Existence of energy-minimal diffeomorphisms between doubly connected domains
Tadeusz Iwaniec, Ngin-Tee Koh, Leonid V. Kovalev, Jani Onninen
aa r X i v : . [ m a t h . C V ] A ug EXISTENCE OF ENERGY-MINIMAL DIFFEOMORPHISMSBETWEEN DOUBLY CONNECTED DOMAINS
TADEUSZ IWANIEC, NGIN-TEE KOH,LEONID V. KOVALEV, AND JANI ONNINEN
Abstract.
The paper establishes the existence of homeomorphisms be-tween two planar domains that minimize the Dirichlet energy.
Among all homeomorphisms f : Ω onto −→ Ω ∗ between bounded doublyconnected domains such that Mod Ω Mod Ω ∗ there exists, unique upto conformal authomorphisms of Ω , an energy-minimal diffeomorphism. No boundary conditions are imposed on f . Although any energy-minimal diffeomorphism is harmonic, our results underline the majordifference between the existence of harmonic diffeomorphisms and theexistence of the energy-minimal diffeomorphisms. The existence of glob-ally invertible energy-minimal mappings is of primary pursuit in themathematical models of nonlinear elasticity and is also of interest incomputer graphics. Contents
1. Introduction 22. Statements 53. Basic properties of deformations 64. Harmonic replacement 125. Reich-Walczak-type inequalities 136. Hopf differentials 177. Monotonicity of the minimum energy function 188. Existence: Theorem 2.3 209. Nonexistence: Theorem 2.4 2110. Convexity of the minimum energy function 2311. Open questions and conjectures 2712. Appendix: Monotone Sobolev mappings 28References 32
Mathematics Subject Classification.
Primary 58E20; Secondary 30C62, 31A05.
Key words and phrases.
Dirichlet energy, Sobolev homeomorphism, deformation, min-imal energy, harmonic mapping, conformal modulus.Iwaniec was supported by the NSF grant DMS-0800416 and the Academy of Fin-land grant 1128331. Koh was supported by the NSF grant DMS-0800416. Kovalev wassupported by the NSF grant DMS-0968756. Onninen was supported by the NSF grantDMS-1001620. Introduction
Throughout this text Ω and Ω ∗ will be bounded domains in the complexplane C . The Dirichlet energy of a diffeomorphism f : Ω onto −→ Ω ∗ is definedand denoted by(1.1) E [ f ] = Z Ω | Df | = 2 Z Ω (cid:0) | ∂f | + | ¯ ∂f | (cid:1) where | Df | is the Hilbert-Schmidt norm of the differential matrix of f . Theprimary goal of this paper is to establish the existence of a diffeomorphism f : Ω onto −→ Ω ∗ of smallest (finite) Dirichlet energy. The behavior of such an energy-minimal diffeomorphism f resembles that of a conformal mapping.Indeed, a change of variables in (1.1) yields(1.2) E [ f ] = 2 Z Ω J f ( z ) dz + 4 Z Ω | ¯ ∂f | > | Ω ∗ | where J f stands for the Jacobian determinant and | Ω ∗ | is the area of Ω ∗ .A conformal mapping of Ω onto Ω ∗ ; that is, a homeomorphic solution ofthe Cauchy-Riemann system ¯ ∂f = 0, would be an obvious choice for theminimizer of (1.2). Unfortunately, for generic multiply connected domainsthere is no such mapping. The existence of an energy-minimal diffeomor-phism f : Ω onto −→ Ω ∗ may be interpreted as saying that the Cauchy-Riemannequation ¯ ∂f = 0 admits a diffeomorphic solution in the least squares sense,meaning that k ¯ ∂f k L assumes its minimum. For this reason energy-minimaldiffeomorphisms are known under the name least squares conformal map-pings in the computer graphics literature [27, 34]. They are also of greatinterest in the theory of nonlinear elasticity due to the principle of nonin-terpenetration of matter [4, 38].An energy-minimal diffeomorphism may fail to exist when a minimizingsequence collapses, at least partially, onto the boundary of Ω ∗ . This phenom-enon was observed in the papers [2, 18] for a pair of circular annuli. A relatedphenomenon occurs in free boundary problems for minimal graphs, where itis called edge-creeping [6, 13, 39]. Since the boundary of Ω ∗ plays a crucialrole in the minimization of energy among diffeomorphisms f : Ω onto −→ Ω ∗ , ourquestions are essentially different from widely studied variational problemsfor mappings between Riemannian manifolds where the target is usually as-sumed to have no boundary [5, 22, 23, 26]. We do not prescribe boundaryvalues of f , nor do we suppose that it has a continuous boundary extension.Any energy-minimal diffeomorphism satisfies Laplace’s equation, sinceone can perform first variations while preserving the diffeomorphism prop-erty. However, the existence of a harmonic diffeomorphism does not implythe existence of an energy-minimal one, see Example 9.1. This is why ournecessary condition for the existence of an energy-minimal diffeomorphism,Theorem 2.4, is more restrictive than the corresponding result for harmonicdiffeomorphisms in [15]. XISTENCE OF ENERGY-MINIMAL DIFFEOMORPHISMS 3
As we have already pointed out, energy-minimal diffeomorphisms for sim-ply connected domains are obtained from the Riemann mapping theorem.The doubly connected case, being next in the order of complexity, is thesubject of our main result.
