A note on arclength null quadrature domains
aa r X i v : . [ m a t h . C V ] M a y A note on arclength null quadrature domains
Dmitry Khavinson and Erik LundbergMay 21, 2020
Abstract
We prove the existence of a roof function for arclength null quadrature domainshaving finitely many boundary components. This bridges a gap toward classificationof arclength null quadrature domains by removing an a priori assumption fromprevious classification results.
A domain Ω ⊂ C is referred to as an arclength null-quadrature domain (arclength NQD)if the identity Z ∂ Ω g ( z ) ds ( z ) = 0 , (1.1)is satisfied for all functions g in the Smirnov space E (Ω) (a class of analytic functionssuitable for integration along ∂ Ω, see below), where ds ( z ) denotes the arclength element.Arclength NQDs are related to a free boundary problem for the Laplace equation (acorrespondence that we will strengthen in this note). Additional motivation for studyingarclength NQDs comes from fluid dynamics [3], [16] and minimal surfaces [15].The following problem was stated in [11] and restated with discussion in [3], [13]. Problem:
Classify arclength NQDs.The related problem of classifying area null-quadrature domains (area NQDs), where theintegration is over Ω with respect to area measure, was completely solved in 1981 by M.Sakai [14] who showed that area NQDs fall into one of the following four cases: • the exterior of an ellipse • the exterior of a parabola • a halfplane • a domain whose boundary is a proper subset of a lineThe halfplane and the exterior of a disk are NQDs for both area and arclength. Theother examples constructed in [3] show that the class of arclength NQDs is quite rich and1ncludes multiply-connected examples with boundary curves parameterized by ellipticfunctions.Area and arclength NQDs have natural generalizations to volume and surface area NQDs(respectively) in higher dimensions using appropriate test classes of harmonic functions.The classification of volume NQDs is an ongoing investigation, see [6], [9], [2], andthe references therein. Classification of surface area NQDs is an interesting unchartedterritory.While the classification of planar area NQDs is completely resolved by Sakai’s results, theclassification problem for arclength NQDs remains open, and progress has been stifledby a nagging question, stated below, concerning the existence of a so-called roof functionfor arclength NQDs. A sufficient condition for a domain Ω to be an arclength NQD is that Ω admits a rooffunction , a positive function u harmonic in Ω such that the gradient ∇ u coincides withthe inward-pointing unit normal vector along ∂ Ω. Note that this boundary condition isstronger than a mere Neumann condition since it is imposed on the gradient (not justthe normal derivative), and it implies that u itself is constant along each component of ∂ Ω (with possibly distinct constants on different boundary components).Domains that admit roof functions are called quasi-exceptional domains , and when theroof function is further assumed to have constant Dirichlet data (not just piecewiseconstant), they are referred to as exceptional domains . It was shown in [3] that quasi-exceptional domains are arclength NQDs, i.e., we have
Theorem 1.1. If Ω admits a roof function then Ω is an arclength NQD. Let us sketch the proof of this result to provide some context for what follows. Suppose u is a roof function for Ω. Then notice that the analytic completion f of u has a single-valued derivative. Namely, f ′ is just the complex conjugate of ∇ u . Thus, if ′ ( z ) providesan analytic continuation of the unit tangent vector thoughout all of Ω. Using the relationbetween the arclength element ds ( z ) and the unit tangent vector, we have ds ( z ) = if ′ ( z ) dz. This allows us to restate the condition (1.1) as requiring, for all g ∈ E (Ω), Z ∂ Ω g ( z ) f ′ ( z ) dz = 0 . That the above integral vanishes is a consequence of Cauchy’s theorem, and we concludethat Ω is an arclength NQD. However, in order to make this argument rigorous, onemust verify that g · f ′ is in the test class E (Ω). In fact, one can show f ′ ∈ H ∞ (Ω) usingpotential theoretic estimates on u (this step relies on the positivity of u ), see [3].Substantial progress has been made classifying exceptional domains and quasi-exceptionaldomains [8, 11, 15, 16, 3]. These results rely on the existence of a roof function, raising2he following question that was posed in [3] (cf. [13]) asking whether the converse toTheorem 1.1 holds. Question 1.
Does every arclength NQD admit a roof function?
Under the assumption that Ω has finitely many boundary components, we give an af-firmative answer to this question in the next section (see Theorem 2.2), thus showingthat the above-mentioned classification results for quasi-exceptional domains representdefinitive progress on the classification problem for arclength NQDs.The proof of Theorem 2.2 has two key ingredients. The first is a characterization ofthe Smirnov space of analytic functions, a result of Havinson and Tumarkin stated asTheorem 2.1 below, that allows reversing the main step in the proof of Theorem 1.1by establishing that the vanishing of integrals stated in the arclength NQD condition(1.1) guarantees the analytic continuation of the tangent vector to all of Ω. This leadsto a candidate roof function, but showing positivity requires a second key idea (in thisinstance potential theoretic) from [12, Thm. II] which is based on the proof of theDenjoy-Carleman-Ahlfors theorem. In order to highlight the utility of this method, wepoint out an interesting comparison with [3]: the result [12, Thm. II] was used in [3]to establish a growth condition on the roof function while using in part its assumedpositivity, whereas here we will need to show the positivity of a candidate roof functionfor which we will have already established a growth condition, see the Claim in the proofof Theorem 2.2 below.
