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International Journal of Mathematics and Mathematical Sciences
Volume 2011 (2011), Article ID 141209, 10 pages
http://dx.doi.org/10.1155/2011/141209
Research Article

Harmonic Deformation of Planar Curves

Mathematisch-Geographische Fakultät, Katholische Universität Eichstätt-Ingolstadt, 85071 Eichstätt, Germany

Received 13 November 2010; Accepted 17 January 2011

Academic Editor: Aloys Krieg

Copyright © 2011 Eleutherius Symeonidis. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We establish a principle of deformation of an arbitrary planar curve, so that the integral of a harmonic function over this curve does not change. The equations of deformation can be derived from a specific “potential.” Several applications are presented.

1. Introduction

Let be an open set. A function with continuous second partial derivatives is called harmonic if for all . As is well known, such a function has the following mean value property: for every disk , , , it holds that

Now let us look at this property from a slightly different point of view. We will not consider the recovering of at the center of the circle as crucial but the fact that the left side of (1.1) is independent of . In other words, has the same mean value over all concentric circles with center .

We give two further examples.

Let us consider a family of confocal ellipses in , centered at . Taking and for their foci , the ellipses are given by equations of the form . For a harmonic function , it then holds (see, e.g., [1]) that (Here, of course, it is necessary that contains the interiors of all ellipses of the family.) Therefore, the left side of (1.2) is independent of , and has the same mean value (in this precise sense) over all confocal ellipses with foci at and .

Finally, let be so big that it contains a family of confocal hyperbolas (together with their interiors) with foci at and . Such hyperbolas emerge from equations of the form , and we restrict ourselves to their “right” branches . For a bounded harmonic function , for which there exists such that is integrable over , it then holds for [2] that Again, the left side is independent of , and has the same mean value over all these confocal hyperbolas.

Thus, naturally the question arises, whether, given a curve in , it can be deformed in such a way that the mean value over it of a harmonic function does not change. In this paper we introduce a general principle of deformation and derive the equations of the family curves explicitly. This principle serves a double purpose: on the one hand it offers a unified reasoning for (1.1), (1.2), and (1.3), and on the other it turns out to be a source of further examples of families of curves that leave the harmonic mean value unchanged.

Now let us set out the framework and specify the notion of “mean value” that we are going to study.

Let be an interval, a smooth curve in an open and simply connected set . Let be a harmonic function for which is integrable. We then search for functions and defined on some , where is an interval with , such that The last equation will be regarded as the invariance of the mean value of over the curves for all .

Having the examples (1.1), (1.2), and (1.3) in mind, we now make the following assumption. If is a harmonic conjugate function to (i.e., is holomorphic on ), then is bounded and This is not really a restriction when is compact and all the curves of the family are closed (the boundedness of can be achieved by shrinking ), but it is crucial in situations like (1.3), where has infinite length. (It seems that the use of is no less than the key, which opens the way to a unified treatment of the cases of bounded and unbounded . It also seems unlikely that the conditions of boundedness and (1.8) for could be formulated in terms of . (Consider, e.g., on and on .)

In this framework we are going to show that if is a conformal mapping satisfying (1.5), (1.6), and (1.8), then (1.7) holds. This will establish the principle of deformation.

In the present paper we restrict ourselves to mean values where the measure is given by the curve parameter . (Note that a parameter change affects the conformality of (1.9) because of (1.6).) However, the same principle of deformation can serve to treat more general cases of measures, where a weight function enters: Since this is far more difficult, it will be studied in a future paper.

2. Principle and Equations of Deformation

The principle of deformation is as follows.

Theorem 2.1. Let be a simply connected open set, a smooth curve defined on an interval . Furthermore, let be an open interval with and a smooth function satisfying and (1.6).
If is a harmonic function for which there exists a bounded harmonic conjugate and (1.8) holds, then (1.7) holds.

Proof. Since is simply connected, a harmonic conjugate function to exists and is determined up to a constant. We fix such an and assume it bounded.
Let . It then holds that Therefore, Now, since is bounded, the limit for and on the right side can be taken under the integral sign, and (1.7) is established.

On the basis of this theorem we derive the equations of deformation.

From (2.2) it follows that there exists a harmonic function on such that For this , (1.6) entails that Thus, satisfies a Cauchy problem with initial data (2.6) and is therefore determined up to an additive constant.

Let us now give a series representation of . From (2.6) and the harmonicity of , it inductively follows that This leads to the series expansion on the basis of which, together with , (2.5) constitutes the equations of deformation. (The -interval of convergence of (2.8) depends, of course, on the initial curve . Furthermore, it clearly follows from the equations of deformation that if is a closed curve (which can be extended to a smooth periodic map for all ), then remains closed for every .)

Such a function (which is determined up to a constant) will be called a -potential (from the word deformation) of the curve . This should not be confused with the Schwarz potential of this curve (see [3]).

3. Applications

We start by briefly revisiting the situations in (1.1), (1.2), and (1.3), now as applications of the -potential, and continue with further interesting examples.

(I) If for , then by (2.8) and the -potential is So (2.5) gives a family parametrized by of circles with center .

(II) For the ellipse with and , we have This entails that , , so with and . This is a family parametrized by of confocal ellipses with foci at and .

(III) For the hyperbola with and , we have which gives , , so with and . This is a family of confocal hyperbolas with foci at and . Of course, here we have to put the conditions of boundedness and (1.8) for . In particular, if is defined on the “right” half-plane and is continuous into the boundary, then it holds (, ) that

(IV) Let be the upper half-plane. For the base curve we take the straight line with . Then, and , . Thus, under the mentioned conditions on , a harmonic function on has the same integral over all straight lines parallel to the boundary.

(V) Consider the starlike curve for . On the basis of the relations and , arbitrary derivatives can be computed as follows: This allows the -potential to be given explicitly as follows: with . This results in the following family of curves: Figure 1 exhibits some distinct curves of the family. (All figures in this paper have been produced with Mathematica.) The one for may be of particular interest, where the double point is : This curve satisfies the equation and resembles a lemniscate. (Bernoulli's lemniscate is given by , .)

fig1
Figure 1

(VI) It seems difficult to compute the -potential of Bernoulli's lemniscate explicitly. The opposite is true in the case of Gerono's lemniscate with equation , . It is parametrized by We have , , so This gives Figure 2 shows some snapshots of the family.

fig2
Figure 2

(VII) For the infinitely extended cycloid ( being a suitable half-plane), we have for , for , so The result is which is a prolate cycloid for and a curtate one for (up to a vertical displacement by ).

References

  1. W. C. Royster, “A Poisson integral formula for the ellipse and some applications,” Proceedings of the American Mathematical Society, vol. 15, pp. 661–670, 1964. View at Publisher · View at Google Scholar
  2. E. Symeonidis, “The Poisson integral for the interior of a hyperbola,” Journal of Nanjing University (Mathematical Biquarterly), vol. 27, no. 2, pp. 242–247, 2010. View at Google Scholar
  3. H. S. Shapiro, The Schwarz Function and Its Generalization to Higher Dimensions, vol. 9 of University of Arkansas Lecture Notes in the Mathematical Sciences, John Wiley & Sons, New York, NY, USA, 1992.