Lemniscates do not survive Laplacian growth

LEMNISCA TES DO NOT SUR VIVE LAPLACIAN GR O WTH D. KHA VINSON, M. MINEEV-WEINSTEIN, M. PUTINAR, AND R. TEODORESCU 1. Intr oduction Man y mo ving b oundary pro cesses in the p lane, e.g., solidification, electro dep osition, viscous fingering, bacterial gro wth, etc., can b e math- ematically mo deled b y the so-called Laplacian growth [9, 13]. In a nut- shell, it can b e describ ed b y the equation (1.1) V ( z ) = ∂ n g Ω( t ) ( z , ζ ) , where V is the normal component of the v elo cit y of the b oundary ∂ Ω( t ) of t he mo ving domain Ω( t ) ⊂ R 2 ≃ C , z ∈ ∂ Ω( t ), t is time, ∂ ∂ n denotes the normal deriv ativ e on ∂ Ω( t ) and g Ω( t ) ( z , ζ ) is the G reen function for the Lapla ce op erator in the domain Ω( t ) with a unit source at the p oin t ζ ∈ Ω( t ). Equation (1.1) can b e elegan tly r ewritten as the area- preserving diffeomorphism (1.2) ℑ ( ¯ z t z θ ) = 1 , where ℑ denotes the imaginary part of a complex n um b er, ∂ Ω( t ) := { z := z ( t, θ ) } is the moving b o undar y parametrized b y w = e iθ on t he unit circle and the conformal mapping fro m, sa y , the exterior o f the unit disk D + := {| w | > 1 } on to Ω( t ) with the normalization z ( ∞ ) = ζ , z ′ ( ∞ ) > 0. The equation (1.2), named L aplacian gr owth or the Polub arinova - Galin equation in mo dern literature, w a s first deriv ed by P olubarino v a- Ko c hina [11] and Ga lin [7 ] in 1945, as a description of secondary oil reco very pro cesses . This equation is known to b e in tegrable [10], and as suc h p ossesses an infinte num b er o f conserv ed quan tit ies. More precisely , it admits Part of this work was done during the first a nd third authors’ visit to LANL a nd was supp orted by the LDRD pro ject 2 0 0704 8 3 “Minimal Description of Complex 2D Shap es” at LANL. Also, D. Khavinson and M. Putinar gratefully acknowledge partial supp or t by the National Science F o undation. 1 2 KHA VINSO N, MINEEV- WEINSTEIN, PUTINAR , A ND TEODORESCU conserv ed momen ts c n = R Ω( t ) z n dx dy , where n runs o v er either all non- negativ e o r all non-p ositive in tegers dep ending on whether domains Ω( t ) a re finite or infinite. A t the same time (1.2) admits an impressiv e n umber of closed-form solutions. F or the back ground, history , generalizations, references, connections to the theory o f quadrature domains and other branche s of mathemat- ical ph ysics w e refer the reader to [4, 8–10, 12, 13] and the references therein. In section § 2 of this pa p er, we sho w that any con tin uo us c hain of p olynomial lemniscates of o rder n : Γ t := {| P ( z , t ) | = 1 } , P ( z , t ) = a ( t ) n Q j =1 ( z − λ j ( t )), where a ( t ) is real-v alued, is destro ye d instan tly under the Laplacian gro wth pro cess describ ed in (1.1), with Ω( t ) = {| P ( z , t ) | > 1 } , ζ = ∞ , unless n = 1, λ 1 ( t ) = const and { Γ t = ∂ Ω( t ) } is simply a family of concen tric circles. Here the ro ots λ j ( t ) of P ( z , t ) are all