Boundary value problems for the one-dimensional Willmore equation

Transcription

1 Boundary value problems for the one-dimensional Willmore equation Klaus Dekelnik and Hans-Christoph Grunau Fakultät für Mathematik Otto-von-Guerike-Universität Postfah D-396 Magdeburg November 7, 6 Abstrat The one-dimensional Willmore equation is studied under Navier as well as under Dirihlet boundary onditions. We are interested in smooth graph solutions, sine for suitable boundary data, we expet the stable solutions to be among these. In the first part, lassial symmetri solutions for symmetri boundary data are studied and losed expressions are dedued. In the Navier ase, one has existene of preisely two solutions for boundary data below a suitable threshold, preisely one solution on the threshold and no solution beyond the threshold. This effet reflets that we have a bending point in the orresponding bifuration diagram and is not due to that we restrit ourselves to graphs. Under Dirihlet boundary onditions we always have existene of preisely one symmetri solution. In the seond part, we onsider boundary value problems with nonsymmetri data. Solutions are onstruted by rotating and resaling suitable parts of the graph of an expliit symmetri solution. One basi observation for the symmetri ase an already be found in Euler s work. It is one goal of the present paper to make Euler s observation more aessible and to develop it under the point of view of boundary value problems. Moreover, general existene results are proved. Introdution Reently, the Willmore funtional and the assoiate L gradient flow, the so alled Willmore flow, have attrated a lot of attention. Given a smooth immersed surfae f : M R 3, the Willmore funtional is defined by Wf := H da, fm where H = κ + κ / denotes the mean urvature of fm. Apart from being of geometri interest, the funtional W is a model for the elasti energy of thin shells or biologial membranes. Furthermore, it is used in image proessing for problems of surfae restoration and image inpainting. In these appliations one is usually onerned with minima, or more generally with ritial points of the Willmore funtional. It is well known that the orresponding surfae Γ has to satisfy the Willmore equation H + HH K = on Γ, Klaus.Dekelnik@mathematik.uni-magdeburg.de Hans-Christoph.Grunau@mathematik.uni-magdeburg.de

2 where denotes the Laplae Beltrami operator on Γ and K its Gauß urvature. A solution of is alled a Willmore surfae. Existene of losed Willmore surfaes of presribed genus has been proved by Simon [Sn] and Bauer & Kuwert [BK]. Also, loal and global existene results for the Willmore flow of losed surfaes are available, see e.g. [KS, KS, KS3, St]. The Willmore flow for one dimensional losed urves was studied by [P, DKS]. If one is interested in surfaes with boundaries, then appropriate boundary onditions have to be added to. Sine this equation is of fourth order one requires two sets of onditions and a disussion of possible hoies an be found in [Nit] along with orresponding existene results. These results, however, are based on perturbation arguments and hene require severe smallness onditions on the data, whih are by no means expliit. Thus the question arises whether it is possible to speify more general onditions on the boundary data that will guarantee the existene of a solution to. Suh a task seems to be quite diffiult sine the problem is highly nonlinear and in addition laks a maximum priniple. In order to gain some insight it is natural to look at the one dimensional ase first, where in some situations, almost expliit solutions an be found for suitable boundary value problems. Critial points of the total squared urvature funtional Γ κ ds are alled elasti urves and the analogue of reads κ ss + κ3 = on Γ. Here, s denotes arlength of Γ. In view of the saling properties of the total squared urvature funtional one often onsiders Γ κ +λds, at least in the ase of losed urves leading to an additional term λκ in. It is possible to desribe the solutions of in terms of ellipti integrals. Many results and losed parametri expressions for solutions an already be found in the Additamentum, De Curvis Elastiis in Vol. of the first series of Euler s Opera omnia [E, pp. 3 97], f. also [Nit, pp ]. More reently, Langer and Singer [LS] gave expliit desriptions for losed urves in manifolds of onstant setional urvature. Formulae for non losed urves in the plane are derived in [LI]. Depending on the onditions presribed at the fixed endpoints, a nonlinear system of up to three equations needs to be solved in order to determine the parameters that appear in the ellipti integrals. In the above papers the solutions are given in terms of arlength parametrizations on a priori unknown parameter domains. In the present work we take a different point of view: the urves under onsideration are given as graphs over a fixed domain whih we take for simpliity as the unit interval [,]. In this ase the Willmore funtional beomes Wu = κx dsx = κx + u x dx, 3 where graphu κx = d dx u x + u x = u x + u x 3/ is the urvature of the graph of u at the point x,ux. The Willmore equation now takes the form d κ x + + u x dx + u x κ3 x =, x,, 5 see Lemma below. We shall fous on two different sets of boundary onditions. Firstly, for given values α,α R we shall onsider Navier boundary onditions u = u =, κ = α, κ = α. 6

