Research Article Harmonic Deformation of Planar Curves

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1 Interntionl Journl of Mthemtics nd Mthemticl Sciences Volume, Article ID 9, pges doi:.55//9 Reserch Article Hrmonic Deformtion of Plnr Curves Eleutherius Symeonidis Mthemtisch-Geogrphische Fkultät, Ktholische Universität Eichstätt-Ingolstdt, 857 Eichstätt, Germny Correspondence should be ddressed to Eleutherius Symeonidis, Received 3 November ; Accepted 7 Jnury Acdemic Editor: Aloys Krieg Copyright q Eleutherius Symeonidis. This is n open ccess rticle distributed under the Cretive Commons Attribution License, which permits unrestricted use, distribution, nd reproduction in ny medium, provided the originl work is properly cited. We estblish principle of deformtion of n rbitrry plnr curve, so tht the integrl of hrmonic function over this curve does not chnge. The equtions of deformtion cn be derived from specific potentil. Severl pplictions re presented.. Introduction Let Ω R be n open set. A function h : Ω R with continuous second prtil derivtives is clled hrmonic if hx, y/ x hx, y/ y forllx, y Ω. Asiswell known, such function hs the following men vlue property: for every disk {z R ; z z r} Ω, z x,y, r, it holds tht π h ( x r cos t, y r sin t ) dt h ( ) x,y. π. Now let us look t this property from slightly different point of view. We will not consider the recovering of h t the center of the circle s crucil but the fct tht the left side of. is independent of r. Inotherwords,h hs the sme men vlue over ll concentric circles with center x,y. We give two further exmples.

2 Interntionl Journl of Mthemtics nd Mthemticl Sciences Let us consider fmily of confocl ellipses in Ω, centeredt,. Tking c, nd c, for their foci c >, the ellipses re given by equtions of the form x /c cosh r y /c sinh r. For hrmonic function h : Ω R,itthenholdssee, e.g., tht π hc cosh r cos t, c sinh r sin tdt π π c c hx, c x dx.. Here, of course, it is necessry tht Ω contins the interiors of ll ellipses of the fmily. Therefore, the left side of. is independent of r, ndh hs the sme men vlue in this precise sense over ll confocl ellipses with foci t c, nd c,. Finlly, let Ω be so big tht it contins fmily of confocl hyperbols together with their interiors with foci t c, nd c, c>. Such hyperbols emerge from equtions of the form x /c cos r y /c sin r, nd we restrict ourselves to their right brnches x >. For bounded hrmonic function h : Ω R, for which there exists r > suchtht t hc cos r cosh t, c sin r sinh t is integrble over R,itthenholdsfor r r tht hx, hc cos r cosh t, c sin r sinh tdt c x c dx..3 Agin, the left side is independent of r,ndhhs the sme men vlue over ll these confocl hyperbols. Thus, nturlly the question rises, whether, given curve in Ω, itcnbedeformed in such wy tht the men vlue over it of hrmonic function does not chnge. In this pper we introduce generl principle of deformtion nd derive the equtions of the fmily curves explicitly. This principle serves double purpose: on the one hnd it offers unified resoning for.,., nd.3, nd on the other it turns out to be source of further exmples of fmilies of curves tht leve the hrmonic men vlue unchnged. Now let us set out the frmework nd specify the notion of men vlue tht we re going to study. Let I R be n intervl, I t ( x t,y t ) Ω,. smooth curve in n open nd simply connected set Ω R.Leth : Ω R be hrmonic function for which I t hx t,y t is integrble. We then serch for functions xs, t nd ys, t defined on some J I,whereJ is n intervl with J,suchtht ( xs, t,ys, t ) Ω s, t J I,.5 x,t x t, y,t y t t I,.6 h ( xs, t,ys, t ) dt h ( x t,y t ) dt s J..7 I I The lst eqution will be regrded s the invrince of the men vlue of h over the curves t xs, t,ys, t for ll s J.

