The Clamped Plate Equation for the Limaçon

Transcription

1 A. Dll Acqu G. Swees The Clmped Plte Eqution fo the Limçon Received: dte / in finl fom: dte Abstct. Hdmd climed in 907 tht the clmped plte eqution is positivity peseving fo domins which e bounded by Limçon de Pscl. We will show tht this clim is flse in its full genelity. Howeve, we will lso pove tht thee e nonconvex limçons fo which the clmped plte eqution hs the sign peseving popety. In fct we will give n explicit bound fo the pmete of the limçon whee sign chnge my occu. Mthemtics Subject Clssifiction 000: 35J40, 35Q7, 35B50 Key wods: clmped plte bihmonic positivity nonconvex. Intoduction Hdmd in [0] sttes tht the clmped plte eqution fo pltes hving the shpe of Limçon de Pscl is positivity peseving. Positivity peseving fo this line eqution on Ω R mens tht in the fouth ode boundy vlue poblem { u = f in Ω, u = ν u = 0 on Ω, the sign of f is peseved by u. Hee f is the foce density nd u the deflection of the plte of shpe Ω. So the sttement eds s, sy fo f L Ω: f 0 implies u 0. Fo pecise cittion of Hdmd let ΓA B = G Ω A, B be the coesponding Geen function, tht is, u x = Ω G Ω x, y f y dy solves. Concening Hdmd in [0] wites: Γ B A M. Boggio, qui, le pemie, noté l significtion physique de ΓA B, en déduit l hypothése que ΓA B étit toujous positif. Mlgé l bsence de Dept. of Applied Mthemticl Anlysis, ITS Fculty, Delft Univesity of Technology PObox 503, 600GA Delft The Nethelnds e-mil: A.Dllcqu@ITS.TUDelft.nl e-mil: G.H.Swees@ITS.TUDelft.nl

2 A. Dll Acqu, G. Swees démonsttion igoueuse, l exctitude de cette poposition ne pâit ps douteuse pou les ies convexes. Mis il étit l intéessnt d exmine si elle est vie pou le cs du Limçon de Pscl, qui est concve. L ėponse est ffimtive. Let us focus on Hdmd s two clims septely. Clim. Thee is no doubt tht Γ B A is positive fo convex domins. This conjectue stood fo long time nd only in 949 fist counteexmple, with Ω long ectngle, ws estblished by Duffin in [3]. This counteexmple ws soon to be followed by numeous othes. A shot suvey cn be found in the intoduction of [3]. So by now it is well known tht convexity is not sufficient condition. Let us emind the ede tht ound 905 Boggio [] did pove tht holds in cse of disk. In fct some believed tht the disk might be the only domin whee holds. Howeve in [5] it is shown tht lso holds in domins tht e smll petubtions of the disk. Since smllness of these petubtions is defined by C -nom non-convex domins e not included. Clim. ΓA B is positive fo some non-convex domins, nmely fo the Limçons de Pscl. Hdmd in [0] stts his poof of this clim by obseving tht:... on constte isément que, si l un de ces deux points est tès voisin du contou, l ptie pinciple de ΓA B est positive. Although we e not cetin wht he ment by ptie pinciple we know by now tht ΓA B cn be negtive when one point is ne the boundy. In fct we will show tht if the Geen function on limçon is negtive somewhee it will be negtive fo some A nd B ne the boundy. Hdmd continues his poof by efeing to the esults in [9]. In this ppe he gives n explicit fomul fo the Geen function fo in cse of limçon. This fomul will llow us to show the theoem below. Since thee is no explicit poof tht his fomul indeed gives the Geen function we will supply such poof in the ppendix. The domins unde considetion e defined fo [ 0, ] by Fo 0 Ω = { ρ cos ϕ, ρ sin ϕ R ; 0 ρ < + cos ϕ }. the cuve ρ = + cos ϕ is non self-intesecting limçon. We will show the following: Theoem. The clmped plte poblem on Ω with [ 0, ] is positivity peseving if nd only if [ 0, 6 6 ].