3 In the first part of the paper we onfine ourselves to the ase of symmetri boundary onditions α = α = α. Note that ritial points of 3 in H, H, will satisfy κ = κ = as natural boundary onditions. In order to obtain the orresponding property for nonzero α one has to replae W by the funtional W α u = graphu κx + ακx dsx = κx + ακx + u x dx, 7 see Corollary below. As a seond set of boundary onditions we shall examine Dirihlet boundary onditions u = u =, u = β, u = β, 8 where β and β are real parameters. Again, we first fous on the symmetri situation, while nonsymmetri solutions are disussed in Set. 6. Note that due to the presene of the length element + u x the nie struture of is lost. Nevertheless it is still possible to obtain formulae for the solutions of the above boundary value problems. The key observation in deriving these expressions is that if u is a solution of 5, then the auxiliary funtion vx := κx + u x / satisfies a seond order differential equation of the form axv x + bxv x =. 9 Here, the oeffiients ax, bx depend on the solution u. A similar observation was already made by Euler, see [E, p. 3, line 3]. In the ase of a symmetri solution we shall be able to onlude that v is a onstant, more preisely we have the following result: Lemma. Let u C [,] be a funtion being symmetri around x = / and define := dτ = B R + τ 5/, 3 = π Γ3/ Γ5/ = Then u solves the Willmore equation 5 iff there exists a onstant, suh that x [,] : κx + u x / =. Having found the above representation we an then examine the abovementioned boundary value problems. Solvability of the Navier boundary value problem in the ase of symmetri data strongly depends on whether the boundary datum α is below a threshold α max or not. Theorem. There exists α max = suh that for < α < α max, the Navier boundary value problem d κ x + + u x dx + u x κ3 x =, x,, u = u =, κ = κ = α has preisely two smooth graph solutions u in the lass of smooth funtions that are symmetri around x =. If α = α max one has preisely one suh solution, for α = one only has the trivial solution and no suh solutions exist for α > α max. 3

4 u x.3 u x.3.5. u x Figure : Solutions of the Navier boundary value problem for α =., α = and α =.3 left to right Figure : Bifuration diagram for : The extremals value of the solution u/ left and of the derivative u right plotted over α The small solutions are ordered with respet to α while the large ones beome smaller for inreasing α, see Figure. For the bifuration diagram, see Figure. For the Dirihlet problem with symmetri data the situation is somehow surprisingly simpler: Theorem. For every β R, the Dirihlet boundary value problem + u x d dx κ x + u x + κ3 x =, x,, u = u =, u = u = β has preisely one smooth graph solution u in the lass of smooth funtions that are symmetri around x =. This solution is the unique minimum of the Willmore funtional in the lass M β := {v H, H, v = v = β}. The solutions are ordered with respet to β, f. Lemma 5. This means that we have a omparison priniple for the Dirihlet problem. For the bifuration diagram, see Figure 3. Looking for orresponding results for nonsymmetri boundary value problems is more involved and less expliit than in the symmetri situation, although the expliit formulae from the latter

