution to (3)–(6) is subject to periodic boundary conditions for at , mod() and the initial condition corresponding to the initial curve .
Iii The Wulff shape and interfacial energy functional
The anisotropic curvature driven flow of embedded closed plane curves is associated with the socalled interfacial energy density (anisotropy) function defined on . It is assumed that is a strictly positive function depending on the tangent angle only. With this notation we can introduce the total interfacial energy
associated with a given anisotropy density function . If then is just the total length of a curve . The Wulff shape is defined as an intersection of hyperplanes:
If the boundary of the Wulff shape is smooth and it is parameterized by , then, it follows from the relation that
Hence and holds and the boundary can be parameterized as follows:
and its curvature is given by . Let us denote by the anisotropic curvature defined by . It means that the anisotropic curvature of the boundary of the Wulff shape is constant, . Moreover, the area of the Wulff shape satisfies:
Clearly, for the case . If we consider the anisotropy density function for then the area of can be easily calculated:
(7)  
In Fig 1 we plot shapes of for various degrees .







Since the global quantities evaluated over the closed curve do not depend on the tangential velocity we may take . Hence and . These identities follow from (4) and (5) with . Recall that . Therefore and so . Hence
holds. For the time derivative of we obtain
because and . From the previous equality we can deduce the following identity:
(8) 
where the family of planar embedded closed curves , evolves in the normal direction with the velocity .
Now, let us consider an evolving family of plane embedded closed curves , homotopicaly connecting a given curve and the boundary of the Wulff shape . The homotopy can be realized by taking a suitable normal velocity (eventually depending on the position vector ). Using such a normal velocity we deduce the identity:
(9) 
It means that is equal to the length of the boundary of the Wulff shape. The same result has been recently obtained by Barrett et al. in [2, Lemma 2.1]. We can say that identity (9) is a generalization of the rotation number: , since .
Remark 1
Identity (9) can be easily shown for convex curves. Indeed, if is convex then its arclength parameterization can be reparameterized by the tangent angle . We have and therefore . Hence
For the length of the boundary of a convex Wulff shape we obtain
Let us denote by the total interfacial energy corresponding to , i.e. . Let be the unit circle. Then, by applying identity (9), we deduce
(10) 
Latter identity can be rephrased as follows: the length of the boundary of the Wulff shape equals to the total interfacial energy of the unit circle. It can be easily generalized to the case of arbitrary two anisotropies and . We have the following proposition:
Theorem 1
Let and be two smooth anisotropy functions satisfying . Then the duality
(11) 
between total interfacial energies of boundaries and of Wulff shapes holds.
P r o o f. Notice that the Wulff shapes and are convex sets because and hold. For the curvature at the boundary we have and so
(12)  
(13) 
arguing vice versa.
Iv Gradient flow for the anisoperimetric ratio.
Recall that for the enclosed area and the total length for a flow of embedded closed plane curves driven in normal direction by the velocity we have
(14) 
(c.f. [18]). Using governing equations (3)–(6), for the total interfacial energy of a curve , we obtain
(15)  
(16)  
(17)  
Here we have used the governing equations (5) and (4) (with ) and the identity . For the anisoperimetric ratio
we have and, in particular, (see Remark 3). Taking into account identities (15) and (14) we obtain
Hence, the flow driven in the normal direction by the nonlocally dependent velocity
(18) 
represents a gradient flow for the anisoperimetric ratio with the property for . Notice that on if and only if , i.e. is homotheticaly similar to .
V A mixed anisoperimetric inequality
The aim of this section is to prove a mixed anisoperimetric inequality of the form
(19) 
which holds for any smooth Jordan curve in the plane. Here is a constant depending only on the anisotropy functions and such that and hold for any . The existence of a minimizer of the mixed anisoperimetric ratio is discussed in Remark 2. The idea of the proof of the inequality (19) is rather simple and consists in solving the constrained minimization problem:
(20) 
where is a given constant. To this end, let us assume that a curve is parameterized by a smooth function . If we denote the local length then, for the derivative of in the direction , we obtain and so
(21) 
Here and here after, for scalarvalued function and vectorvalued function we denote their derivatives in the direction by
respectively.