3 Interntionl Journl of Mthemtics nd Mthemticl Sciences 3 Hving the exmples.,., nd.3 in mind, we now mke the following ssumption. If h : Ω R is hrmonic conjugte function to h i.e., h i h is holomorphic on Ω, then h is bounded nd lim h [ xs, t,ys, t ] lim h [ xs, t,ys, t ] for every s J..8 t inf I t sup I This is not relly restriction when I is compct nd ll the curves of the fmily re closed the boundedness of h cn be chieved by shrinking Ω, but it is crucil in situtions like.3, wherei hs infinite length. It seems tht the use of h is no less thn the key, which opens the wy to unified tretment of the cses of bounded nd unbounded I. It lso seems unlikely tht the conditions of boundedness nd.8 for h could be formulted in terms of h. Consider, e.g., hx, y x on Ω : R, nd on Ω : {x, y ; y < x }. In this frmework we re going to show tht if s, t ( xs, t,ys, t ),.9 is conforml mpping stisfying.5,.6, nd.8, then.7 holds. This will estblish the principle of deformtion. In the present pper we restrict ourselves to men vlues where the mesure is given by the curve prmeter t. Note tht prmeter chnge ffects the conformlity of.9 becuse of.6. However, the sme principle of deformtion cn serve to tret more generl cses of mesures, where weight function w enters: h ( xs, t,ys, t ) ws, tdt.. Since this is fr more difficult, it will be studied in future pper. I. Principle nd Equtions of Deformtion The principle of deformtion is s follows. Theorem.. Let Ω R be simply connected open set, t x t,y t Ω smooth curve defined on n intervl I R. Furthermore, let J R be n open intervl with J nd J I Ω, s, t ( xs, t,ys, t ),. smooth function stisfying x s y t, x t y s,. nd.6.

4 Interntionl Journl of Mthemtics nd Mthemticl Sciences If h : Ω R is hrmonic function for which there exists bounded hrmonic conjugte h : Ω R nd.8 holds, then.7 holds. Proof. Since Ω is simply connected, hrmonic conjugte function h to h exists nd is determined up to constnt. We fix such n h nd ssume it bounded. Let, b I.Itthenholdstht d ds h ( xs, t,ys, t ) dt d ds h( xs, t,ys, t ) dt ( h x ( x, y ) x h s y ( x, y ) y ) s, tdt s ( h y ( x, y ) y h t x ( x, y ) x ) s, tdt t.3 d dt h ( xs, t,ys, t ) dt h ( xs, b,ys, b ) h ( xs,,ys, ). Therefore, h ( xs, t,ys, t ) dt s h ( x t,y t ) dt [ h( x ( s,b ),y ( s,b )) h ( x ( s, ),y ( s, ))] ds.. Now, since h is bounded, the limit for b sup I nd inf I on the right side cn be tken under the integrl sign, nd.7 is estblished. On the bsis of this theorem we derive the equtions of deformtion. From. it follows tht there exists hrmonicfunction v vs, t on J I such tht x v t, v y s..5 For this v,.6 entils tht v t,t x t, v s,t y t..6 Thus, v stisfies Cuchy problem with initil dt.6 nd is therefore determined up to n dditive constnt.

5 Interntionl Journl of Mthemtics nd Mthemticl Sciences 5 Let us now give series representtion of v. From.6 nd the hrmonicity of v, it inductively follows tht k v s k,t k x k t for k N, k v s k,t k y k t for k N {}..7 This leds to the series expnsion k x k t vs, t v,t s k k y k t s k, k! k! k k.8 on the bsis of which, together with v,t/ t x t,.5 constitutes the equtions of deformtion. The s-intervl of convergence of.8 depends, of course, on the initil curve x,y. Furthermore, it clerly follows from the equtions of deformtion tht if I t x t,y t is closed curve which cn be extended to smooth periodic mp for ll t R, then I t xs, t,ys, t remins closed for every s J. Such function v which is determined up to constnt will be clled d-potentil from the word deformtion of the curve x t,y t. This should not be confused with the Schwrz potentil of this curve see Applictions We strt by briefly revisiting the situtions in.,.,nd.3, now s pplictions of the d-potentil, nd continue with further interesting exmples. I If x t,y t R cos t, b R sin t for t, π, thenby.8 nd v,t/ t x t the d-potentil is vs, t v,t k R sin t k! sk R sin t k! sk b R sin ts k t R sin t d R sin t cosh s R sin t sinh s bs 3. t bs Re s sin t d with d R. So.5 gives xs, t Re s cos t, ys, t b Re s sin t, 3. fmily prmetrized by s of circles with center, b.

