A Distortion Theorem for Quadrature Domains for Harmonic Functions

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1 JOURNAL OF MATEMATICAL ANALYSIS AND APPLICATIONS 202, ARTICLE NO A Distortion Theorem for Quadrature Domains for armonic Functions Born Gustafsson* Deartment of Mathematics, Royal Institute of Technology, S Stockholm, Sweden Submitted by Bruce C. Berndt Received June 26, 1995 We rove that any finitely connected domain in the lane can be distorted so that it becomes graviequivalent to a signed measure with arbitrarily small suort. Precisely: if D is a bounded, finitely connected domain with analytic boundary then for any a D and r 0, 0 with Ba,r Dthere exists a univalent function g in D with gž z. z Ž zd. and a signed measure with suort in Ba,r such that for every integrable harmonic function h in gž D. we have hdxdyhd Academic Press, Inc. 1. INTRODUCTION The urose of this aer is to rove the following theorem, stated, but not roved, in 8. ŽNotation. Ba,r z:zar. 4. TEOREM 1. Let D be a bounded domain in the comlex lane with D consisting of finitely many disoint analytic Jordan cures, let a D, and let r 0, 0 be any numbers satisfying BŽ a, r. D. Then there exists a unialent function g in D with gž z. z Ž z D. such that the image domain gž D. has the following roerty: there exists a signed measure * address: gborn@math.kth.se X96 $18.00 Coyright 1996 by Academic Press, Inc. All rights of reroduction in any form reserved.

2 170 BJORN GUSTAFSSON with suort in B a, r such that hdxdy hd Ž 1.1. for eery integrable harmonic function h in. That an equality of tye Ž 1.1. holds for Ž e.g.. all harmonic integrable h is often referred to by saying that is a quadrature domain for the measure Žsee, e.g., 11, 17 and references therein.. By taking hz logz with one sees that Ž 1.1. means Ž or at least imlies. that regarded as a body with density one roduces the same gravitational Žor logarithmic. otential outside as does the signed mass distribution. Thus the theorem roughly says that the set of bodies graviequivalent to signed mass distributions suorted in any fixed ball Ba,r are dense in the set of all bodies containing Ba,r. One main oint with this result is that if one instead asks for ositie mass distribution doing the same ob, i.e., if one in the theorem requires also 0, then it becomes tremendously false. Indeed, one may show that if Ž 1.1. holds with 0 and with 2 area 4r then is essentially ball shaed Že.g., is simly connected, is analytic, and for any z the inward normal of at z intersects Ba,r.. This follows from results in 11, 6, 7, 16, 10. The corresonding theorem with Ž 1.1. required to hold for all integrable analytic functions h was Ž essentially. roved in 5. From this follows the theorem for harmonic h in case is simly connected. A short direct roof for this case is also given in 8. In the resent aer we shall extend the results of 5 to rove Theorem 1. Notation. C f : : f is continuous 4, A f : : f is analytic 4, M f : 4 : f is meromorhic 4, h: : h is harmonic 4, 1 AL f AŽ. : f dxdy, ½ 5 ½ 5 1 L h Ž. : h dxdy, a Dirac measure at the oint a, 1 if k, k ½ 0 if k, area of.

3 QUADRATURE DOMAINS QUADRATURE DOMAINS FOR ANALYTIC AND ARMONIC FUNCTIONS Let D be a finitely connected bounded domain with D analytic, as in Section 1. Then D can be comleted to a comact Riemann surface D, ˆ called the Schottky double of D. It consists of D, a coy D Žthe back-side. of D rovided with the oosite conformal structure, and their common boundary D through which they are welded together. See 9, 5. If z D we denote by z the corresonding oint on D. There is a natural anticonformal involution on Dˆ given by Ž z. z, Ž z. z if z D Ž zd. and by Ž z. z if z D. Together, the air Ž D, ˆ. constitutes a so-called symmetric Riemann surface. If f is any meromorhic function on Dˆ then f * f is also meromorhic on D. ˆ OnD, f*f. Setting f f, f f * 1 D 2 D it follows that f and f are meromorhic functions in D continuous Ž 1 2 in the extended sense. u to D and satisfying f1 f2 on D. Ž 2.1. Conversely, any air of meromorhic functions f, f on D satisfying Ž reresents a meromorhic function on D. ˆ Note that if f is reresented as above by Ž f, f., then f * is reresented by Ž f, f We shall first recall a result for quadrature domains for analytic functions which has been roved in various arts in 3, 1, 5. See also, e.g., 17, 13, 14. PROPOSITION 1. With D and Dˆ as aboe, let be a bounded domain conformally equialent to D and let g : D be a conformal ma. Then the following assertions are equialent. Ž. i There exists a comlex-alued distribution with suort in a finite number of oints in such that ² : 1 fdxdy, f for all f AL Ž.. Ž 2.2. ii There exists a meromorhic function S z in, continuously extendible u to with SŽ z. z on. Ž 2.3. Ž iii. g extends to a meromorhic function on D. ˆ