5 Figure 3: Bifuration diagram for : The extremal value of the solution u/ plotted over β ase are intensively exploited. For the formulation and proofs of suh results, whih we think are nevertheless somehow exhaustive, we refer to 6. This paper is organized as follows. In we shall onsider the Euler Lagrange equations for W and W α. 3 is devoted to the derivation of 9 and to provide a proof of Lemma. This result is subsequently used in to obtain a representation for the solution u itself from whih we an easily dedue several qualitative properties. In 5, we present the proofs of Theorems and. Finally, 6 is onerned with nonsymmetri boundary value problems. Euler-Lagrange equation For the reader s onveniene, we alulate the first variation of the Willmore funtional. Lemma. Let u C [,] and κ denote the orresponding urvature. Then, for all ϕ C,, we have d dt Wu + tϕ t= = d κ x + κ 3 x ϕx dx. 3 + u x dx + u x Proof. We assume only that ϕ = ϕ =. d dt Wu + tϕ t= = d u x + tϕ x dt + u x + tϕ x 5/ dx ϕ x = κx + u x dx 5 κx u x xdx + u x ϕ [ κxϕ ] x = + u x κ ϕ x x + u x dx + κxϕ x u xu x + u x +5 κx 3 ϕxdx + t= κxκ u x x + u x ϕxdx 5

6 [ κxϕ x = + u x +5 [ κxϕ x = + u x = +8 ] + κ x u xu x + u x ϕx κx 3 ϕxdx + ] + κ x ϕx + u x + u x + u x 3/ + κxκ u x x + u x ϕxdx κ x ϕxdx + u x + u x κx ϕ u x x + u x dx κxκ u x x + u x ϕxdx 8 κxκ u x x + u x ϕxdx [ κxϕ x + u x κx 3 ϕxdx + 5 κx 3 ϕxdx ] κ x + ϕxdx + + u x + u x κx 3 ϕxdx. Corollary Let u C [,] H, and κ denote the orresponding urvature. We assume that u is a ritial point of the modified Willmore funtional Wα for some fixed α R. So for all ϕ C [,] with ϕ = ϕ = one has d dt W α u + tϕ t= =. Then u is a solution of the Willmore equation 5, subjet to the Navier boundary onditions: u = u =, κ = κ = α. Proof. It remains to alulate d u x + tϕ x dt + u x + tϕ x dx t= = Combining this result with Lemma, we obtain ϕ x + u x xu xϕ [ x ϕ x u + u x dx = + u x ]. = d dt W α u + tϕ t= [ κx + αϕ x = + u x ] + κ x ϕxdx + + u x + u x κx 3 ϕxdx. Taking first arbitrary ϕ C,, we see that u solves the Euler Lagrange equation 5 and that for all ϕ C [,] with ϕ = ϕ = : [ κx + αϕ ] x + u x =. This implies κ = κ = α. 6

7 3 The differential equation for the auxiliary funtion v We assume that u C [,] solves the Willmore equation 5 and reall the definition of the fundamental auxiliary funtion vx := κx + u x /. The ruial point is that v satisfies a seond order differential equation without term of order zero, see also [E, pp. 33 3]. Lemma 3. For x [,] we have: d + u x 3/ v x + κxu x dx + u x /v x =. Proof. By straightforward alulations we obtain: v x = κ x + u x / + κx u x + u x 3/ ; + u x 3/ v κ x x = + u x + κx u x. By making use of the Willmore equation 5 we onlude: d + u x 3/ v x = d κ x + dx dx + u x κx u x + κxκ xu x = κx3 u x + u x + κxκ xu x = κxu x + u x /v x. Corollary. By the preeding lemma we know that v satisfies a maximum priniple. In partiular we may onlude for any solution u of the Willmore equation 5: If κ,κ <, then κ < in [,]. If κ = κ =, then κ = in [,] and the solution u is a straight line segment. If we additionally assume u = u = then ux in [,]. That means that we have uniqueness for the homogeneous Navier boundary value problem in the lass of smooth graphs without assuming a priori any smallness on the solution. Proof of Lemma. Let again vx := κx + u x /. In order to prove neessity of ondition, we observe first that v = v by our symmetry assumption on u. Sine v solves a seond order linearized differential equation without term of order zero, we onlude that there exists R suh that x [,] : vx =. The additional statement on the admissible range, follows from Lemma below. For proving suffiieny, we start with κx = + u x / 7