First, we recall the definitions of the Hardy spaces H p ( D ) and the Smirnov spaces E p ( D ).A function g analytic in D is said to belong to E p ( D ) if there exists a sequence of cycles γ k homologous to zero, rectifiable, and converging to the boundary ∂D (in the sense that γ k eventually surrounds each compact sub-domain of D ), such that:sup γ k Z γ k | g ( z ) | p | dz | ≤ ∞ . On the other hand, a function g analytic in D is said to belong to H p ( D ) if the function | f | p admits a harmonic majorant in D . Basic properties of these spaces can be found in[1], [5], [18].We recall a key result from the theory of Smirnov spaces (see [1, Ch. 10] for a moredetailed overview). The following result due to Havinson and Tumarkin [18] provides anextension (to the multiply-connected setting) of a result of Smirnov [1, Thm. 10.4]. Theorem 2.1 (Havinson, Tumarkin) . Let D be a domain with rectifiable boundary.Suppose g ∈ L ( ∂D ) , and the function h defined by h ( w ) := Z ∂D g ( ζ ) ζ − w dζ anishes for all w ∈ C \ D . Then h ∈ E ( D ) and has boundary values g almost everywhereon ∂D . We now state our result addressing Question 1.
Theorem 2.2.
Suppose Ω is an arclength NQD and that the boundary ∂ Ω consists offinitely many smooth curves. Then Ω admits a roof function.Remark. Note that Ω is necessarily unbounded, since otherwise the constant functionsare in the test class E (Ω) and fail to satisfy the null quadrature condition. Also, Ω mayhave boundary components that are unbounded. Proof.
From the arclength null quadrature condition we have, for an arbitrary function g ∈ E (Ω), Z ∂ Ω g ( z ) ds = 0 , where ds denotes the arclength element. Write Z ∂ Ω g ( z ) ds = Z ∂ Ω g ( z ) T ( z ) dz, where T ( z ) denotes the unit tangent vector to ∂ Ω.Let φ : K → Ω be a conformal mapping from a bounded circular domain K to Ω. Recallthat a circular domain is a finitely-connected domain whose boundary components areall circles, and also recall that each finitely-connected domain is conformally equivalentto a circular domain [7, Ch. 3]. Then we have for each g ∈ E (Ω) Z ∂K g ( φ ( w )) φ ′ ( w ) T ( φ ( w )) dw = 0 . By Havinson and Tumarkin’s extension [17] (to the multiply-connected setting) of a resultof Keldysh and Lavrentiev [10] we have that g ∈ E (Ω) is equivalent to g ( φ ( w )) φ ′ ( w ) ∈ H ( K ) = E ( K ), and in particular the functions g ( φ ( w )) φ ′ ( w ) generate all of E ( K ).Hence, Z ∂K G ( w ) T ( φ ( w )) dw = 0 , for all G ∈ E ( K ) . This implies that the function κ defined by κ ( w ) := Z ∂K T ( φ ( ξ )) ξ − w dξ vanishes for all w ∈ C \ K . By Theorem 2.1 we have that κ ∈ E ( K ) and has boundaryvalues T ( φ ( w )) almost everywhere on ∂K . Since the boundary components of K arereal-analytic (they are circles), we have E ( K ) = H ( K ) [1, p. 182]. Since H ⊂ N + ,the Smirnov class, we conclude [1, Thm. 2.11] that κ ∈ H ∞ since it has boundary valuesin L ∞ ( ∂K ).Let ψ denote the inverse of φ and define h ( z ) = κ ( ψ ( z )). Then h ∈ H ∞ (Ω) and hasboundary values T ( z ) almost everywhere on ∂ Ω.4s a candidate for the roof function we take u ( z ) = ℜ{ f ( z ) } + C , where f ( z ) = − i Z zz h ( ζ ) dζ, (2.1)where z ∈ Ω is fixed, and C is an appropriate constant to be specified below. Weverify from the boundary values of h that ∇ u = f ′ ( z ) = ih ( z ) coincides with the inward-pointing unit normal vector. Indeed, h has boundary values T ( z ), so that ∇ u hasboundary values iT ( z ), which is the unit normal vector. Furthermore, we notice that u is single-valued, since the integral R γ h ( ζ ) dζ = R γ T ( ζ ) dζ = R γ T ( ζ ) T ( ζ ) ds is purelyreal for each subarc γ of ∂ Ω. The constant C is chosen to ensure non-negativity of thepiecewise-constant boundary values of u . Since h ∈ H ∞ (Ω) we have |∇ u | = O (1) as z → ∞ which implies u ( z ) = O ( | z | ) , as z → ∞ . (2.2)Indeed, we can express u ( z ) as an integral u ( z ) = Z ℓ h∇ u, r i| dz | + u ( z ) (2.3)along a line segment ℓ obtained by taking the connected component containing z of theline segment running from the origin to z , and h∇ u, r i denotes the inner product of ∇ u with the unit vector r in the direction of ℓ . Hence ℓ is a line segment from z to z , where z is either a point on ∂ Ω or z = 0. This gives the estimate | u ( z ) | ≤ | z − z ||∇ u | + | u ( z ) | = O ( | z | ) , (2.4)since |∇ u | = O (1) and u ( z ) is either u (0) or one of the finitely many Dirichlet boundaryvalues.It remains only to prove the following claim concerning positivity of u throughout Ω. Claim.
