Evolution equations for polynomials and rational functions which are conformal on the unit disk

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(1)Evolution equations for polynomials and rational functions which are conformal on the unit disk Citation for published version (APA): Graaf, de, J. (1999). Evolution equations for polynomials and rational functions which are conformal on the unit disk. (RANA : reports on applied and numerical analysis; Vol. 9939). Technische Universiteit Eindhoven.. Document status and date: Published: 01/01/1999 Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement: www.tue.nl/taverne. Take down policy If you believe that this document breaches copyright please contact us at: openaccess@tue.nl providing details and we will investigate your claim.. Download date: 10. Sep. 2021.

(2) Evolution equations for polynomials and rational functions which are conformal on the unit disk J. de Graaf. Abstract On the unit disk D in the complex plane C two evolution equations for con-. formal mappings (z t) z 2 D t 0 are studied: The quasi-linear LownerKufarev equation and the quasi-linear Hopper equation. The rst one has 'Hamiltonian', say, f , the second one F : The L-K-equation has the property that for any initial condition 0(z) which is conformal on D, the solution (z t) remains conformal as long as it exists (Section 1). The H-equation has the property that for any initial condition 0(z) which is polynomial/rational on D, the solution (z t) remains polynomial/rational as long as it exists (Section 2). We nd conditions on the pair of Hamiltonians ff F g, such that both the L-K- and the H-equations describe one and the same evolution phenomenon. This implies that both the properties of being conformal and of being polynomial/rational persist (Section 3). We show that 'compatible pairs' ff F g are not rare. They can both be found in physics and be 'arti cially' constructed. In this paper the emphasis is on algebraic properties, although analysis cannot be avoided altogether. October 1999 AMS Subject Classi cation: 26C99, 30C20, 47H20. Keywords: Lowner-Kufarev equation, Hopper equation, conformal polynomials, conformal rational functions.. Home institution:. Department of Mathematics and Computing Science Eindhoven University of Technology P.O. Box 513 5600 MB Eindhoven The Netherlands j.d.graaf@tue.nl.

(3) 1 The (quasi-linear) Lowner-Kufarev equation. On the unit disk D C we rst consider the linear Lowner-Kufarev initial value problem (. @ @t (z t). = f (z t) z @z@ (z t) (z 0) = 0(z). t 2 I R z 2 D C. (1). with analytic initial condition 0(z). The apriori given coecient function f is supposed to be analytic in z and continous in t. This linear- LK-equation plays an important role in 'pure' complex analysis. See, for example, Pomerenke 5]. By applying the method of characteristics the solution of the initial value problem (1) can locally be written (z t) = 0(' (z t)): Here ' denotes the inverse of z 7! '(z t), which is the solution of the initial value problem ( '_ (z t) = ;f ('(z t) t)'(z t) '(z 0) = z: (The dot _ denotes dierentiation with respect to t ). To prove this we calculate (write 0 instead of @z@ ) 0 = ddt ' ('( t) t) = ' _('( t) t) + ' ('( t) t)'_ ( t) = ' _('( t) t) ; f ('( t))'( t)' ('( t) t): 0. 0. With = ' (z t) this reads. @ ' (z t) = f (z t) z @ ' (z t): @t @z Therefore (z t) = 0(' (z t)) locally solves the initial value problem (1).. From the theory of ordinary dierential equations we gather the following Properties 1.1 Suppose that 8t 0 the mapping z 7! f (z t) is analytic on an open set Ut D. I. 8t 0 the mapping z 7! '(z t) is an analytic bijection from an open set V0t U0 to an open set Vt Ut. The sets V0t Vt are a maximal pair and both contain 0. Also '(0 t) = 0: II. If U0 D, then for t 0, suciently small, we have V0t D and Vt D. III. If 0: V0t ! C is conformal then (z t) = 0(' (z t)) is a conformal map on Vt. In 4] a quasi-linear version of the LK-equation has been introduced for modelling the behaviour of drops of viscous uids with surface tension. The function f may now depend on in a 'functional' way and we obtain the Quasi-linear Lowner-Kufarev equation. @ (z t) = f (z) z @ (z t) t 2 I R z 2 D C: (2) (t) @t @z It is assumed that for every conformal : D ! PC the mapping 7! f is well de ned and f : D ! C is analytic. Notation f (t)(z) = 1j=0 f j (t)zj . With the Ansatz (z t) = 2.