Theorem 1.1.
Suppose that Ω and Ω ∗ are bounded doubly connected do-mains in C such that Mod Ω Mod Ω ∗ . Then there exists an energy-minimal diffeomorphism f : Ω onto −→ Ω ∗ , which is unique up to a conformalchange of variables in Ω . Hereafter Mod Ω stands for the conformal modulus of Ω. Any boundeddoubly connected domain Ω ⊂ C is conformally equivalent to some circularannulus { z : r < | z | < R } with 0 r < R < ∞ . The ratio R/r , being inde-pendent of the choice of conformal equivalence, defines Mod Ω := log
R/r .The conformal modulus is infinite precisely when the bounded componentof C \ Ω degenerates to a point. We call such domain a punctured domain .Theorem 1.1 has the following corollary.
Corollary 1.2.
For any bounded doubly connected domain Ω and any punc-tured domain Ω ∗ there exists an energy-minimal diffeomorphism f : Ω onto −→ Ω ∗ , which is unique up to a conformal change of variables in Ω.In the converse direction we show (Theorem 2.4) that there exists noenergy-minimal diffeomorphism when Mod Ω ∗ Φ(Mod Ω). Here Φ : (0 , ∞ ) → (0 , ∞ ) is a certain function asymptotically equal the identity at infinity,lim t →∞ Φ( t ) /t = 1. It is in this asymptotic sense that Theorem 1.1 is sharp.It is rather surprising that our existence result for energy-minimal diffeo-morphisms relies only on the conformal modulus of the target. Indeed, theenergy minimization problem is invariant only with respect to a conformalchange of variable in the domain, not in the target.Yet in other perspectives, the classical Teichm¨uller theory is concernedwith the existence of quasiconformal mappings g : Ω ∗ onto −→ Ω with smallest L ∞ -norm of the distortion function K g ( w ) = | Dg ( w ) | J g ( w ) , a.e. w ∈ Ω ∗ . Analogous questions about L -norm of K g lead to minimization of theDirichlet energy of the inverse mapping via the transformation formula(1.3) k K g k L (Ω ∗ ) = E [ f ] , where f = g − : Ω onto −→ Ω ∗ For rigorous statements let us recall that a homeomorphism g : Ω ∗ onto −→ Ω ofSobolev class W , (Ω ∗ ) has integrable distortion if(1.4) | Dg ( w ) | K ( w ) J g ( w ) a.e. in Ω ∗ for some K ∈ L (Ω ∗ ). The smallest such K : Ω ∗ → [1 , ∞ ), denoted by K g ,is referred to as the distortion function of g . IWANIEC, KOH, KOVALEV, AND ONNINEN
It turns out that the inverse of any mapping with integrable distortionhas finite Dirichlet energy and the identity (1.3) holds. As a consequence ofTheorem 1.1 we obtain the following result.
Theorem 1.3.