8 and obtain by differentiating κ x = κxu x + u x / = κx u x + u x ; κ x = u κ x xκx + u x + u x + u x u x κx + u x = u x κx 3 κx3 + u x = κx3 so that the Willmore equation 5 is satisfied. Expliit form of symmetri solutions of the Willmore equation In what follows, the funtion G : R, s, Gs := dτ 5 + τ 5/ plays a ruial role. It is straightforward to see that G is stritly inreasing, bijetive with G s >. So, also the inverse funtion G :, R 6 is stritly inreasing, bijetive and smooth with G =. Lemma. Let u C [,] be a funtion symmetri around x = /. Then u solves the Willmore equation 5 iff there exists, suh that x [,] : u x = G x. 7 For the urvature, one has that κx = + G. 8 x Moreover, if we additionally assume that u = u =, then one has ux = + G x + G. 9 Proof. First we show the neessity of the representation formula 7. Let u C [,] be a symmetri solution of 5. By Lemma we know that there exists a onstant R suh that for all x [,]: u x = + u x 5/. Integration yields x x = / u ξ dξ = + u ξ 5/ 8 u x dτ + τ 5/ = Gu x.

9 Now it beomes apparent that neessarily /, / GR so that < dτ R = +τ 5/. The suffiieny of 7 is obtained by diret alulation and with the help of Lemma. Formula 8 follows by differentiating 7, while for 9 we perform several hanges of variables: x ux = G s ds = / G σdσ = G / G / x / x t + t 5/ dt = + G x + G The expliit formulae of the preeding lemma allow for preise statements on the qualitative behaviour of solutions: Corollary 3. Let u be a smooth solution in [,] of the Willmore equation 5, being symmetri around x = /. For any nononstant solution we have that κ is of fixed sign, x, : κx > κ, x u x is either stritly dereasing or stritly inreasing. The following bound for u is independent of and is valid for all symmetri solutions of the boundary value problems,, independent of the data α or β: x [,] : ux < π = Γ3/ = Proof. Only the last statement requires a proof. It suffies to show that for, u = + G <. By hanging the variable = Gd, this is equivalent to showing for all d, that Hd := Gd + + d / >.. We have H =, lim d Hd = and H d = d + d 5/. This shows that H is inreasing first and then dereasing so that d, : Hd >. Moreover, from Lemma, we obtain that the smooth symmetri solutions are ordered with respet to, and hene with respet to β = u = G / R: Lemma 5. For the solutions u = u in Lemma, we have that for, x, : u x >. Proof. We obtain from the proof of Lemma x ux = G s ds = x s + G 5/ s ds >, sine the integrand is odd with respet to s = /, positive first and negative then. 9

10 5 Boundary value problems The Navier boundary value problem We obtain all smooth solutions u = u to 5 being symmetri around / and satisfying u = u = by formulae 9, 7, 8. This family is parametrised over,. It remains to onsider the dependene of α = κ = + G on. For this purpose, it is enough to study the funtion h :, R, h = + G. The range of h is preisely the set of α, for whih the Navier boundary value problem has a solution. The number of solutions of the equation α = h is the number of symmetri solutions of the boundary value problem. Lemma 6. We have h > in,, h < in,, lim ր h = lim ց h =. The funtion h is odd and has preisely one loal maximum in max = and one loal minimum in min = max. The orresponding value is α max = h max = Proof. First of all we observe that lim ր h = lim ց h = by definition of G = /. Seondly we alulate h = + G + lim ր G G = h = G + G 5/ < in,, 8, so that h is stritly onvex in, and stritly onave in,. This shows that there exists preisely one loal maximum max in, and preisely one loal minimum min = max in,. These are determined as zeroes of : hene h max = h min =, max = min =.88..., α max = h max = Thanks to onvexity and onavity we see that every number α α max,α max \ {} has preisely two preimages under h. Remark. From the previous alulations we see that for the extremal parameter α max the boundary slope of the orresponding solution is u = u = G max / =