As for the tangent vector we have and so . As , for the derivative of the tangent angle , we obtain
(22) 
Recall that and . Since we obtain
Hence
(23) 
For the area enclosed by a Jordan curve we have . Therefore
Since we obtain
(24) 
In order to solve the constrained minimization problem (20) we introduce the Lagrange function with .
Then the first order condition for to be a minimizer of (20) reads as follows: at . Latter equality has to be satisfied for any smooth function . Taking into account (23) and (24) we obtain
It means that
(25) 
In other words, (up to an affine translation in the plane ). The Lagrange multiplier can be computed from the constraint . It follows from duality (11) (see Proposition 1) that
To calculate the enclosed area we make use of the identity . Clearly, as we obtain
Since we end up with the identity
Since the Lagrange multiplier it is given by . Furthermore,
Now, let be an arbitrary smooth Jordan curve in the plane. Set . Then
Remark 2
The proof of existence of a minimizer of the mixed anisoperimetric ratio is as follows: let be a sequence of Jordan curves minimizing this ratio. As , and for each , without lost of generality, we may assume for all . We can also fix the barycenter of at the origin. Since where , then, by the isoperimetric inequality, the value of the infimum is positive. Moreover, the parameterization of can be chosen in such a way that . As a consequence, the position vectors are uniformly bounded. By the ArzelàAscoli theorem there is a convergent subsequence converging to some function which is the minimizer of the mixed anisoperimetric ratio.
In summary, we have shown the following mixed anisoperimetric inequality:
Theorem 2
Let be a smooth Jordan curve in the plane. Then
(26) 
where .The equality in (26) holds if and only if the curve is homothetically similar to the boundary of a Wulff shape corresponding to the mixed anisotropy function .
Remark 3
If we obtain the well known isoperimetric inequality . If we obtain the anisoperimetric inequality . Finally, if we obtain the mixed anisoperimetric inequality
Remark 4
In the case , the anisoperimetric inequality in the plane has been stated in a paper by G. Wulff [24] from 1901. Later, it was proved by Dinghas in [4] for a special class of polytopes. Recently, Fonseca and Müller [5] proved the anisotropic inequality in the plane. Later Fusco et al. [6] proved it in arbitrary dimension. Giga in [10] pointed out that the anisotropic inequality where are periodic function is the isoperimetric inequality in a suitable Minkowski metric. It is a useful tool in the proof of anisotropic version of the socalled Gage’s inequality (c.f. [9, Corollary 4.3]).
However, in all aforementioned proofs, the surface energy was associated with a functional where is an absolute homogeneous anisotropy function of degree one, i.e. for any . The relation between our description of anisotropy and the latter one is: and, conversely, where . Since we do not require periodicity of , in our approach of description of anisotropy we therefore allow for nonsymmetric anisotropies, like e.g. functions with odd degree (see Fig 1) corresponding thus to anisotropy function which are positive homogeneous only, i.e. for any .
In the case of general anisotropy functions , the mixed anisoperimetric inequality derived in Theorem 2 is, to our best knowledge, new even in the case of symmetric (periodic) anisotropy functions.
Vi Convexity preservation. Temporal area and length behavior
In this section we analyze behavior of the enclosed area of a curve evolved in the normal direction by the anisoperimetric ratio gradient flow, i.e. .
First we prove the preservation of convexity result stating that the anisoperimetric ratio gradient flow preserves convexity of evolved curves. In the case of the isoperimetric ratio gradient flow of convex curves with , the convexity preservation has been shown by Jiang and Pan in [14]. However, similarly as Mu and Zhu in [16], they utilized the Gauss parameterization of the curvature equation (3) by the tangent angle and this is why their results are applicable to evolution of convex curves only. In our paper we first prove convexity preservation based on the analysis of the curvature equation (3) with arclength parameterization. Moreover, we show the anisoperimetric ratio gradient flow may initially increase the total length and decrease the enclosed area. This phenomenon cannot be found in the isoperimetric ratio gradient flow (c.f. [14, 22]).
Theorem 3
Let , be the anisoperimetric ratio gradient flow of smooth Jordan curves in the plane evolving in the normal direction by the velocity . If the curve is convex at some time then remains convex for any .
P r o o f. Since , and we have
where