Let Ω and Ω ∗ be bounded doubly connected domains in C such that Mod Ω Mod Ω ∗ . Among all homeomorphisms g : Ω ∗ onto −→ Ω thereexists, unique up to a conformal automorphism of Ω , mapping of smallest L -norm of the distortion. We conclude this introduction with a strategy of the proof of Theorem 1.1.The natural setting for our minimization problem is the Sobolev space W , (Ω). In this paper functions in the Sobolev spaces are complex-valued.Let us reserve the notation H , (Ω , Ω ∗ ) for the set of all sense-preserving W , -homeomorphisms h : Ω onto −→ Ω ∗ . When this set is nonempty, we define(1.5) E H (Ω , Ω ∗ ) = inf {E [ h ] : h ∈ H , (Ω , Ω ∗ ) } . By virtue of the density of diffeomorphisms in H , (Ω , Ω ∗ ), see [16], theminimization of energy among sense-preserving diffeomorphisms leads tothe same value E H (Ω , Ω ∗ ). A homeomorphism h ∈ H , (Ω , Ω ∗ ) is energy-minimal if it attains the infimum in (1.5). Let us emphasize that the set H , (Ω , Ω ∗ ) ⊂ W , (Ω) is unbounded. Even bounded subsets of H , (Ω , Ω ∗ )are lacking compactness, due to the loss of injectivity in passing to a limitof homeomorphisms. One way out of this difficulty is to consider the weakclosure of H , (Ω , Ω ∗ ) ∩ B where B is a sufficiently large ball in W , (Ω)whose size depends only on E H (Ω , Ω ∗ ). This is the approach undertakenin [17, 24]. However, the presence of B creates problems of its own. Forinstance, the resulting class of mappings is not closed under compositionswith self-diffeomorphisms of Ω; inner variation of such mappings would beinadmissible.That is why we introduce the class of so-called deformations . These aresense-preserving surjective mappings of the Sobolev class W , that can beapproximated by homeomorphisms in a certain way. The precise definitionis given in §
3. A deformation is not necessarily injective. In addition, anenergy-minimal deformation need not be harmonic, since one cannot performfirst variations f + ǫϕ within the class of deformations. This is why we relyon inner variations, which yield that the Hopf differential ( §
6) of an energy-minimal deformation is holomorphic in Ω and real on its boundary. We gainadditional information about the Hopf differential from the Reich-Walczak-type inequalities ( §
5) which is where the conformal moduli of Ω and Ω ∗ enterthe stage.The crucial idea of the proof of Theorem 1.1 is to consider a one-parameterfamily of variational problems in which Ω changes continuously while Ω ∗ remains fixed. We establish strict monotonicity of the minimal energy asa function of the conformal modulus of Ω ( § § § XISTENCE OF ENERGY-MINIMAL DIFFEOMORPHISMS 5
The interested reader is invited to look upon the open questions collectedin §
11. 2.
Statements
A homeomorphism of a planar domain is either sense-preserving or sense-reversing. For homeomorphisms of the Sobolev class W , (Ω) this impliesthat the Jacobian determinant does not change sign: it is either nonnegativeor nonpositive at almost every point [3, Theorem 3.3.4], see also [12]. Thehomeomorphisms considered in this paper are sense-preserving unless statedotherwise.Let Ω and Ω ∗ be bounded domains in C . To every mapping f : Ω → Ω ∗ we associate a boundary distance function δ f ( z ) = dist( f ( z ) , ∂ Ω ∗ ) which isset to 0 on the boundary of Ω.The following concept, which interpolates between c -uniform (i.e., uniformon compact subsets) and uniform convergence, proves to be effective. Definition 2.1.
A sequence of mappings h j : Ω → Ω ∗ is said to converge cδ -uniformly to h : Ω → Ω ∗ if • h j → h uniformly on compact subsets of Ω and • δ h j → δ h uniformly on Ω.We designate it as h j cδ −→ h . Definition 2.2.