We have u > u ( z ) < z ∈ Ω. Let R denote the connected component containing z ofthe set where u <
0. Notice that R is unbounded (otherwise u < L be an unbounded component of ∂ Ω, and let m denote the value of u along L .The set Ω m of points for which u > m is contained in Ω \ R , and Ω m contains pointsnear each point on L by positivity of the inward normal derivative of u . Let γ denotea path from a point on L to z and consider Ω m \ γ which consists of two regions Ω a and Ω b . In each of these regions the boundary values of u are bounded, and u is notconstant, which implies that u → ∞ along a path to infinity in each of the regions. Thus,choosing M to be the maximum of u along γ , the region R := { z ∈ Ω a : u ( z ) > M } and R := { z ∈ Ω b : u ( z ) > M } are each nonempty.We thus have three disjoint regions R , R , R each unbounded, with u ( z ) < R and u ( z ) > M in R , R . 5or k = 1 , , θ k ( t ) to be the length of { z ∈ R k : | z | = t } . Let M ( r ) := max | z |≤ r | u ( z ) | , and for k = 1 , , M k ( r ) := max | z |≤ r,z ∈ R k | u ( z ) | . By the Phragmen-Lindelof principle [4, Thm. 6.1] (the theorem is stated for an analyticfunction f but only relies on the subharmonicity of log | f | and thus can easily be adaptedreplacing log | f | with | u | which is harmonic in each of the regions R k ), we have for k = 1 , , M k ( r ) ≥ π Z r θ k ( t ) dt. We also have for k = 1 , , M ( r ) ≥ log M k ( r ), and hence3 log M ( r ) ≥ π Z r X k =1 θ k ( t ) dt. (2.5)Since X k =1 θ k ( t ) ≤ πt, we have (using the Cauchy-Schwarz inequality)2 πt X k =1 θ k ≥ X k =1 θ k X k =1 θ k ≥ X k =1 p θ k r θ k ! = 3 , which implies π X k =1 θ k ( t ) ≥ t . Integrating gives π Z r X k =1 θ k ( t ) dt ≥ Z r t dt = (9 /
2) log r, and combining this with (2.5) we obtainlog M ( r ) ≥ (3 /
2) log r. This contradicts (2.2) which states that | u ( z ) | = O ( | z | ) as z → ∞ , and we conclude that u > Concluding Remarks
Theorem 2.2 shows that in the definition of the roof function (at least for domains withfinitely many boundary components) the condition of positivity can be replaced by agrowth condition u ( z ) = O ( | z | ) while only imposing positivity on the boundary values,and the positivity of u follows automatically. Indeed, the boundary condition along withthe growth condition imply the arclength NQD condition (1.1) as explained in Section1.1. Then Theorem 2.2, with some attention to the details of its proof, implies positivityof u .Theorem 2.2 allows immediate application of several results from [3] to the classificationof arclength NQDs that we shall summarize below.Assume that ∂ Ω has finitely many connected components. Then Theorem 2.2 shows thatΩ is a quasi-exceptional domain. This implies [3] that the number of unbounded com-ponents of ∂ Ω is either zero, one, or two, and we have the following partial classification(see [3]). • ∂ Ω compact = ⇒ Ω is the exterior of a disk • exactly one component of ∂ Ω is unbounded = ⇒ Ω is a halfplane • two components of ∂ Ω are unbounded and Ω is simply-connected = ⇒ Ω is theHauswirth-Helein-Pacard example [8]This leaves open the case that two components of ∂ Ω are unbounded and Ω is multiply-connected. This category appears to be the most interesting. Doubly-connected exam-ples were constructed using elliptic functions in [3], where it is conjectured that thereexist examples with every connectivity.
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