6 6 Interntionl Journl of Mthemtics nd Mthemticl Sciences hve II For the ellipse x t,y t cos t, b sin t with >b>ndt, π, we vs, t v,t k sin t k! sk b sin t k! sk k 3.3 sin t cosh s b sin t sinh s d with d R. This entils tht xs, t cosh s b sinh s cos t, ys, t sinh s b cosh s sin t, so xs, t c coshs s cos t, ys, t c sinhs s sin t, 3. with c b nd s rsinhb/c. This is fmily prmetrized by s of confocl ellipses with foci t c, nd c,. III For the hyperbol x t,y t cosh t, b sinh t with, b > ndt R,we hve vs, t v,t k k sinh t k! s k k k b sinh t s k k! 3.5 sinh t cos s b sinh t sin s d with d R, which gives xs, t cos s b sin s cosh t, ys, t b cos s sin s sinh t, so xs, t c coss s cosh t, ys, t c sins s sinh, 3.6 with c b nd s rcsinb/c. This is fmily of confocl hyperbols with foci t c, nd c,. Of course, here we hve to put the conditions of boundedness nd.8 for h. In prticulr, if h is defined on the right hlf-plne nd is continuous into the boundry, then it holds s s π/, s s tht h,csinh tdt hc cosh t, dt, h (,y ) hx, dy c y c x c dx. or 3.7 IV Let Ω be the upper hlf-plne. For the bse curve we tke the stright line x t,y t t, with t R. Then, vs, t v,t s s t s s d with d R, 3.8 nd xs, t t, ys, t s. Thus, under the mentioned conditions on h, hrmonic function h on Ω hs the sme integrl over ll stright lines prllel to the boundry.

7 Interntionl Journl of Mthemtics nd Mthemticl Sciences 7 V Consider the strlike curve x t,y t cos 3 t, sin 3 t for t, π. Onthe bsisofthereltionscos 3 t cos3t 3cost/ ndsin 3 t 3sint sin3t/, rbitrry derivtives cn be computed s follows: x k t k [ ] 3sint 3 k sin3t y k t k [ ] 3sint 3 k sin3t for k, for k. 3.9 This llows the d-potentil to be given explicitly s follows: 3sint 3 k sin3t vs, t v,t k! sk s k k! k k k 3sint k! sk 3 k sin3t k! sk v,t 3sint e s sin3t ( ) e 3s k 3. 3es sin t e 3s sin3t d with d R. This results in the following fmily of curves: xs, t 3es cos t e 3s cos3t ys, t 3es sin t e 3s sin3t 3e s e 3s( ) sin t 3es e 3s( cos t ) cos t, sin t. 3. Figure exhibits some distinct curves of the fmily. All figures in this pper hve been produced with Mthemtic. The one for s ln 3/ my be of prticulr interest, where the double point is, : ( x ln 3 ),t 3 3 ( cost cos t, y ln 3 ) 3,t 3 cost sin t. 3. This curve stisfies the eqution x y 3 3 3/x y nd resembles lemniscte. Bernoulli s lemniscte is given by x y cx y, c>. VI It seems difficult to compute the d-potentil of Bernoulli s lemniscte explicitly. The opposite is true in the cse of Gerono s lemniscte with eqution x x y, >. It is prmetrized by ( x t,y t ) sin t, sin t cos t for t, π. 3.3

8 8 Interntionl Journl of Mthemtics nd Mthemticl Sciences s.3 s ln b s. s c d s..5 s e f Figure

9 Interntionl Journl of Mthemtics nd Mthemticl Sciences 9 s s s b c s d.. s e s f s s g h Figure We hve x k t k cos t, y k t k / sint,so vs, t v,t k cos t k! sk k sint k! sk k cos t cosh s sint sinhs d with d R. 3.

10 Interntionl Journl of Mthemtics nd Mthemticl Sciences This gives xs, t sin t cosh s cost ys, t cos t sinh s sint sinhs cosh s sin t sinh s cos t, coshs cos t sinh s sin t cosh s. 3.5 Figure shows some snpshots of the fmily. VII For the infinitely extended cycloid ( x t,y t ) t sin t, cos t, t R, 3.6 Ω being suitble hlf-plne, wehvex k t k cos t for k, y k t k cos t for k, so vs, t v,t s cos t cos t k! sk s k! sk t s k k s e s cos t d with d R. 3.7 The result is xs, t t e s sin t, ys, t s e s cos t, 3.8 which is prolte cycloid for s<ndcurtteonefors> up to verticl displcement by s. References W. C. Royster, A Poisson integrl formul for the ellipse nd some pplictions, Proceedings of the Americn Mthemticl Society, vol. 5, pp , 96. E. Symeonidis, The Poisson integrl for the interior of hyperbol, Journl of Nnjing University (Mthemticl Biqurterly), vol. 7, no., pp. 7,. 3 H. S. Shpiro, The Schwrz Function nd Its Generliztion to Higher Dimensions, vol. 9 of University of Arknss Lecture Notes in the Mthemticl Sciences, John Wiley & Sons, New York, NY, USA, 99.

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