assumed to b e inside Ω ′ t := {| P ( z , t ) | < 1 } , so Ω and Ω ′ are simply connected. This res ult sho ws that unlik e quadrature domains (cf. [4, 12]) that are preserv ed under the La placian gro wth pro cess, lemniscates for whic h all the ro o ts of the defining p olynomial are in Ω ′ t are instantly destro ye d, except for the trivial case of concen tric circles . This, incide n tally , ag r ees with a well-kno wn fact — cf. [5] — that lemniscates whic h are also quadrature domains m ust b e circles. The pro of of the theorem for the case o f La placian gro wth is giv en in § 2. In § 3 w e extend the result of § 2 to all the g ro wth pro cesses that are in v aria nt under time-rev ersal and for whic h the b oundary v elo cit y is giv en by (1.3) V ( z ) = χ ( z ) ∂ n g Ω( t ) ( z , ζ ) , with χ ( z ) is a b o unded, real, p ositive function on Γ t . Inv aria nce under time-rev ersal is defined here in the following w ay : if the b o undary Γ t + dt is the image of Γ t under a map f ( t,dt ) : z t ∈ Γ t 7→ z t + dt ∈ Γ t + dt , then f ( t + dt, − dt ) ◦ f ( t,dt ) = I . W e conclude with a few remarks in § 4. 2. D estruction of Lemnisca te s Theorem 2.1. Supp ose that a fa mily of movin g b oundaries Γ t , ( w her e t > 0 is time), pr o duc e d by a L apla cian gr owth pr o c ess, is a fa m ily of p olynomial lemnisc ates {| P ( z , t ) | = 1 } , wher e P ( z , t ) = a ( t ) n Q j =1 [ z − λ j ( t )] , LEMNISCA TES DO NOT SUR VIVE LAPLACIAN GROWTH 3 and al l λ j ( t ) ar e a ssume d to b e inside Γ t . T hen, n = 1 and λ 1 = const , i.e., Γ t is a family o f c onc entric cir c l e s. Pr o of. Let Ω t = { z : | P ( z , t ) | > 1 } , D + = {| w | > 1 } . The function ϕ ( t ) : Ω t → D + , w = ϕ ( z , t ) = n p P ( z , t ), where w e c ho ose the branch for the n − th ro ot so that ϕ ′ ( t, ∞ ) > 0, maps Ω t conformally o n to D + , ϕ ( t, ∞ ) = ∞ . It is useful to note that on Γ t , P ( z , t ) = w n , | w | = 1 and do es not dep end on t . This is b ecause for an y tw o momen t s of time t, τ , w e hav e w ( t )( . ) = w ( τ ) ◦ κ ( t, τ )( . ) , where κ is a M¨ obius automorphism of the disk. In our case, κ ( t, τ )( ∞ ) = ∞ , so κ ( t, τ )( z ) = e iα z , α ∈ R , but since it a lso fixes the argumen t a t ∞ , κ is the iden t ity . Therefore, w e ha v e (where, as is customary , we denote the partia l t -deriv ativ e by a “dot” ) : (2.1) ˙ P + P ′ z ˙ z = 0 . Since ϕ ( t ) maps Γ t on to the unit circle, w e ha ve z ( t ) = Ψ( t, w ), where Ψ( t, w ) = ϕ − 1 ( t, z ). W e also ha v e on Γ t , by differen tiating P ( z ( w ) , t ) = w n with respect to w , (2.2) P ′ z · z w = nw n − 1 or (2.3) w z w = nw n P ′ z = nP P ′ z . F rom (2.1), conjugating, w e infer (2.4) ˙ z = − ˙ P P ′ z . P ara metrize the unit circle b y w = e iθ , 0 ≤ θ ≤ 2 π . Then, from (2.3), it f o llo ws tha t w e hav e on Γ t (since ( z ( t ) = z ( w , t ) = z ( w ( θ , t )))), (2.5) 1 i z θ := ∂ z i∂ θ = z w w = nP P ′ z . Com bining (2.4) and (2 .5) yields ( ℜ stands