4 172 BJORN GUSTAFSSON Ž. When i iii hold,, S, and g are related by Ý ² : 1, f Res SŽ z. fž z. fal z 2.4 SŽ gž.. g* Ž. Ž D. and SŽ z. is called the Schwarz function of 3, 17. As to the roof we ust mention that Ž ii. Ž i. follows from artial integration and the residue theorem, 1 fdxdy 2i fdzdz 1 1 fž z. zdz fž z. SŽ z. dz 2i 2i Ý z Res fž z. SŽ z., and that Ž iii. simly is the statement Ž ii. ulled back to D by means of g. Note that, in terms of the reresentation Ž 2.1., Eq. Ž 2.3. says that the air Žz,SŽ z.. reresents a meromorhic function on. ˆ Any distribution D with suort in a finite number of oints is of the form m n i ² :, Ý Ý a ki i Ž z k. Ž C. Ž 2.5. xy k1 i, 0 for suitable m, n and z,..., z, a. Since f x iž fy. 1 m ki when f is analytic, many different distributions of the form Ž 2.5. have the same action on AL 1 Ž.. It is always ossible to take aki to be real whenever i 1, but this still does not make uniquely determined in general, since, e.g., 2 f x 2 2 fy 2. One way to write the action of on AL 1 in a canonical form is to write Ž 2.5. as, f m 2 n Ý Ý c k f z k k1 0 ² : for f AL 1 Ž.. m m When 2.2 holds we see, by taking f 1, that Ýk1 ak 00 Ýk1 ck0 is real and ositive, but the individual ak 00 ck0 may very well be nonreal. To be recise, we have the following. Suose that Ž 2.2. holds and choose

5 QUADRATURE DOMAINS 173 aki for i 1 to be real. Then ak00 for all k if and only if the identity udxdy²,u: Ž 2.6. holds for every harmonic function u which is the real art of some fal 1 Ž.. Indeed, if a then is real-valued Ž k 00 i.e., takes real values on real-valued test functions. and Ž 2.6. follows from Ž 2.2. by taking real arts. Conversely, Ž 2.6. alone imlies Ž 2.2. and then for every f AL 1 Ž., ²,Re f: Ž Re f. dxdyre fdxdyre², f :, which imlies that a when a, i 1. Ž k 00 ki If, e.g., a100 then we get a contradiction by taking f to be a olynomial with fž z. 1, fž z. 1 k 0 for k 1.. EXAMPLE 1. a 0 It is ossible to construct such that for a suitably large fdxdyaf Ž 0. iž fž 1. fž 1.. Ž 2.7. for all f AL 1 Ž.. Ž See Proosition 4 below.. ere the right member is ², f: with a0 i1 i 1. This is nonreal and its action on AL 1 does not coincide with that of any real-valued distribution with suort in a finite number of oints. Therefore, the identity Ž 2.6. does not hold for all u Re f, f AL 1 Ž.. In order to get a quadrature identity for this and holding for all ure f, f AL 1 Ž., write f u i and let be the straight line segment from 1 to1. Then f iž fž 1. fž 1.. i dx x so that, taking real arts in 2.7, u i dx dx x x u u i dx dx x y u udxdyauž 0. dx Ž 2.8. y