11 .5.5 Figure : Admissible values of the boundary datum α plotted over the parameter Energy of small and large solution We onsider α suh that α α max and determine the orresponding values of desribing the solutions u to aording to α = + G. Then the modified Willmore energy an be easily alulated as: W α u = α artan G. 3 In Figure 5 left, we display the Willmore energy of the small and the large solution as a funtion of the boundary datum α Figure 5: Energy of small and large solution of the Navier boundary value problem plotted over α left and energy of the funtions u for α =. plotted over [, right Looking at the Willmore energy 3 from a different point of view exhibits a somewhat unexpeted feature: Let us onsider some α < α max relatively lose to α max and keep this α fixed. Then it turns out that even in the family {u, [, }, the energy of the small solution with boundary datum α is not a global minimum. The infimum is approahed by W α u when ր and hene, it is not attained. See Figure 5 right for α =..

12 The Dirihlet boundary value problem The first part of Theorem follows from 7: u = u = G, the strit monotoniity and ontinuity of G and from G /, / = R. Next, let v M β be arbitrary. Realling we have Wv Wu = = = = κ v x + v x dx κ u x + u x dx κ v x + v x κ u x + u x dx v x dx + v x 5 κ v x + v x κu x + u x dx Gv Gv κ v x + v x κu x + u x dx sine Gv Gv = Gβ =. From this we infer Wv > Wu unless v u. Open Problem. Can one show that suitable symmetry of Navier or Dirihlet boundary data implies symmetry of the solution? 6 Nonsymmetri boundary value problems Navier boundary onditions As a starting point we take the large expliit solution of the Navier problem whih we obtain formally when we let ր, i.e. α ց : U x = + G x. One should observe that this solution is no longer smooth as a graph near x = and x =, and for this reason, it was not inluded in Theorem. However, as a urve in R, it smoothly extends to the points, and, as a solution of the Navier problem with α =. Moreover, for its ontinuous odd extension see Figure 6 + G x if x, U x = + G + x if x, 5 if x {,,} we an prove: Lemma 7. The urve [,] x x,u x R is a smooth urve and a ritial point of the Willmore funtional 7 with α =.

13 Proof. Only the points,,,,, have to be onsidered. By symmetry it is enough to study the urve in a neighbourhood of,. Close to x =, the funtion x U x is ontinuous and stritly monotone and hene has an inverse funtion y V y := U y for y lose to, the expliit form of whih ist given by G if y > y V y = + G if y < y 6 if y =. By ontinuity, lim x U x = and oddness, V C is obvious. For y lose to, we alulate V y = y y, 7 from whih it is immediate that V C in a neighbourhood of y = Figure 6: Graph of U In order to onstrut solutions to the Navier boundary value problem 5, 6 we selet two points < x < x < and take the line through x,u x and x,u x as the new x-axis, the new y-axis being orthogonal. In this oordinate system we obtain a solution of the Navier boundary value problem on an interval of length Lx,x := x x + U x U x, 8 3

14 whih takes on -boundary values for the solution itself and κ x and κ x 9 for the urvature. Here, the urvature funtion κ aording to U is given by + G x. if x, κ x = + G + x. if x, = U x. 3 if x {,,} In order to find out whether we have obtained a smooth graph, we have to hek whether the angle between the new x-axis and the tangent vetor of the urve when passing from the left to the right lies in π/,π/, i.e. whether for x [x,x ], one has x x + U xu x U x >. 3 If x < < x, this is only a ondition on those points x with U x <, beause in this ase U x U x >. In these regions, we know from Corollary 3 that U is inreasing resp. dereasing. If x,x have the same sign, then U is dereasing resp. inreasing on the whole interval [x,x ]. Hene, it is enough to hek whether x x + U x j U x U x >, j =, 3 is satisfied. In view of 7 and 3 and sine U is an even funtion, this ondition is equivalent to x x + G x j κ x κ x >, j =,. 33 We emphasize that for some suitable δ >, this ondition is obviously satisfied for δ < x < x < + δ. In what follows, we exlude the symmetri ase U x = U x, for whih we may refer to Theorems and. For points x,x subjet to 3 the funtion x x x Lx,x x x + U x U x U x U x, x x x Lx,x is stritly inreasing and hene has an inverse funtion φ : [,Lx,x ] [x,x ]. It is not diffiult to verify that in the rotated oordinate system graph U [x,x ] is given by Lx,x x x Ũy = φy x + U x U x U x U x y, y Lx,x. Resaling to the interval [,] by setting ux = ux = Lx,x Ũ Lx,x x we then find that x φ Lx,x x + x x x, x [,]. U x U x