(4) P1. k=0 !k (t)z. equations. 2 66 66 d 66 d t 666 64. k. the Ql-L-K-equation (2) becomes an in nite system of ordinary dierential. !1 !2 !3. 3 2 77 66 77 66 77 66 77 = 66 77 66 5 4. 32. f 0(t) f 1(t) f 2(t). 3. 0 0 0 0 7 6 1 !1 7 f 0(t) 0 0 0 77 66 2 !2 77 f 1(t) f 0(t) 0 0 77 66 3 !3 77 77 66 .. 77 : (3) ... ... ... ... . . . 77 66 . 77 !n f n;1 (t) f n;2 (t) f 0(t) 75 64 n !n 75 ... ... ... ... ... ... ... ... ... For some local and global existence results of this problem see 4]. Note that in the classical problem (1) with f NOT a rational function of z, a polynomial/rational initial condition will NEVER give a solution of that type. Even if f is rational this will rarely happen. In our quasi-linear case however, we have straightforwardly Theorem 1.2 Consider the initial value problem 8 @ > < @t (z t) = f (t)(z ) z @z@ (z t) t 2 I R z 2 D C (4) > : (z 0) = 0(z ) = PN an z n n 2 N: n=1 Suppose that the solution (z t) exists for z 2 D and t 2 0 T ). I. 8t 2 0 T )8N 2 N the solution z 7! (z t) is a polynomial of degree N i 1 (! )z + !2 2 (! )z 2 + + !N N (! )z N + : (5) 7! f (z) = !1z + 2!2 z2 + + N!N zN + j Here (z) = P1 j =1 !j z and ! = (!1 !2 ): II. (z t) = PNk=1 ak (t)zk solves (4) i 8 d a1 (t) > = 1(a1(t) aN (t) 0 0 ) > dt > < d ad2t(t) = 2 (a1(t) aN (t) 0 0 ) (6) ... ... ... > > > : d aN (t) = N (a1(t) aN (t) 0 0 ): dt Condition (5) is not easy to verify. The dynamics of a Stokes drop as studied in 3] however, has to be of this form. A similar but more complicated condition could be given in order that rational solutions persist. For the purpose of this paper the main attraction of the quasi-linear Lowner-Kufarev equation lies in the following Theorem 1.3 Consider the initial value problem 8 @ > < @t (z t) = f (t)(z ) z @z@ (z t) t 2 I R z 2 D C (7) > : (z 0) = 0(z ) is conformal on D: Suppose (z t) for z 2 D and 0 t < T: Then 8t 2 0 T ) the mapping z 7! (z t) is conformal on D: Proof The proof is reduced to Properties 1.1 by taking there f (z t) = f (t)(z) with our special solution (z t) substituted. 2 3.

(5) 2 The (quasi-linear) Hopper equation In 3] Hopper introduced an evolution equation which describes the behaviour of a drop of a Stokes uid driven by surface tension. The unknown function in Hopper's equation is a conformal map from the unit disk D C to the region occupied by the uid. For mathematical considerations on Hopper's equation see 2] and 1]. In this section we look at a slightly modi ed and more general version of this equation. First some notation. Notation 2.1 Let g be an analytic function on an open set W C. The analytic function gy is de ned by gy(z) = g( 1z ). Clearly z 2 W , the domain of gy, i z1 2 W . Observe the following. Properties 2.2. If @D W , then @D W and g y(z ) = g (z ) if jz j = 1. If g is analytic on D , then g y is analytic on Dc

(6) f1g. P1 1 k y If g (z ) = P1 k=;1 gk z then g (z ) = k=;1 g k zk : (z ddz g (z ))y = ;z ddz g y(z ) and (g 0)y(z ) = ;z 2 (g y)0(z ): R 2 1 g conformal on D ] ) 21i jzj=1 g 0 (z )g y(z ) dz = P1 k=1 k jg (k )j = Areag (D)] ]:. We now formulate our version of Hopper's initial value problem for the evolution of a conformal map ( t): D ! C. 8 @ 0 > (t)(z ) < @ t (z (z t)y(z t)) ; z @@z (F (t) (z )z 0(z t)y(z t)) = with t 2 I R z 2 D C: (8) > : (z 0) = 0(z):. Remarks 2.3. F is an analytic function on D. It 'regulates the dynamics' and may depend on in a functional way. In 3] Hopper takes F Rsuch that Re F (z) = 2j 1(z)j if jzj = 1 and Im F (0) = 0. Hence in his case F (z) = 21i jj=1 2j ()+jz(;z) d :. is required to be analytic on D. Therefore the singularities on the lefthand side of (8) have to cancel 8t 2 I . This gives 'the dynamics'. In 3] the constant term in the Tay0. 0. lorseries of vanishes. Division of both sides by z in this case leads to Hopper's original equation as introduced in 3]. Put (t)(z ) = (t) (z ) + 2zX 0 (t) (z ). Here is the constant term in the Taylor expansion for . For later convenience the remaining term is written with the derivative 1 the growth of a suitable analytic function X . Note that the functional (t) gives the R rate of the area of (D t). Indeed, evaluation of the contour integral 21i jzj=1 z dz of both sides of the equality (8) leads to ddt Area(D t)] = (t), use Properties 2.2. Because of this, has to be a R;valued functional. In 3] = 0 because of area conservation for incompressible Stokes uids. (Rotation symmetry) If for some constant