3 The Clmped Limçon 3 Fig.. Limçons fo esp. =.,.75,.5,.35, 6 6,.45,.5. The fifth one with = 6 6 is citicl fo positivity. Remk. The limçon is convex pecisely if 0 4. Notice tht 4 < 6 6. So Hdmd is ight in the sense tht convexity is not necessy condition. He is wong in climing the positivity peseving popety fo ll limçons. Remk. A elted question is if the fist eigenfunction hs fixed sign fo ll limçons compe the Boggio-Hdmd-conjectue vesus the Szegö-conjectue in [4], see lso [3]. Since one cnnot expect n explicit fomul fo the eigenfunction this seems much hde question. One does know tht the numbe whee positivity of the fist eigenfunction beks down is stictly lge thn the numbe whee fils. See [7]. Finlly we would like to mention some ppes tht conside explicit solutions fo the clmped plte eqution. Schot constucted in [], see lso Boggio in [], n explicit Geen function on the disk nd on the hlf-plne. Dube in [4] gives seies solution fo the Geen function on limçon.. Poofs Any limçon cn be seen s the imge of cicle though the confoml mp z z combined with two shifts. It will be convenient in the following to use complex nottion fo the unit disk: B = {z C ; z < }. The ppopite confoml mp fom B C to Ω R is then given by h : B Ω, η x = Re η + η, Im η + η. 3 The fct tht this confoml mp is qudtic, nd hence tht Ω is qutic cuve, seems to llow n explicit Geen function. This mkes the limçon specil cse. Fo the clmped plte eqution with constnt f on domins bounded by qutic cuves see []... Behviou of the Geen function In [9, Supplement] one finds the explicit fomul of the Geen function fo, which we will denote with G. Fo x, y Ω we my ewite this

4 4 A. Dll Acqu, G. Swees function s follows G x, y = s [log + s ], 4 whee, with η, ξ B such tht x = h η nd y = h ξ, the, nd s e given by = η ξ, = η ξ, s = η + ξ +. 5 We wnt to study when the function G is of fixed sign in Ω Ω. Fo estblishing this positivity we will need to conside the function q F β, q := log + q β. 6 q q Note tht q = /. Lemm. Set I β := {q : F β, q 0}. It holds tht: I β = {} fo β [ 0, ] ; I β = [, q β ] with q β > fo β, ; I β = [, fo β [,. 0 β q β < β < β Fig.. Gphs of q F β, q. Remk 3. Note tht β F β, q is decesing nd hence tht β q β is incesing. It will be convenient to wok with functions defined in the disk. If f is function defined on Ω, then f will denote the function f := f h defined on the disk. We fix the uxiliy function H η, ξ := s = η ξ η + ξ +, 7

5 The Clmped Limçon 5 nd hence the Geen function in 4 becomes G η, ξ := s F H η, ξ, = s η ξ F H η, ξ,. 8 ηξ The peceding Lemm gives tht if sup η,ξ B H η, ξ, 9 then F nd hence G e positive. Note tht 9 gives condition on the pmete which is sufficient condition fo the positivity of the function. In the following we will see tht this condition is lso necessy. Fist we will educe the dimension of the poblem. The following lemm sttes tht it is sufficient to study the behviou of H fo couples of conjugte points. Lemm. Let < nd define the sets l η,ξ fo η, ξ B B by l η,ξ = { } χ = χ + iχ B : χ = η+ξ, χ mx { η, ξ }, 0 whee η = η + iη nd ξ = ξ + iξ. If H η, ξ >, then fo evey χ l η,ξ it holds tht H χ, χ >. l η,ξ η ξ Fig. 3. A set l η,ξ nd its imge within limçon. Poof. By hypothesis one hs: H η, ξ = η ξ η ξ + η ξ η ξ η + ξ + + η + ξ >,

6 6 A. Dll Acqu, G. Swees which is equivlent to + η ξ + η ξ η ξ η ξ + η ξ + η ξ > η + ξ + + η ξ + η + ξ + η + ξ + η ξ, o similly + η + ξ > η + ξ + η + ξ + + η + ξ 4η + ξ. Fo χ l η,ξ, we hve χ H χ, χ = + χ + 4χ χ χ + χ + + χ χ + = +χ 4 +χ4 χ +χ χ χ +4χ χ χ 4 3 χ+4χ χ χ + = + χ χ χ χ 4χ. By the definition of l η,ξ nd it follows tht the lst tem is positive: + χ + η + ξ > χ + + χ 4χ. Remk 4. Note tht implies: H χ, χ is incesing in χ. We e now ble to pove tht 9 lso gives necessy condition fo the positivity of F nd hence of G. Lemm 3. Let <. i. If H v, v > then thee is χ l v, v such tht F H χ, χ, χ < 0. 3 ii. If 3 holds, then F χ χ H z, z, z z z < 0 fo evey z l χ, χ. Poof. Fist clim: Since the function β F β, q is decesing, see 6, nd the function H z, z is incesing in z, by Remk 4, one gets tht F H z, z, z F H v, v, z fo evey z l z z z z v, v. 4 In F H v, v, z the fist gument does not depend on z; it is z z fixed coefficient which is lge then / by hypothesis. Hence, pplying Lemm, one hs tht thee exists q H v, v > such tht F H v, v, z < 0, z l z z v, v with z < q z z Hv, v. 5