fo r the real pa r t ): (2.6) ℜ  ˙ z 1 i z θ  = ℜ − ˙ P P ′ z · nP P ′ z ! . Also, (2.7) ℜ  ˙ z ∂ z i∂ θ  = ℑ  ˙ z z θ  = 1 , 4 KHA VINSO N, MINEEV- WEINSTEIN, PUTINAR , A ND TEODORESCU where in the last equality w e used the h yp o t hesis that t he lemniscates Γ t := { z ( t, θ ) } satisfy the main equation (1.2 ) of Lapla cian growth pro cesses— cf. [8, § 4]. Hence, (2.7), (2.4) and (2 .5) imply that (2.8) 1 n ℜ ˙ P P ′ z iz θ ! = ℜ ˙ P P ′ z P P ′ z ! = − 1 n . Or, w e can rewrite (2.8) as (2.9) ℜ  ˙ P P  = − 1 n | P ′ z | 2 . Th us, we are finally arriving at (2.10) d dt  | P | 2  = − 1 2 n | P ′ z | 2 . Therefore, (2.10) holds on the lemniscates Γ t = {| P ( z , t ) | = 1 } that are assumed to b e in terfa ces of a Laplacian gro wth pro cess. Now the theorem follo ws from the follo wing. Lemma 2.1. L et t b e the time variable, P ( z , t ) = a ( t ) n Q 1 ( z − λ i ( t )) , b e a “flow” of n -de gr e e p olynomials. Assume that the le mnisc ates Γ t := {| P ( z , t ) | = 1 } al l h a ve c onne cte d interiors { | P ( z , t ) | < 1 } and a gener alize d e quation (2 .10) ho lds on Γ t ; i.e., (2.11) d dt  | P ( z , t ) | 2  − c ( t ) | P ′ z ( z , t ) | 2 = 0 , wher e the f unction c ( t ) is r e al- v alue d, dep ends on t only and, h enc e, is a c onstant on Γ t . Then, n = 1 , λ 1 = λ 1 ( t ) = const and Γ t is a family of c onc entric cir cles c enter e d at λ 1 . Pr o of o f the L emma. Our hy p othesis implies that all polynomials | P ( z , t ) | 2 − 1 are irreducible. Hence, using Hilb ert’s Nullstellensatz (e.g., c.f. [2 ], Prop osition 3.3.2), w e infer from (2.11) that (2.12) d dt  | P ( z , t ) | 2  − c ( t ) | P ′ z ( z , t ) | 2 = B ( t )  | P ( z , t ) | 2 − 1  . Equation (2.12) ho lds f or all z ∈ C and fo r an in terv al of time t , and for eac h t , b oth sides are real-ana lytic functions in z and z . Hence, w e can “p olarize” (2.12 ), i.e., replace z b y an indep enden t complex v ariable ξ . (This is due to a simple o bserv ation: r eal- analytic functions of tw o v ariables are nothing else but restrictions of holomorphic functions in z , ξ -v ariables to the pla ne { ξ = z } . Hence, if tw o real-analytic f unctions coincide on that plane, they coincide in C 2 as w ell.) Denoting b y P # the p olynomial whose co efficien ts are o btained from P by complex LEMNISCA TES DO NOT SUR VIVE LAPLACIAN GROWTH 5 conjugation, we hav e (2.1 2) in a “ p olarized” form holding for ( z , ξ ) ∈ C 2 : (2.13) d dt  P ( z , t ) P # ( ξ , t )  − c ( t )  P ′ z ( z , t ) ·  P #  ′ ξ ( ξ , t )  = B ( t )  P ( z , t ) P # ( ξ , t ) − 1  . No w let us denote b y k j the multiplic it y of t he ro o t λ j ( t ) of the p olyno- mial P ( z , t ), so that there are m ≤ n distinct ro ots and P m j =1 k j = n . Since P ( z , t ) = a ( t ) m Y 1 ( z − λ j ( t )) k j , P # ( ξ , t ) = ¯ a ( t ) m Y 1  ξ − λ j ( t )  k j , dividing b y P ( z , t ) P # ( ξ , t ) w e obtain: (2.14) 2 ℜ  ˙ a a  − m X 1 k j ˙ λ j ( t ) z − λ j ( t ) + k j ˙ λ j ( t ) ξ − λ j ( t ) ! − c ( t ) " m X 1 k j z − λ j ( t ) # · " m X 1 k j ξ − λ j ( t ) # = B ( t )  1 − 1 P ( z , t ) P # ( ξ , t )  . In tegra ting (2.14) along a small circle cen tered at λ j ( t ), so that it do es not enclose o ther zeros o f P , yields for all ξ : (2.15) − k j ˙ λ j ( t ) − c ( t ) m X 1 k i k j ξ − ¯ λ i ( t ) ! = − B ( t ) P # ( ξ , t ) q j , where q j = 1 ( k j − 1)!  ∂ ∂ z  k j − 1 h ( z − λ j ) k j P ( z ,t ) i z = λ j . Letting ξ → ∞ in (2.15) implies that ˙ λ j ( t ) = 0 for all j = 1 , . . . , n . In other w ords, the “no des” λ j ( t ) of all the lemniscates Γ t are fixed, i.e. do not mov e with time. So, (2.16) P ( z , t ) = a ( t ) n Y 1 ( z − λ j ) = a ( t ) Q ( z ) . 6 KHA VINSO N, MINEEV- WEINSTEIN, PUTINAR , A ND TEODORESCU Substituting (2 .16) in to (2.13 ), we o btain (2.17) d dt  | a | 2  Q ( z ) Q # ( ξ ) − c ( t ) | a | 2 Q ′ z  Q #  ′ ξ = B ( t )  | a | 2 Q ( z ) Q # ( ξ ) − 1  . Comparing the leading terms (i.e., t he co efficien ts at z n ξ n ) in (2.17) yields (2.18) d dt  | a | 2  = B ( t ) | a | 2 . Therefore, (2.19) c ( t ) | a | 2 Q ′ z  Q #  ′ ξ = B ( t ) , and t hus deg Q ′ z = 0, i.e., n = deg P = 1. The pro of s of the Lemma and the Theorem a re no w complete.  3. Extending the theorem to gro wth processes inv ariant under time reversal First, let us note t ha t any b oundary Γ t is an equip otential line of the logarithmic p oten tial (3.1) Φ( z ) = log | P n ( z , λ i ( t )) | 2 . The b oundary v elo city of the general g r o wth pro cess defined in (1.3) can now b e expressed as ~ V ( z ) = χ ( z ) ~ ∇ Φ , z ∈ Γ t , χ ( z ) ∈ R + . As indicated in t he In tro duction, inv ariance under time-r eve rsal is defined here in the fo llo wing w ay: if t he b oundary Γ t + dt is the imag e of Γ t under a map f ( t,dt ) : z t ∈ Γ t 7→ z t + dt ∈ Γ t + dt , then f ( t + dt, − dt ) ◦ f ( t,dt ) = I . That means t ha t the normal at z t + dt ∈ Γ t + dt m ust b e para llel to the normal at z t ∈ Γ t , which sho ws that Γ t + dt is p erp endicular at ev ery p oin t to g r a dien t lines of Φ, and is therefore a lev el line of Φ. The displacemen t of the p oin t z t b ecomes z t + dt − z t = χ ( z ) ~ ∇ Φ( z t ) dt. Denoting b y ~ E = ~ ∇ Φ = 2 ¯ ∂ Φ = 2 · P ′ n ( z , λ i ( t )) P n ( z , λ i ( t )) the gradien t of the logar it hmic p oten tia l and b y ~ r = z t , conserv ation of the normal (o r gradient) direction b ecomes ~ E ( ~ r + χ ~ E ( ~ r ) dt ) = µ ( z ) ~ E ( ~ r ) , LEMNISCA TES DO NOT SUR VIVE LAPLACIAN GROWTH 7 where µ ( z ) = 1 + m ( z ) dt, m ( z ) = O (1) , m ( z ) ∈ R , so after expanding in the infinitesimal time in terv al dt , ( ~ E · ~ ∇ ) ~ E ( ~ r ) = m ( z ) χ ( z ) ~ E ( ~ r ) . R emark 3.1 . The prop or t ionalit y relat io n indicated ab o v e carries a lso the follo wing ph ysical significance: the dynamical system that w e study is of frictional ty p e, where the ac c eler ation field (prop ortional to the force, or gradien t o f Green’s function) is also prop ortional to the velo