(7) 2 R it happens that Fei (z ) = F (z ) and also ei = for all , then ei(z t) is a solution if (z t) is. In 3] there is rotation symmetry. (Reection symmetry) Write e(z ) for (z ). If it happens that F e(z ) = F e(z ) and. e = for all , then e(z t) is a solution if (z t) is. Reection symmetry implies that 4.

(8) all Taylor coecients of F are real if all Taylor coecients of are real. In 3] there is reection symmetry. The function H (z ) = z 0 (z )y(z ) plays a prominent role in (8). In fact (8) can be considered as an evolution equation for the non-positive Laurent part of H . For the interesting problem 'how to recover from H ' see 1]. j j P1 Write F (t)(z) = P1 j =0 FPj (t)z = j =0 Fj (! )z , with ! = (!1 !2 !3 ). Substituk tion of the Ansatz (z t) = 1 k=1 !k (t)z , with !1 (t) > 0 and !j (t) 2 C j > 1, in the left hand side of (8) and demanding that the negative part of the Laurent series should vanish, turns the H-equation in to an in nite system of ordinary dierential equations which is now of upper diagonal type: 2 66 d 666 d t 66 4. 32. 3. !1 2!2 3!3 4!4 7 6 !1 7 0 !1 2!2 3!3 77 66 !2 77 0 0 !1 2!2 777 666 !3 777 = 0 0 0 !1 75 64 !4 75 .... ... ... 2 66 00 F 0(! ) 0 66 0 0 = ; 66 64 0 0 ... .... (9). ... ... ... 32 0 0 7 6 !1 F1 (! ) F2(! ) 77 66 0 2F0(!) 2F1(!) 77 66 0 0 3F0(!) 775 664 0 ... ... ... .... 2!2 3!3 !1 2!2 0 !1 0 0 ... .... 32. 3. 4!4 7 6 !1 7 2 (!) 3 3!3 77 66 !2 77 666 0 777 2!2 77 66 !3 77 + 66 0 77 !1 775 664 !4 775 64 0 75 ... ... ... 0. Theorem 2.4 Consider the initial value problem (8) with initial condition 0. I. If 0(z) = PNj=1 !j (0)zj , with !1P(0) > 0, then the solution(z t) remains a poly-. nomial of degree N : (z t) = Nj=1 !j (t)z j . Here the coecient functions !j (t) satisfy the system of ordinary dierential equations obtained from (9) by taking there !N +1(t) = !N +2 (t) = = 0. Note that we obtain in this way 1 real and (N ; 1) complex ordinary dierential equations. Note also that !1 (t) > 0 as long as the solution exists. II. If we take 0(z) = !1(0)z + !N (0)zN , then (z t) = !1(t)z + !N (t)zN . The system of ordinary dierential equations for !1 !N is obtained by equating to 0 in (9) all functions !j with 1 < j < N and j > N . III. If F has reection symmetry, as e.g. in 3], then a polynomial initial condition with real coecients leads to a polynomial solution with real coecients.. Proof All straightforward veri cations.. 2. Next we turn to the persistence of rational initial conditions. Hopper's equation (8) can be reformulated as

(9) z 00(z t) i h _ 0 t) y(z t) 0((z ; F (t)(z ) 1 + 0 (10) ; z F 0 (t)(z ) + z t) (z t) 0 + z1 (0)y(z t)F (t)(z) + _ y(z t) ; z(t) 0(1z t) = 2 X0((zz tt)) 5.