7 The Clmped Limçon 7 Note tht the function z z z z is decesing, since Hence, since z z z χ l v, v such tht z z z z = z + z z z. 6 3 is equl to t the boundy, it follows tht thee exists χ χ χ < q Hv, v. 7 Combining 4, 5 nd 7 the fist clim follows. Second clim: If F H χ, χ, χ χ χ < 0 we cn deduce fom Lemm tht H χ, χ > nd χ < q χ χ Hχ, χ. 8 Since H z, z is incesing in z Remk 4 nd the function z z z z is decesing, see 6, fom 8 one gets tht H z, z > nd z z z < q H χ, χ fo evey z l χ, χ. 9 Since β q β is incesing Remk 3, fom 9 we hve tht z z z < q Hz, z fo evey z l χ, χ. 0 By 9, 0 nd Lemm it follows tht F H z, z, z z z < 0 fo evey z l χ, χ. The pevious esults show tht if the function G x, y is negtive fo some x, y Ω then G will be negtive somewhee ne opposite boundy points. To be pecise: Coolly. Suppose tht G x, y < 0 fo some x, y Ω, then fo ll ε > 0 thee is x ε Ω with d Ω x ε < ε such tht: G x ε, x ε, x ε, x ε < 0. By d Ω x we denote the distnce of x to the boundy of Ω: d Ω x = inf { x x ; x Ω}. Poof. If G η, ξ < 0, Lemm gives tht necessily H η, ξ >. Hence, one hs fom Lemm tht H z, z > fo evey z l η,ξ. The clim follows diectly fom Lemm 3.

8 8 A. Dll Acqu, G. Swees.. Positivity of the Geen function Using the esults of the pevious section, we hve seen tht the function H in 7 plys cucil ole fo the positivity of the Geen function. Let us collect this esult. Coolly. The Geen function fo the clmped plte eqution on limçon is positive if nd only if sup η,ξ B H η, ξ = sup η,ξ B η ξ η + ξ +. Condition gives n uppe bound fo the pmete. In the following Lemm we give the explicit vlue of this uppe bound. Lemm 4. Inequlity is stisfied if nd only if 6 6. Poof. Lemm implies tht it is sufficient to veify fo couples of conjugte points, tht is: sup χ B H χ, χ = sup χ B χ χ + χ +. By we find H χ, χ = + χ + + 4χ χ +, which gives sup χ B H χ, χ = sup χ B χ χ +. 3 A stightfowd computtion shows tht the mximum in 3 is ttined fo χ = nd χ =. We obtin sup χ B H χ, χ = 4 + = 6 4, which is non-negtive fo > 6 6.

9 The Clmped Limçon 9.3. Shp estimtes fo the Geen function The Geen function fo the bihmonic poblem in two dimensions does not hve singulity in L -sense: x, y Gx, y is unifomly bounded. Howeve, ntul solution spce concening the Diichlet boundy condition u = ν u = 0, see [], is the Bnch lttice with the ntul odeing: C e Ω = {u C Ω; u e := sup ux d Ω x < }, whee d Ω. is s in. Howeve x G x,. fom Ω into C e Ω does show singulity when x Ω. Pecise infomtion fo the singulity of polyhmonic Diichlet Geen functions on blls in R n, whee the Geen function is known to be positive, cn be found in [8]. In the next theoem one finds how the estimte of G fom below chnges depending on. It is inteesting to see tht lthough the Geen function becomes negtive, no boundy-singulity fom below ppes. Note tht Theoem is diect consequence of Theoem. x Ω Theoem. Fo evey η, ξ B B, the following estimtes hold: i. fo ll [ 0, ] thee exists c > 0 such tht { G η, ξ c d B η d B ξ min, d B η d B ξ η ξ }, 4 ii. fo ll [ ] 0, 6 6, thee exists c > 0 such tht { } G η, ξ c 6 6 d B η d B ξ min, d B η d B ξ η ξ, 5 iii. fo 6 6, ] thee exists η, ξ B B such tht iv. fo ll 6 6, G η, ξ < 0. ], thee exists c3 > 0 such tht G η, ξ c 3 6 whee the constnts c nd c do not depend on. 6 d B η d B ξ, 6 Remk 5. Let us obseve tht fo evey ε > 0 thee exists two constnts m ε, M such tht fo evey η, ξ B nd [0, ε] it holds: m ε. η ξ h η h ξ M. η ξ, m ε. d η, B d h η, Ω M. d η, B. 7 Using 7 one cn pove estimtes fo G simil to the one poven fo G in Theoem. Ne the cusp when the estimte fom below in 7 beks down.