A mapping h : Ω → Ω ∗ is called a deformation if • h ∈ W , (Ω); • The Jacobian J h := det Dh is nonnegative a.e. in Ω; • R Ω J h | Ω ∗ | ; • there exist sense-preserving homeomorphisms h j : Ω onto −→ Ω ∗ , calledan approximating sequence , such that h j cδ −→ h on Ω.The set of deformations h : Ω → Ω ∗ is denoted by D (Ω , Ω ∗ ).The first thing to note is H , (Ω , Ω ∗ ) ⊂ D (Ω , Ω ∗ ). Outside of some de-generate cases, the set of deformations is nonempty by Lemma 3.15 and isclosed under weak limits in W , (Ω) by Lemma 3.13. Define(2.1) E (Ω , Ω ∗ ) = inf {E [ h ] : h ∈ D (Ω , Ω ∗ ) } where E [ h ] is as in (1.1). A deformation that attains the infimum in (2.1) iscalled energy-minimal . It is obvious that E H (Ω , Ω ∗ ) > E (Ω , Ω ∗ ), but whetherthe equality holds is not clear. We are now in the position to state theexistence Theorem 1.1 more precisely. Theorem 2.3.
Suppose that Ω and Ω ∗ are bounded doubly connected do-mains in C such that Mod Ω Mod Ω ∗ . There exists a diffeomorphism h ∈ H , (Ω , Ω ∗ ) that minimizes the energy among all deformations; that is, E [ h ] = E (Ω , Ω ∗ ) and hence, E H (Ω , Ω ∗ ) = E (Ω , Ω ∗ ) . Moreover, h is uniqueup to a conformal automorphism of Ω . IWANIEC, KOH, KOVALEV, AND ONNINEN
In opposite direction, for every ǫ > , Ω ∗ with Mod Ω ∗ > log cosh Mod Ω − ǫ , forwhich there is no energy-minimal homeomorphism in H , (Ω , Ω ∗ ). See [2,Corollary 3] or Example 9.1. More generally, we have the following counter-part to Theorem 2.3. Theorem 2.4.
There is a nondecreasing function
Υ : (0 , ∞ ) → (0 , suchthat lim τ →∞ Υ( τ ) = 1 and the following holds. Whenever two bounded doublyconnected domains Ω and Ω ∗ in C admit an energy-minimal diffeomorphism h : Ω onto −→ Ω ∗ , we have (2.2) Mod Ω ∗ > (Mod Ω) · Υ(Mod Ω) . Specifically, one can take Υ( τ ) = exp (cid:18) − π τ (cid:19) · Λ (cid:18) coth π τ (cid:19) , whereΛ( t ) = log t − log(1 + log t )2 + log t , t > . (2.3)In §
11 we conjecture that (2.2) can be specified as Mod Ω ∗ > log cosh Mod Ω,which would be the sharp bound, known to be true for circular annuli [2].3. Basic properties of deformations
In this section we establish the essential properties of the class of deforma-tions D (Ω , Ω ∗ ) introduced in Definition 2.2. Among them is that D (Ω , Ω ∗ )is sequentially weakly closed and its members satisfy a change of variableformula (3.9).Deformations enjoy two distinct properties, both of which are commonlyknown in literature as monotonicity. The topological monotonicity is thesubject of Lemma 3.7. To avoid confusion, in the following definition we usethe term oscillation property . Definition 3.1.
Let U be an open subset of C . A continuous function f : U → C is said to have oscillation property if for every compact set K ⊂ U we have(3.1) diam f ( K ) = diam f ( ∂K ) . Note that for real-valued functions (3.1) can be stated asmin K f = min ∂K f max ∂K f = max K f. The relevance of this property to Sobolev mappings hinges on the fol-lowing continuity estimate. If f ∈ W , ( U ) has the oscillation property,then(3.2) | f ( z ) − f ( z ) | C R D | Df | log (cid:0) e + diam D | z − z | (cid:1) , XISTENCE OF ENERGY-MINIMAL DIFFEOMORPHISMS 7 with z , z ∈ D for a pair of concentric disks D ⊂ D ⊂ U . The constant C is universal. See, e.g., Corollary 7.5.1 [19].The oscillation property (3.1) obviously holds for all homeomorphisms andis preserved under c -uniform limits. Therefore, deformations satisfy (3.1)and consequently (3.2): the local modulus of continuity of a deformation iscontrolled by its energy.Our approach to energy-minimal deformations involves the comparison ofenergies of h and h ◦ f , where f is a diffeomorphism or (more generally) aquasiconformal homeomorphism [1, 3, 25]. It is important to observe that h ◦ f is also a deformation. Lemma 3.2.