15 Figure 7: Admissible pairs of boundary urvatures in and for smooth graph solutions left and for Willmore urves right obtained by the desribed proedure and refletions solves 5. In partiular we have κ u = Lx,x κũ = Lx,x κ x, κ u = Lx,x κ x so that, employing also refletions, we an realize the following urvatures on the interval [, ] as Navier boundary data for graph solutions of the Willmore equation } C := C {a,a : a,a C } } 3 {a,a : a, a C {a,a : a, a C where C := { } κ x Lx,x,κ x Lx,x : < x < x < ; ondition 3 is satisfied. Theorem 3. Let C R be defined aording to 3. Then for every α,α C = C, the Navier boundary value problem d κ x + + u x dx + u x κ3 x =, x,, 35 u =, u =, κ = α, κ = α has a smooth graph solution. The set C is plotted on the left of Fig. 7, while those resulting pairs of urvatures, where we drop the ondition on the urve being a graph is plotted on the right of this figure. In Figure 8 we display the solution left obtained by the desribed proedure when hoosing x = / and x = /. On the right its urvature is displayed. In Figure 9 we display the solutions left obtained by the desribed proedure when hoosing x =, x =.7 and x =, x = On the right their urvatures are shown. Open Problem. We think that stable solutions are found among smooth graphs and that the set of admissible boundary urvatures, for whih the Navier boundary value problem 5, 6 has a stable solution, oinides with C. This onjeture is strongly supported by numerial evidene. 5

16 Figure 8: Graph left and urvature right of the solution obtained by hoosing x = / and x = / Figure 9: Graph left and urvature right of the solutions obtained by hoosing x =, x =.7 and x =, x = Remark. As a Willmore urve, U may be extended further by adding an odd refletion at the left or right end of the interval. Again, the above proedure may be applied to obtain Willmore urves, satisfying ertain boundary onditions. These solutions, however, are no longer smooth graphs. Sine in this proedure, the length in 8 may beome arbitrarily large while we ahieve the same urvatures as above, the Navier boundary data may beome arbitrarily large. Dirihlet boundary onditions Here, similar as in Theorem, we have an existene result without restritions on the data. The proof, however, is not ompletely onstrutive. Theorem. For every β,β R, the Dirihlet boundary value problem d + u x dx κ x + + u x κ3 x =, x,, 36 u =, u =, u = β, u = β for the Willmore equation has a smooth graph solution. 6

17 Proof. The key observation is that the derivative of a solution is the artan of the enlosed angle, whih is invariant under rotation and under the resaling u L x := Lu L x, whih transforms solutions of the Willmore equation into solutions and whih we always employed in this setion in order to resale solutions to the unit interval. In what follows, we develop the method from above introduing x,x and orresponding rotated and resaled parts of the graph of U over the interseting straight line from x,u x to x,u x. For our purposes, it is onvenient to extend U oddly around the point, to be defined on the interval [,]. We first look at the ase where β,β. In what follows we onsider angles between the graph of U and ertain straight lines interseting in x,u x and x,u x with the onvention that in both points angles in π,π are ounted positive, if the graph of U is above the straight line in a right neighbourhood of x,u x and a left neighbourhood of x,u x. We proeed in two steps and start by proving that we an always solve for extreme values of the boundary data. The proof of all laims below is based upon ontinuity and the mean value theorem. Claim. For given γ [,π/], we an find / < x < x suh that the angle is γ in x,u x and π/ in x,u x. We start with x = and draw the straight line perpendiular to the graph of U, whih intersets U again in x = under the angle π/. Now we derease x, onsider the straight line through x,u x perpendiular to the graph of U and look for the largest intersetion point x < x with U. Before x reahes /, this will no longer exist. So, there will be a limiting situation ˆx < ˆx where the left angle is and the right angle is π/. For the intermediate situations, all left angles between and π/ are attained. See Figure Figure : Left angle... π/, right angle π/ Claim. For given γ,γ,π/, we an find / < x < x < 3/ suh that the angle is γ in x,u x and γ in x,u x. 7