10 0 A. Dll Acqu, G. Swees Remk 6. One my deive tht fo [ ] 0, 6 6 thee exist constnts c4, c 5, independent on, such tht c Dx, y G x, y c 5 Dx, y, { } whee Dx, y = d x d y min, dxdy nd d = d xy Ω. Remk 7. Note tht the Geen function is positive on the digonl. This follows fom the eigenfunction expnsion nd tking x = y: Gx, y = i λ i ϕ i xϕ i y. Hee λ i, ϕ i e the eigenvlues/functions of the coesponding eigenvlue poblem. Note tht λ i > 0 holds fo ll i. Poof. We will pove the sttements septely. i. One hs fom 4 tht ] G η, ξ s [ log + [ log + ]. 8 The tem inside the bckets in the ight hnd side of 8 is the Geen function fo the clmped plte eqution on the disk. Inequlity 4 follows using the estimte in [6, Pop..3iii]. ii. Let 0 = 6 6 nd s0 = η + ξ + 0. Using tht s is decesing in fo ll η, ξ B when <, one finds fo [ ] 0, 6 6 tht G η, ξ s 0 log + s 0 = 4 0 G 4 0 η, ξ 0 + s 0 s [ log [ ] log + + ], since G 0 η, ξ 0, see Coolly nd Lemm 4. Fo [ ] 0, 6 6 one hs s , hence by using [6, Pop..3iii] one gets { } G η, ξ c 6 6 d B η d B ξ min, d B η d B ξ η ξ.

11 The Clmped Limçon iii. This clim follows fom Coolly nd Lemm 4. iv. Let 0 = 6 6 nd s0 = η + ξ + 0. We hve G η, ξ = s s s 0 log 0 + s s 0 s s 0 s 0 0 4, 9 since G 0 is positive in the entie domin. Using tht s 0 6 one gets tht s s 0 4 = s s 0 s , 30 Hence, fom 9 nd 30 it follows tht thee exists constnt c 3 > 0 such tht G η, ξ c 3 6 6dB η d B ξ, fo 6 6,. A. The Geen function fo the limçon As pomised in the intoduction this ppendix will contin poof tht the function supplied by Hdmd is indeed the Geen function fo the limçons. Fo x, y R let R = x y. The function U = R logr stisfies U = δ y in R. Then witing the function G x, y = R log R + J x, y, 3 J x, y := R log s + s 4, 3 should be bihmonic nd such tht G stisfies the boundy condition. Note tht 3 follows fom 4 using tht s = R. In fct one hs R = η + η ξ + ξ = η + η ξ ξ = = η ξ η + ξ + = s.

12 A. Dll Acqu, G. Swees A.. Boundy condition: Let us ewite 3 s [ G x, y = s log + ] When x Ω, then η D nd it holds =. It follows fom 33 tht G x, y = 0 t the boundy. Now we e inteested in ν G x, y on Ω. One obseves tht the tem gives no contibution becuse it is zeo of ode two t the boundy. The emining tem is poduct of two fctos: one tht is non-zeo t the boundy nd the othe tht is identiclly zeo. Hence, when we look t the noml deivtive t the boundy the only elevnt tem will be ν [ log + ]. 34 Using tht the tem inside the bckets in 34 is the Geen function fo the disk, see [], one gets tht lso the second Diichlet boundy condition is stisfied. A.. The function J x, y is bihmonic on Ω. To pove the bihmonicity of J it is convenient to conside septely the tem with the logithm nd the emining pt. We fist obseve tht log s is hmonic function on Ω. Fom this, the identity R log s = 0 follows using tht if v is hmonic function then R v is bihmonic. Lemm 5. It holds tht x s = 0. Poof. Next to h : B C R we will use h η : C C defined by h η = η + η with η = η + iη. Let us conside K x, y := h x + h h y + h x x h y, nd then s = h y Kx, y, nd Y η, ξ := K h η, h ξ = η + ξ + η η ξ.