Let Ω , Ω ∗ and Ω ◦ be bounded domains in C . If f : Ω ◦ onto −→ Ω is a quasiconformal mapping then for any h ∈ D (Ω , Ω ∗ ) we have h ◦ f ∈ D (Ω ◦ , Ω ∗ ) .Proof. Since a K -quasiconformal mapping distorts the Dirichlet integralonly by a factor up to K , it follows that h ◦ f ∈ W , (Ω). That the Ja-cobian of h ◦ f is nonnegative follows from the chain rule det D ( h ◦ f ) =(det Dh )(det Df ). Finally, observe that if h j cδ −→ h in Ω, then h j ◦ f cδ −→ h ◦ f inΩ ◦ ; this purely topological fact only requires f to be a homeomorphism. (cid:3) Let us recall a change of variable formula for Sobolev mappings, foundin [3, Corollary 3.3.6], [19, Theorem 6.3.2] and [9].
Lemma 3.3.
Let Ω and Ω ∗ be bounded domains in C . Suppose that h : Ω → Ω ∗ is continuous and belongs to W , (Ω) . Then for any measurable function v : Ω ∗ → [0 , ∞ ) we have (3.3) Z Ω v (cid:0) h ( z ) (cid:1) | J h ( z ) | dz Z C v ( w ) N Ω ( h, w ) dw. where N Ω ( h, w ) is the cardinality of the preimage h − ( w ) . If, in addition, h satisfies Lusin’s condition ( N ) then the equality holds in (3.3) . Lusin’s condition ( N ) means that | f ( E ) | = 0 whenever | E | = 0.Hereafter deg Ω ( h, w ) stands for the degree of a mapping h with respectto a point w [28]. The degree is well-defined provided that h ∈ C (Ω) and w / ∈ h ( ∂ Ω). However, we work with mappings that are not necessarilycontinuous up to the boundary. In that case deg Ω ( h, w ) still makes senseas long as the values of h near ∂ Ω are bounded away from w . Specifically,deg Ω ( h, w ) := deg e Ω ( h, w ) where e Ω ⋐ Ω is any compactly contained domainsuch that inf Ω \ e Ω | h − w | > Lemma 3.4.
For any h ∈ D (Ω , Ω ∗ ) we have h (Ω) ⊃ Ω ∗ .Proof. We will prove the stronger statement(3.4) deg Ω ( h, w ) = 1 for all w ∈ Ω ∗ . IWANIEC, KOH, KOVALEV, AND ONNINEN
Pick a point w ∈ Ω ∗ and let δ = dist( w, ∂ Ω ∗ ). Consider the open set U = n z ∈ Ω : δ h ( z ) > δ o ⋐ Ω . Let { h j } be an approximating sequence for h . For sufficiently large j we have | δ h j − δ h | < δ in Ω and | h − h j | δ in U . For all z ∈ Ω \ U , δ h j ( z ) δ/ | h j ( z ) − w | > δ z ∈ Ω \ U. Since h j is a homeomorphism it attains the value w at some point z ◦ in U .Let U ◦ be the component of z ◦ in U . Clearly, deg U ◦ ( h j , w ) = 1. On theboundary ∂U ◦ we have | h j − h | δ/
4, which together with (3.5) implydeg U ◦ ( h, w ) = deg U ◦ ( h j , w ) = 1 . It remains to observe that h ( z ) = w for z ∈ Ω \ U ◦ . Indeed, by (3.5)the preimage of the open disk D ( w, δ/
2) under the homomorphism h j is aconnected subset of U , hence a subset of U ◦ . It follows that | h ( z ) − w | > | h j ( z ) − w | − δ > δ , z ∈ Ω \ U ◦ as desired. (cid:3) Definition 3.5.