6 174 BJORN GUSTAFSSON for all u Re f, f AL 1 Ž.. Thus Ž 2.6. holds, but with a distribution of a more general form. Next we wish to discuss quadrature identities hdxdy²,h: Ž 2.9. with of the form Ž 2.5. holding for all h L 1 Ž.. Clearly, if Ž 2.9. holds with comlex-valued it also holds for Re Žand Im annihilates L 1 Ž.., so we may as well assume from the beginning that is realvalued. Recall from the revious discussion that Ž 2.9. then holds for all hl 1 of the form h Re f, f AL 1 if and only if the equivalent conditions in Proosition 1 hold. PROPOSITION 2. With D and Dˆ as in the beginning of this section, let be a bounded domain conformally equialent to D and let g : D be a conformal ma. Then the following assertions are equialent. Ž. i There exists a real-alued distribution with suort in a finite number of oints in such that Ž 2.9. holds for all h L 1 Ž.. Ž ii. There exists a meromorhic function SŽ z. in, continuously extendible u to with and satisfying in addition SŽ z. z on Ž ReŽ z SŽ z.. dz0 Ž for eery smooth cure aoiding the oles of SŽ z. and satisfying Ž i.e., either is closed or oins two comonents of.. Ž iii. g extends to a meromorhic function on Dˆ satisfying the additional condition that Re g* dg 0 Ž for eery closed cure in Dˆ aoiding the oles of g*dg. Proof. We first rove that Ž. i and Ž ii. are equivalent. To large arts this is actually well-known Žsee 12, 14, e.g.., but since there are some subtle oints we refer to outline the full roof.

7 QUADRATURE DOMAINS 175 Let EŽ z. 12 logz be the standard fundamental solution of Ž so that E.. If Ž i. holds, set 0 u Ž. E. Then u is a real-valued distribution, continuously differentiable outside su and satisfying u in, Ž u u c u 0 on. Ž x y The latter equation follows by taking hz EŽ z., Ž Ex.Ž z., c Ey z for in Ž Define SŽ z. by u SŽ z. z4, z. z Then S z 1 u in so that SŽ z. is meromorhic in. By Ž 2.14., SŽ z. is continuous u to with Ž holding there. Finally, Ž holds since u ReŽ z SŽ z.. dz4re dz z 2 which is zero since u 0on Ž or is emty.. Thus Ž i. imlies Ž ii.. Conversely, suose Ž ii. holds. Let z,..., z be the oles of SŽ z. 1 m, ick a oint z, and define uz on z,..., z 4 by du, 0 1 m 1 už z. Re SŽ. d z 2 z 0 and extend it by zero outside. Then, due to Ž 2.10., Ž 2.11., u is a well-defined continuously differentiable real-valued function in z,..., z 4 satisfying Ž Moreover, 1 m u in z,..., z 4. Ž m This follows from the definition of u together with the known fact 1, 13 that is iecewise smooth Ž algebraic. when Ž holds.

8 176 BJORN GUSTAFSSON At z 1,..., z m, u has olar singularities. Unfortunately, u need not be integrable over these, so u need not, in a canonical way, be a distribution on. Therefore one cannot define right away by Ž 2.13., as one is temted to do, with u the distributional Lalacian of u on. Nevertheless, with B BŽ z,. BŽ z,. 1 m for 0 small it follows, using Ž 2.14., Ž 2.15., and the cutting-off technique in 11,. 60, that for h L 1 hdxdy lim hudxdy 0 B h u lim u ds h ds. n n ž / 0 B B Since the singularities of u at z 1,..., zm are of finite order it follows that the right member above can be written ², h: for some real-valued distribution Ž not uniquely determined in general, as a distribution. with suort at z,..., z. Thus Ž ii. imlies Ž i. 1 m. It remains to rove that Ž ii. and Ž iii. are equivalent. As in Proosition 1 Žor. 5 the existence of a meromorhic SŽ z. satisfying Ž is equivalent to the meromorhic extension of g to D. ˆ Thus we only need to show that condition Ž is the same as Ž Pulling Ž back to D it becomes ReŽ g g*. dg 0 Ž for every D with D Ž g.. We shall rove that Ž and Ž are equivalent. There are two kinds of curves which need to be considered in Ž and Ž 2.16.: Ž a. closed curves in D; Ž b. for Ž 2.16., curves in D oining two comonents of D and, for Ž 2.12., the corresonding closed curves on Dˆ obtained by going back along the same track on the back-side D. As for Ž. a, we note that 1 1 Re gdg Ž gdggdg. dž gg when is a closed curve in D, so then 2.16 is really the same as 2.12.