(10) or, equivalently, omitting the arguments of , X , and F , 00 i 0 h _0

(11) h i 0 y 0 ; zF 0 + _ y ; y zF ; z 10 = 2 X0 : (11) Note that the functions between ] are analytic on D. The requirement is again, that _ t) has to be such that the left hand side of (10) for given ( t) the time derivative ( extends to an analytic function on D. Thereby the evolution of is determined. Theorem 2.5 Consider the initial value problem (8) with the Ansatz N X (z t) = A0(t)z + z 1 ;A

(12) n(t()t)z (12) n n=0 P with N 2 N, A0(0) 2 R An (t) 2 C

(13) n (t) 2 C such that 0 (z 0) = A0(0)+ Nn=1 An(0) > 0 and j

(14) n (t)j < 1. I. If 6= 0 and A0(0) 6= 0 the initial value problem (8) is locally solved by (12). The A0 A1 AN

(15) 1

(16) N satisfy a coupled system of 1 real and 2N complex ordinary dierential equations. II. If = 0 as in 3], then a solution (12) exists with A0 identically zero. III. There exists a solution of the more general form (z t) = c1(t)z + + cN. (t)zN. +. (m) M NX X. m=1 n=1. n cmn (t) (1 ; z (t)z)n :. Also in this general case the t-dependent coecients dierential equations. Proof (sketch) I. Calculate 0(z) = A0 + PNn=1 (1;Annz)2 _ 0(z) = A_0 + PNn=1 (1;A_ nnz)2 + PNn=1 (12;_ nAnnz)z3 _ y(z) = A_z0 + PNn=1 zA;_ n n + PNn=1 (zA;n_nn)2 :. m. (13). satisfy a system of ordinary. 00(z) = PNn=1 (12;nnAzn)3 y(z) = Az0 + PNn=1 z;Ann . (14) Substitute (14) in (11) and equate to 0, respectively, the 1rst-order pole at z = 0, the 1rst-order poles at z =

(17) k 1 k N and the 2nd-order poles at z =

(18) k 1. k N . This leads to the following system of ordinary dierential equations where we employ the notations FA(z), A instead of F ,. PN d d t A0A0 + n=1 An ] = A h i (15) A_ k + Ak _ ((kk )) ;

(19) k FA(

(20) k ) ((kk)) = 0 1 k N

(21) _ k + FA(

(22) k )

(23) k = 0 1 k N: Because of the initial conditions PN and A being real it follows that the functions 0 A0(t) and (z t) = A0(t) + n=1 An(t) are real valued. The latter remains positive as long as remains conformal. 0. 00. 0. 0. 6.

(24) II. This is presisely the case dealt with hitherto in 4] and 1]. III. For the special case = 0 and c1 cN taken 0 see again 4] and 1].. 2. We observe that Hopper's equation has complete sets of polynomial and rational solutions. The natural question arises whether those solutions are conformal mappings on there existence interval. The answer to this question is yes if they are simultaneously solutions of a (quasi-linear) Lowner-Kafarev equation. This is the subject of the next section.. 3 From Lowner-Kufarev to Hopper Our starting point is the quasi-linear Lowner-Kufarev equation (2) with a given Hamiltonian 7! f : @ @ (16) @ t = f z @ z : We apply the y-operation to (16), cf. Properties 2.2, and multiply the result by 0, which gives _ y0 = ;(f y z(y)0)0. Add to this expression y_ 0 = y(f z0)0. This leads to a kind of 'balance equation for the area' y 0 y y0 0 @ 0 y @ (17) @ t ( ) = @ z (f z ) ; z ( ) ff + f g: By integration along the unit circle D t)] = d 1 Z (0y) dz = 1 Z 0y0 (f + f y ) dz: (18). = ddt Area(. d t 2 i jzj=1 2 i jzj=1 z 1R Remark 3.1 Because of Gauss' theorem, (18) should be equal to @ (Dt) Vn (s) ds = 1 R 2 V ((ei ))j0 ((ei ))j d: Here s denotes an arclength parametrization of @ (D t) 0 n and Vn is the 'normal velocity eld' at the boundary @ (D t). A surface motion law attaches to any domain (D) out of a given class, a normal vector eld Vn at its boundary @ (D), thereby determining the evolution of the domain. The above formulae strongly i )) V ( ( e n i suggest that the relation between f and Vn is given by Re f (e ) = j (ei )j . Cf.4]. In order to link (17) to Hopper's equation we introduce an -dependent analytic function F on D which will be speci ed later. With F we rewrite (17) as y 0 y 0 y 0 y 0y 0 @ @ @ (19) @ t (z ) ; z @ z (F z ) = z @ z (f ; F )z ] + (f + f ): We put (f ; F )z0 = ;2 and ;2z @@z y] + 0y0(f + f y ) ; = 2z @@z X . With these notations (19) becomes 0 y 0 y @ @ @ (20) @ t (z ) ; z @ z (F z ) = z @ z X + : This LOOKS like Hopper's equation. However it IS Hopper's equation ONLY if we have been able to nd 7! F such that X is ANALYTIC. In general this not possible because Hopper's equation always has polynomial/rational solutions which is not true for the Lowner-Kufarev equation. We don't know conditions on f such that all solutions of a Lowner-Kufarev equation with Hamiltonian f are also solutions of some Hopper equation with a suitable Hamiltonian F . In the next theorem we put conditions on the pair f X g, such that it leads to a 'compatible' pair ff F g. 0. 7.