A continuous mapping f : X → Y between metric spaces X and Y is monotone if for each y ∈ f ( X ) the set f − ( y ) is compact andconnected. Proposition 3.6. [40, VIII.2.2] If X is compact and f : X onto −→ Y is mono-tone then f − ( C ) is connected for every connected set C ⊂ Y . See [31, 32, 40] for the background on monotone mappings. Deforma-tions are closely related to monotone mappings of S onto itself. Given two k -connected bounded domains Ω and Ω ∗ in C , we choose and fix homeomor-phisms(3.6) χ : Ω onto −→ S \ P and χ ∗ : Ω ∗ onto −→ S \ P where P ⊂ S consists of k points referred to as punctures. A homeomor-phism h : Ω onto −→ Ω ∗ induces unique homeomorphism ⊲⊳ h : S onto −→ S such that(3.7) ⊲⊳ h ◦ χ = χ ∗ ◦ h. Note that ⊲⊳ h takes punctures into punctures in a one-to-one correspon-dence, though it may permute the elements of P . We claim that if a se-quence of homeomorphisms h j : Ω onto −→ Ω ∗ converges cδ -uniformly, then themappings ⊲⊳ h j converge uniformly on S . Indeed, fix a small ǫ > ǫ -neighborhood of the punctures, denoted N i ( ǫ ), i = 1 , . . . , k , are dis-joint. The uniform convergence of { δ h j } allows us to choose σ > N i ( σ ) is mapped by ⊲⊳ h j into the union S i N i ( ǫ ) when j XISTENCE OF ENERGY-MINIMAL DIFFEOMORPHISMS 9 is large. Being connected, the set ⊲⊳ h j ( N i ( σ )) must be contained in N π ( i ) ( ǫ )where π is a permutation of the set { , . . . , k } , possibly dependent on j .But in fact, π does not depend on j when j is large enough, due to uniformconvergence of ⊲⊳ h j on the boundaries of N i ( σ ).Thus we conclude that the sequence { ⊲⊳ h j } converges uniformly to a sur-jective mapping, denoted by ⊲⊳ h , which leaves the set P invariant but maypermute its elements. In a summary, for any deformation h ∈ D (Ω , Ω ∗ ) thereexists a unique mapping ⊲⊳ h : S onto −→ S satisfying (3.7). Being a uniform limitof self-homeomorphisms of S , this mapping is monotone [40, IX.3.11]. Themonotonicity of ⊲⊳ h has direct implications for h , which we state as a lemmafor future references. Lemma 3.7.
Let Ω and Ω ∗ be bounded k -connected domains in C , k =1 , , . . . and h ∈ D (Ω , Ω ∗ ) . Then ⊲⊳ h is monotone. Consequently, for anyconnected set C ⊂ Ω ∗ the preimage h − ( C ) is also connected. Moreover, forevery continuum C ⊂ Ω ∗ the set h − ( C ) ∪ Γ ∪ · · · ∪ Γ ℓ is a continuum, where Γ , . . . , Γ ℓ are those selected components of ∂ Ω which intersect the closureof h − ( C ) . Concerning Lemma 3.7 we remark that h : Ω → Ω ∗ is not necessarilymonotone; however, the restriction of h to h − (Ω ∗ ) is.Next we turn to analytic properties of deformations. Lemma 3.8.
Let Ω and Ω ∗ be bounded domains in C . If h ∈ D (Ω , Ω ∗ ) ,then h satisfies Lusin’s condition ( N ) and N Ω ( h, w ) = 1 for almost every w ∈ Ω ∗ . Also J h = 0 almost everywhere in Ω \ h − (Ω ∗ ) .Proof. By Theorem A in [29] Lusin’s condition ( N ) is true for all continuous W , -mappings that satisfy the oscillation inequality (3.1). Since the latterholds for any deformation (Lemma 3.12), the condition ( N ) is satisfied.By the definition of a deformation, Z Ω J h ( z ) dz | Ω ∗ | Invoking Lemmas 3.3 and 3.4 we arrive at(3.8) | Ω ∗ | > Z Ω J h ( z ) dz = Z C N Ω ( h, w ) dw > Z Ω ∗ N Ω ( h, w ) dw > | Ω ∗ | . Therefore, equality holds throughout in (3.8). This yields N Ω ( h, w ) = 1 a.e.in Ω ∗ and J h = 0 a.e. in Ω \ h − (Ω ∗ ), as claimed. (cid:3) Corollary 3.9.