9 QUADRATURE DOMAINS 177 As for Ž b., let be a nonclosed curve on D with endoints on D and let Ž. be the corresonding Ž oriented. closed curve on D. ˆ Then ½ 5 ½ 5 ½ Ž. 5 ½ 5 Re g* dg Re Ž g. dg Ž g. dg Re g dg gd g Re g dg g g g dg Re g* dg g* dg gg 2Re g*dg Re dž gg. 2ReŽ g*g. dg. Thus Ž is the same as Ž also in case Ž b.. This comletes the roof of Proosition 2. Remark. If in Ž ii. or Ž iii. of Proosition 2, Ž Žor Ž resectively. is required to hold only for all closed curves in Ž or D resectively. then one gets exactly the condition that Ž. i of Proosition 1 holds for some real-valued or equivalently the condition that Ž. i of Proosition 2 holds for all h of the form h Re f, f AL 1 Ž.. 3. PROOF OF TE MAIN RESULT Using Proosition 2 together with an aroximation argument we shall rove the following, which readily imlies Theorem 1 Žstated in the Introduction.. PROPOSITION 3. With D, a D, 0, and r 0 as in Theorem 1 there exists a meromorhic function g on Dˆ satisfying Ž such that, moreoer, g is unialent, gž z. z D for z D, and such that all oles of g are located in ŽBa,r. D. Before turning to the roof let us see how Theorem 1 follows from Proosition 3. With g as in Proosition 3 it follows from Proosition 2 that

10 178 BJORN GUSTAFSSON there exists a real-valued distribution with suort in finite number of oints such that Ž 2.9. holds for gž D.. The suort of is the singular set of SŽ z., hence is contained in gžbž a, r.. Ba,r. Now, everything in Theorem 1 is fulfilled excet that may be a more general distribution than a measure. But clearly can be relaced by a signed measure with suort in Ba,r such that ², h: hd for all h L 1 by mollifying it with a radially symmetric test function with small suort. Thus Theorem 1 follows from Proositions 2 and 3. Proof of Proosition 3. Take D, a D, 0, r 0 as in the state- and else regular. Let be ment. We shall choose g with a ole at a D the number of comonents of D minus one. As for the condition Ž there are then two tyes of curves to consider: Ž. a is homologous to one of the comonents of D Ž b. where is a curve in D a4 from the th inner comonent of D to the outer comonent Ž 1,...,.. Note that, e.g., a small loo around a or a is homologous Žin Dˆ a 4. to Dand hence is covered by case Ž. a. Now for of tye Ž.Ž a, is automatically fulfilled. Indeed, taking to be a comonent of D we have Ž since g* g on D. 2Re g*dg gdg gdg dž gg. 0. For Ž 1,...,. as in Ž b. we introduce the bilinear functional BŽ f, h. f*dh for functions f and h defined on. It is readily verified that B Ž f, h. B Ž h, f.. In articular B Ž f, f. is automatically real and, as at the end of the roof of Proosition 2, we have 2 B f, f Re f * df 2Re f*df f. 3.1

11 QUADRATURE DOMAINS 179 Now to construct g we shall use the Mergelyan aroximation theorem for comact Riemann surfaces Ž 4. For the corresonding Runge theorem, see, e.g.,. 2. Let U be a small neighbourhood of D in Dˆ and set K U. 1 Then Dˆ K is connected and contains a Ž if U is chosen roerly.. It is easy to see that it is ossible to choose functions f, f,..., f CŽ K. 1 AU satisfying fž z. fž z. z for z D, Ž 3.2. BŽ f, f. 0, Ž 3.3. B f, f Ž 3.4. k k Ž k, 1,...,.. Indeed, Ž 3.2. determines f and f on U and then it is clear from Ž 3.1. that f and f can be adusted on U so that Ž 3.3., Ž 3.4. hold. By the Mergelyan aroximation theorem there exist functions h, h,...,h MŽ Dˆ. AD Ž ˆ a4. 1 which aroximate f, f 1,..., f uni- formly on K, say hf2 on K, h f 2 on K. Recall that 0 was given already in the formulation of the roosition, but we shall relace it by smaller values as necessary. We shall choose our g to be g h h h 1 1 for suitable small,..., Ž and.. Thus g MŽ Dˆ. AD Ž ˆ a 4. 1, and whenever and are small enough g will be univalent on D and satisfy gž z. z for z D. It remains to show that g satisfies condition Ž 2.12., i.e., in view of the discussion at the beginning of the roof, to show that, can be chosen so that B g, g 0, k 1,...,. 3.5 k