c- ity . In other w ords, the transp ort deriv ative (or Lie deriv ativ e) o f the v elo cit y field m ust b e parallel to the v elo cit y itself: L ~ V ~ V = [ i ~ V ◦ d − d ◦ i ~ V ] ~ V = ( ~ V · ~ ∇ ) ~ V = χ [ χ ( ~ E · ~ ∇ ) ~ E + ( ~ E · ~ ∇ χ ) ~ E ] is parallel to ~ V and therefore, to ~ E . In complex notation, using the fact that ( ~ E · ~ ∇ ) = ¯ E ¯ ∂ + E ∂ , w e obtain P ′ n ( z , λ i ( t )) P n ( z , λ i ( t )) = δ ( z ) P ′ n ( z , λ i ( t )) P n ( z , λ i ( t )) ·  P ′ n ( z , λ i ( t )) P n ( z , λ i ( t ))  ′ , δ ( z ) ∈ R , whic h (after m ultiplying b oth sides b y E ( z )) reduces to  P ′ n ( z , λ i ( t )) P n ( z , λ i ( t ))  − 2  P ′ n ( z , λ i ( t )) P n ( z , λ i ( t ))  ′ ∈ R , or (3.2) ℑ  P n ( z , λ i ( t )) P ′ n ( z , λ i ( t ))  ′  = 0 , ( ∀ ) z ∈ Γ t . W e note that, since E ( z ) = 2 ¯ P ′ n / ¯ P n is the gradient of the Green’s function fo r Ω t and Ω t is simply connected, it cannot v anish an ywhere in Ω t ∪ Γ t , so all the zeros of P ′ n ( z ), denoted by ξ k , k = 1 , 2 , . . . , n − 1, are f o und inside the domain Ω ′ t . Then  P n ( z , λ i ( t )) P ′ n ( z , λ i ( t ))  ′ = 1 n + n − 1 X k =1 A k ( z − ξ k ) 2 , ξ k ∈ Ω ′ t , with A k constan ts. T he imaginary part of this expression coincides with the imag inary part o f an analytic function in Ω t , that is b ounded there, so the condition (3.2) can only b e satisfied if the function is a constan t. Since at z → ∞ it v anishes, it follo ws t hat  P n ( z , λ i ( t )) P ′ n ( z , λ i ( t ))  ′ = 1 n , whic h means t ha t b oundaries Γ t can only b e concen tric circles. 8 KHA VINSO N, MINEEV- WEINSTEIN, PUTINAR , A ND TEODORESCU 4. Concluding Remarks (1) It is plaus ible that the result can b e exte nded to ratio na l lemnis- cates Γ t := {| R ( z , t ) | = 1 } , where R ( z , t ) are rational functions of degree n where all the zeros are inside Γ, while all p oles are in the un b ounded comp onen t of C \ Γ t . (2) It is well-kno wn that arbitrary “shap es”, i.e. Jordan curv es can b e ar bitr arily close appro ximated b y b oth lemniscates (Hilb ert’s theorem – cf. [14]) and quadrature do mains [1]. A t t he same time our r esults imply that there are fundamen tal differences b et w een these tw o classes of curv es. W e think it is in teresting to pursue these observ a tions in greater depth. (3) F r o m the argumen t in § 3 w e can extract more. Supp ose a f a mily of Jordan curv es, { Γ t } t> 0 ev olve s b y the flow a long the v elo city field V ( z ) according to (1.3). Assuming the in v aria nce under time-rev ersal, the argumen t of § 3 can b e used to prov e that χ = const, i.e. t he pro cess is that of La placian gro wth. In v oking no w w ell-known results on standard Hele-Sha w flo ws, w e can at once conclude, e.g., that t he pro cess (1.3) contin ues for all times t > 0, i.e., the curv es { Γ t } mov e out to infinit y suc h that ∪ t>t 0 Γ t = C \ Ω t 0 , if and only if the initial curv e Γ 0 is an ellipse and a ll the curv es { Γ t } are a lso ellipses homotetic with Γ 0 - cf. [3], also cf. [6]. Reference s [1] S. R. Bell, Density of quadr ature domains in one and sever al variables , Compl. V ar. Elliptic E quations 54 (2009), no. 3 -4, 165– 171. [2] N. Bour baki, ´ El ´ ements de math´ ematique. Fasc. XX X . Alg` ebr e c ommu tative. Chapitr e 5: Entiers. , Actualit´ es Scientifiques et Industrielles, No. 1308 , Her- mann, Paris, 196 4. [3] E mma n uele DiBenedetto and Avner F rie dman, Bubble gr owth in p or ous me dia , Indiana Univ. Math. J. 35 (19 86), no. 3, 5 73–60 6. [4] Peter Eb enfelt, Bj¨ o r n Gustafsson, Dmitry Khavinson, and Mihai Putinar (eds.), Quadr atur e domains and their applic ations , Op erato r Theo ry: Adv ances and Applications, vol. 156 , Birkh¨ auser V er lag, Basel, 2 005. [5] Peter Eb enfelt, Dmitry Khavinson, and Ha rold S. Shapir o, An inverse pr oblem for the double layer p otent ial , Comput. Metho ds F unct. Theo ry 1 (2001), no. 2, 387–4 01. [6] Avner F riedman and Ma koto Sak ai, A char acterization of n u l l quadr atu r e do- mains in R N , Indiana Univ. Math. J. 35 (198 6 ), no. 3, 607 –610 . [7] L. A. Galin, Dokl. Acad. Nauk SSSR 47 (1945), no. 1-2 , 250– 3. [8] Dmitry Khavinson, Mark Mineev-W einstein, and Mihai Putinar, Planar el lip- tic gr owth , Compl. Anal. Op er . Theor y 3 (2009 ), no. 2, 425 –451. LEMNISCA TES DO NOT SUR VIVE LAPLACIAN GROWTH 9 [9] M. Mineev-W einstein, M. P utinar, and R. T eo dor escu, R andom matric es in 2D, La placian gr owth and op er ator the ory , Jour nal o f Ph y s ics A Mathematica l General 41 (2008 ), no. 26, 26 3001. [10] M. Mineev-W einstein, P .B . Wiegmann, and A. Zabro din, Inte gr able stru ct ur e of interfac e dynamics , Physical Review Letters 84 (200 0), 5 106. [11] P . Y a. Polubarinov a-Ko china, Dokl. Acad. Nauk SSSR 47 (19 45), 254–7 . [12] Haro ld S. Shapiro, The Schwarz function and its gener alization to higher di- mensions , Univ ersity of Ark a nsas Lecture Notes in the Mathematical Sciences, vol. 9, John Wiley & Sons Inc., New Y o rk, 1992 . [13] A. N. V archenk o and P . I. ` Etingof, Why the b oundary of a r ound dr op b e c omes a curve of or der four , University Lecture Series, vol. 3, American Mathematical So ciety , P rovidence, RI, 1 9 92. [14] J. L. W als h, Interp olation and appr oximation by r ational functions in the c om- plex domain , Amer. Math. So c. Collo quium Publ., vol. XX, American Mathe- matical So ciety , P rovidence, R.I., 1965 . Dep ar tment of Ma thema tics & St a tistics, University of South Florida, T amp a, FL 33620-5700, U SA E-mail addr ess : dkhav ins@ca s.usf.edu Los Al amos Na tional L abora tor y, MS-365, Lo s Alamo s, NM 875 45, USA E-mail addr ess : marin er@lan l.gov Dep ar tment of Ma thema tics, U niversity of California, Sant a Bar- bara, CA 93106, USA E-mail addr ess : mputi nar@ma th.ucsb.edu Dep ar tment of Ma thema tics & St a tistics, University of South Florida, T amp a, FL 33620-5700, U SA E-mail addr ess : razva n@cas. usf.edu