13 The Clmped Limçon 3 Since h is confoml mp, it holds tht: η Y η, ξ = h η x K h η, h ξ, 35 ηy η, ξ = η h η x K h η, h ξ + η i h η η i x K h η, h ξ i= + h η 4 xk h η, h ξ. 36 The ide is to use 36 in ode to clculte xk h η, h ξ in tems of ηy η, ξ. Since η = 4 η η, one hs η Y η, ξ = η + ξ + η η η + ξ + + η η ξ, η η Y η, ξ = η η η + ξ + η + ξ + η η + ξ + + η ξ ηη ξ, 3 η η η Y η, ξ = η η + ξ + 4 η ξ, 4 η η η η Y η, ξ = 4 4 ξ, which gives η Y η, ξ = 4 3 η 4η ξ + 4 η + ξ + 4 η ξ η ξ, ηy η, ξ = ξ. By the definition of the confoml mp h in 3 nd fom 35 we obtin tht h η = η +, η h η = 6 nd 4 x K h η, h ξ = 3 η η ξ η+ + η + ξ + We find i= + 4 η+ η ξ + η i h η η i x K h η, h ξ + η ξ = 64 4η η+ 4η 4η ξ 4 η 4η ξ 4 ξ + ξ η ξ η+ 8η 4ξ 4 6η 8η ξ 8η 64 η+ + 6 η+ 8η 4η 8η ξ. 3 η+ 8η 4ξ η

14 4 A. Dll Acqu, G. Swees = 64 η η+ ξ η ξ ξ + ξ Hence fom 36 we get 64 η+ + η ξ. ξ = = η+ 4η 4η 4η ξ 4 η 4η ξ 4 ξ + ξ + η+ η ξ η+ + η ξ η+ η ξ η ξ ξ + ξ + h η 4 xk h η, h ξ, ξ = = η+ η + 4 η+ η ξ + η ξ + ξ + η+ ξ 4 η+ η ξ η+ ξ ξ η+ + h η 4 xk h η, h ξ, η+ η ξ η+ η ξ η ξ ξ + η+ ξ ξ = = ξ η+ η+ ξ η + + η+ ξ η+ ξ + h η 4 xk h η, h ξ, 0 = + ξ + η+ + h η 4 xk h η, h ξ, which gives the clim. Acknowledgment:We would like to thnk H.Ch. Gunu fo cefully eding the fist vesion of ou mnuscipt.

15 The Clmped Limçon 5 Refeences. Amnn, H.: Fixed point equtions nd nonline eigenvlue poblems in odeed Bnch spces, Sim Review, 8, Boggio, T.: Sulle funzioni di Geen d odine m, Rend. Cic. Mt. Plemo, 0, Duffin, R.J.: On question of Hdmd concening supe-bihmonic functions, J. Mth. Phys., 7, Dube, R.S.: Geen s function of n elstic plte in the shpe of Pscl s Limcon nd clmped long the boundy, Indin J. Pue Appl. Mth.,, Gunu, H.C., Swees, G.: Positivity fo petubtions of polyhmonic opetos with Diichlet boundy conditions in two dimensions, Mth. Nch., 79, Gunu, H.C., Swees, G.: Positivity fo equtions involving polyhmonic opetos with Diichlet boundy condition, Mth. Ann., 307, Gunu, H.C., Swees, G.: Sign chnge fo the Geen function nd the fist eigenfunction of equtions of clmped-plte type, Ach. Rtion. Mech. Anl., 50, Gunu, H.C., Swees, G.: Shp estimtes fo iteted Geen function, Poc. Roy. Soc. Edinbugh Sect. A, 3, Hdmd, J.: Memoie su le pobleme d nlyse eltif l equilibie des plques elstiques encstees, in: Œuves de Jcques Hdmd, Tomes II, Éditions du Cente Ntionl de l Recheche Scientifique, Pis 968, 55-64, epint of: Mèmoie couonne p l Acdèmie des Sciences Pix Villnt, Mèm. Sv. Etng., 33, Hdmd, J.: Su cetins cs inteessnts du pobleme bihmonique, in: Œuves de Jcques Hdmd, Tomes III, Éditions du Cente Ntionl de l Recheche Scientifique, Pis 968, 97-99, epint of: IV Cong. Mth., 908 Rome.. Schot, S.H.: The Geen s Functions Method fo the Suppoted Plte Boundy Vlue Poblem, Z. Anl. Anwendungen,, Sen, B.: Note on the bending of thin unifomly loded pltes bounded by cdioids, lemnisctes nd cetin othe qutic cuves, Z. Angew. Mth. Mech., 0, Swees, G.: When is the fist eigenfunction fo the clmped plte eqution of fixed sign?, Electon J. Diffe. Equ., 6, Szegö, G.: On membnes nd pltes, Poc. Nt. Acd. Sci. U. S. A., 36,