18 Here, aording to our first laim, we start with x < x suh that the angle on the left is γ and on the right π/. Now we inrease x and keep the left angle γ between the graph of U and the interseting straight line fixed. Sine for large enough x, this straight line will beome perpendiular, there is a limiting situation ˆx < ˆx, where the left angle is still γ and where the angle in the intersetion point on the right has dereased to. As before, the laim follows by virtue of the mean value theorem. See Figure Figure : Left angle γ, right angle... π/ We briefly omment on the remaining ases for β,β. When both are negative, we simply reflet the previously obtained solutions with respet to the x-axis. If β < β, we perform a similar proedure as above. We start within the first laim with x = and subsequently obtain x < x, where a presribed negative angle is attained in the left intersetion point and where we have a right angle in the intersetion point on the right. See also Figure. Then, analogously to Claim, we keep the angle in the left point fixed and inrease x. The angle in the right intersetion point will finally derease to. In the ase that β < β, the previous ase and refletion with respet to the u-axis yield the laim. Aknowledgement. The seond author is grateful to G. Huisken Max-Plank-Institute Golm for suggesting to study boundary value problems for the Willmore equation. Moreover, we are grateful to the referee for a very areful reading of the manusript. Referenes [BK] [DD] [DKS] M. Bauer, E. Kuwert, Existene of minimizing Willmore surfae of presribed genus, Int. Math. Res. Not. 3, No., K. Dekelnik, G. Dziuk, Error analysis of a finite element method for the Willmore flow of graphs, Interfaes Free Bound. 8, G. Dziuk, E. Kuwert, R. Shätzle, Evolution of elasti urves in R n : Existene and omputation, SIAM J. Math. Anal. 33,

19 [E] L. Euler, Opera Omnia, Ser.,, Zürih: Orell Füssli, 95. [KS] [KS] [KS3] [LS] E. Kuwert, R. Shätzle, The Willmore flow with small initial energy, J. Differ. Geom. 57, 9-. E. Kuwert, R. Shätzle, Gradient flow for the Willmore funtional, Commun. Anal. Geom., E. Kuwert, R. Shätzle, Removability of point singularities of Willmore surfaes, Annals Math. 6, J. Langer, D.A. Singer, The total squared urvature of losed urves, J. Differ. Geom., [LI] A. Linnér, Expliit elasti urves, Ann. Global Anal. Geom. 6, [Nit] [MS] [P] [Sn] J.C.C. Nitshe, Boundary value problems for variational integrals involving surfae urvatures, Quarterly Appl. Math. 5, U.F. Mayer, G. Simonett, A numerial sheme for axisymmetri solutions of urvaturedriven free boundary problems, with appliations to the Willmore flow, Interfaes Free Bound., A. Polden, Curves and Surfaes of Least Total Curvature and Fourth-Order Flows, Ph.D. dissertation, University of Tübingen, 996. L. Simon, Existene of surfaes minimizing the Willmore funtional, Commun. Anal. Geom., [St] G. Simonett, The Willmore flow near spheres, Differ. Integral Equ., 5-. [W] T.J. Willmore, Total urvature in Riemannian geometry, Ellis Horwood Series in Mathematis and its Appliations, Ellis Horwood Limited & Halsted Press: Chihester, New York et