(25) De nition 3.2 The pair ff F g is called a compatible pair if all solutions of a Lowner-. Kufarev equation with Hamiltonian f are also solutions of a Hopper equation with F as Hamiltonian and de ned by (18). Theorem 3.3 Consider two -dependent analytic functions X and on D with (0) = 0. If X (z) + y(z) (z) 2 iR for jz j = 1 (21) or, equivalently, 1 X (z) + (z) 2 iR for jz j = 1 (22) (z)y(z) (z) then for any given 2 R the functions f and F are well de ned by 8 < (f ; F )z0 = ;2 (23) : 2z @ (X + y ) 1 + = f + f y @z and the pair ff F g is a compatible pair. Proof Because of (21) the lefthand sidePof the 2nd equation in (23) is real valued if jz j = 1. Therefore it has a Laurent series, 1n=;1 n zn say, with ;n = n . Now just take P f = 12 0 + 1n=1 nzn . Next F can be obtained with the 1rst equation in (23). With these choices for f and F all solutions of (16) are solutions of (20). 2 Next we formulate a main consequence of the results of this paper Corollary 3.4 Consider the intial value problem for the quasi-linear Lowner-Kufarev equation (2),(16) with a Hamiltonian f which is the rst member of a compatible pair ff F g. For the initial condition 0 (z) we only consider functions which are analytic and injective (=conformal) on D. I. If, in addition, 0(z) is a polynomial, then the solution 0(z t) will remain a polynomial which is conformal on D for all t 2 0 T ) the whole existence interval. II. If, in addition, 0(z) is a rational function, then the solution 0(z t) will remain a rational function which is conformal on D for all t 2 0 T ) the whole existence interval. Example 3.5 If (D) = W is a domain in the complex ;plane and are analytic functions on W such that ((z)) = X (z) and ((z)) = (z), then condition (21) corresponds to ( ) + ( ) 2 iR for 2 @W: (24) This condition is, in fact, satis ed in 3]. A function of type (24) represents the general solution of Stokes' equations in 2 dimensions. Mathematical details to this are mentioned in 2],1]. Example 3.6 From conditions (21), (22) it follows that if one of the mappings 7! 7! X is freely chosen, then the other one can, in principle, be found: One has to solve a Dirichlet problem for nding its real part and then apply harmonic conjugation for nding its imaginary part (up to a real constant). An arti cial but elegant class of examples of this type is constructed by letting the domain (D) 'evolve in an external eld' given by a xed entire function ( ) with (0) = 0. Just de ne by (z) = ((z)). Then X follows in the above described way. Of course a similar game can be started with a xed entire function ( ) and taking X (z) = ((z)). 0y. 0. 8. 0y. 0.

(26) References 1] Anthonissen, M. J. H. and Graaf, J. de, Hopper's shape evolution equation. Research Report 20, Departamento de Matem"atica, Univeridade de Coimbra, Coimbra, Portugal, 1999. 2] Graaf, J. de, Mathematical addenda to Hopper's model of plane Stokes ow driven by capillarity on a free surface. In Geometric and quantum aspects of integrable systems (Scheveningen, 1992), 167{185. Springer, Berlin, 1993. 3] Hopper, R. W., Plane Stokes ow driven by capillarity on a free surface. J. Fluid Mech. 213 (1990), 349{375. 4] Klein Obbink, B., Moving boundary problems in relation with equations of LownerKufareev type. Technische Universiteit Eindhoven, Eindhoven, 1995. Dissertation, Technische Universiteit Eindhoven, Eindhoven, 1995. 5] Pommerenke, C., Boundary behaviour of conformal maps. Springer-Verlag, Berlin, 1992.. 9.

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