12 180 BJORN GUSTAFSSON We have, setting B Ž f. B Ž f, f., k k k kž Ý i i/ 1 Bk fý f hfý Ž h f ž i i i i i. / 1 1 B g B h h ž Ý / ž Ý / 1 1 B f f B hf Ž h f. k i i k i i i kž Ý i i Ý iž i i. / 1 1 2ReB f f, hf h f. Using 3.3, 3.4 we see that this is of the form Ý B Ž g. 2Re a Re b Re c, k k k ki i ki i i1 i, 1 Ý where a Ž., b Ž. OŽ., and c Ž. OŽ 1. k ki ki as 0. From this it follows, e.g., from the imlicit function theorem that if ust is chosen small enough then the system Ž 3.5. will have a solution in,..., Ž 1 with these small.. Thus the roosition follows. Finally, we wish to ustify a statement made in Examle 1. PROPOSITION 4. Let be any real-alued distribution with comact suort. Then for a 0 large enough there exists a bounded domain containing the origin and su such that hdxdyahž 0. ², h: for all h L 1 Ž.. Remark. This result is valid in any number of dimensions 2. Proof. Choose R 0 so that su BŽ 0, R.. Then there is a signed measure on BŽ 0, R. such that ², h: BŽ0, R. hd for all h harmonic in a neighbourhood of BŽ 0, R.. ŽIt is ust to take to be the measure with density equal to the normal derivative of the solution u of u in BŽ 0, R., u 0onB0, Ž R...

13 QUADRATURE DOMAINS 181 Doing the same thing with a0 and adding it follows that there is a measure on BŽ 0, R. such that a ahž 0. ², h: hd Ž 3.6. BŽ 0, R. for all h harmonic in a neighbourhood of BŽ 0, R..If a0 is sufficiently large will be ositive and we may even take a so large that Ž moreover. a d 4 BŽ 0, R. Žin N dimensions: 2 N BŽ 0, R.. a. Under these conditions it is known 11, 6, 7, 15, 16, 10 that there exists a domain containing BŽ 0, R. such that hdxdyhda for all h L 1 Ž.. By combining with Ž 3.5. the desired conclusion follows. Using this roosition we may construct, for a 0 large, a domain such that Ž 2.8. in Examle 1 holds. Going backwards in the examle it follows that also Ž 2.7. holds, as claimed there. Note added in roof. Using a different method of roof, M. Sakai 18 has recently generalized the main result of this aer to higher dimensions. a REFERENCES 1. D. Aharonov and. S. Shairo, Domains on which analytic functions satisfy quadrature identities, J. Analyse Math. 30 Ž 1976., C. Anderseth, Sur le theoreme ` d aroximation de Runge, Enseign. Math. 26 Ž 1980., P. Davis, The Schwarz Function and Its Alications, Carus Math. Monograhs, Vol. 17, Math. Assoc. of America, Washington, DC, S. Y. Gusman, Uniform aroximation of continuous functions on Riemann surfaces, Soiet Math. Dokl. 1 Ž 1960., B. Gustafsson, Quadrature identities and the Schottky double, Acta Al. Math. 1 Ž 1983., B. Gustafsson, On quadrature domains and an inverse roblem in otential theory, J. Analyse Math. 55 Ž 1990., B. Gustafsson and M. Sakai, Proerties of some balayage oerators, with alications to quadrature domains and moving boundary roblems, Nonlinear Anal. 22 Ž 1994., B. Gustafsson, M. Sakai, and. S. Shairo, On domains in which harmonic functions satisfy generalized mean value roerties, Potential Analysis, Ž to aear Farkas and I. Kra, Riemann Surfaces, Sringer-Verlag, New York, A. S. Margulis, The moving boundary roblem of otential theory, Ad. Math. Sci. Al., Ž to aear..

14 182 BJORN GUSTAFSSON 11. M. Sakai, Quadrature Domains, Lecture Notes in Math., Vol. 934, Sringer-Verlag, Berlin, M. Sakai, Alications of variational inequalities to the existence theorem on quadrature domains, Trans. Amer. Math. Soc. 276 Ž 1983., M. Sakai, Regularity of a boundary having a Schwarz function, Acta Math. 166 Ž 1991., M. Sakai, Regularity of boundaries of quadrature domains in two dimensions, SIAM J. Math. Anal. 24 Ž 1993., M. Sakai, Regularity of free boundaries in two dimensions, Ann. Scuola Norm. Su. Pisa Cl. Sci. () 4 20 Ž 1993., M. Sakai, Shar estimates of the distance from a fixed oint to the frontier of ele-shaw flows, rerint, S. Shairo, The Schwarz Function and Its Generalization to igher Dimensions, Wiley, New York, M. Sakai, Linear combinations of harmonic measures and quadrature domains of signed measures with small suorts, rerint, 1995.