Let Ω and Ω ∗ be bounded domains in C . If h ∈ D (Ω , Ω ∗ )and v : Ω ∗ → [0 , ∞ ) is measurable, then(3.9) Z Ω v (cid:0) h ( z ) (cid:1) J h ( z ) dz = Z Ω ∗ v ( w ) dw. Proof.
Let G = h − (Ω ∗ ). Combining Lemmas 3.3 and 3.8 we have Z Ω v (cid:0) h ( z ) (cid:1) J h ( z ) dz = Z G v (cid:0) h ( z ) (cid:1) J h ( z ) dz = Z C v ( w ) N G ( h, w ) dw = Z Ω ∗ v ( w ) dw. (cid:3) In general, a deformation may take a part of Ω into ∂ Ω ∗ . This is thesubject of our next lemma. Lemma 3.10.
Suppose that h ∈ D (Ω , Ω ∗ ) where Ω and Ω ∗ are bounded dou-bly connected domains. Let G = { z ∈ Ω : h ( z ) ∈ Ω ∗ } . Then G is a domainseparating the boundary components of Ω . Precisely, the two components of ∂ Ω lie in different components of C \ G .Proof. The set G is open by the continuity of h , and connected by Lemma 3.7.Let ∂ I Ω ∗ and ∂ O Ω ∗ be the inner and outer components of the boundary ofΩ ∗ . The function δ ( z ) := dist( h ( z ) , ∂ I Ω ∗ )dist( h ( z ) , ∂ I Ω ∗ ) + dist( h ( z ) , ∂ O Ω ∗ ) , z ∈ Ω , extends continuously to C by setting the values 0 and 1 in the componentsof C \ Ω. The disjoint open sets { z ∈ C : | δ ( z ) | < / } and { z ∈ C : | δ ( z ) | > / } cover C \ G in such a way that each of them contains one and only oneboundary component of Ω. Thus G separates the components of ∂ Ω. (cid:3) In order to prove that D (Ω , Ω ∗ ) is sequentially weakly closed, we need anestimate near the boundary stated as Proposition 3.11 below. For Sobolevhomeomorphisms a similar result was proved in [17] in all dimensions. Theextension beyond homeomorphisms is deferred to § Proposition 3.11.
Let Ω and Ω ∗ be bounded k -connected domains, k < ∞ . Denote their boundary components by Γ i and Γ ∗ i , i = 1 , . . . , k .Assume that diam Γ i > for all i k . Then there exist functions η i , i k , continuous in C and vanishing on Γ i , such that the followingholds. If h : Ω → Ω ∗ is a continuous W , -mapping such that h (Ω) ⊃ Ω ∗ , h is monotone on the set h − (Ω ∗ ) , and (3.10) dist( h ( z ) , Γ ∗ i ) → as dist( z, Γ i ) → , i = 1 , . . . , k, then (3.11) dist( h ( z ) , Γ ∗ i ) η i ( z ) p E [ h ] , i = 1 , . . . , k. Lemma 3.12.
Let Ω and Ω ∗ be bounded k -connected domains, k < ∞ .Assume that the boundary components of Ω do not degenerate into points.If a family of deformations F ⊂ D (Ω , Ω ∗ ) is bounded in W , (Ω) then it isequicontinuous on compact subsets of Ω and the family ∆ F := { δ h : h ∈ F } is equicontinuous on Ω . XISTENCE OF ENERGY-MINIMAL DIFFEOMORPHISMS 11
Proof.
The equicontinuity of F on compact subsets of Ω is readily seenfrom (3.2). It follows that ∆ F is equicontinuous on compact subsets as well.To show that it is actually equicontinuous on Ω it remains to prove that forany ǫ > K ⊂ Ω such that δ h < ǫ on Ω \ K for all h ∈ F . This is exactly what Proposition 3.11 delivers. (cid:3) Lemma 3.13.
Let Ω and Ω ∗ be bounded k -connected planar domains, k < ∞ . Assume that the boundary components of Ω do not degenerate intopoints. If a sequence { h j } ⊂ D (Ω , Ω ∗ ) converges weakly in W , (Ω) , thenits limit belongs to D (Ω , Ω ∗ ) Proof.
Let h be the weak limit of h j ∈ D (Ω , Ω ∗ ). Its Jacobian determinant J h is nonnegative a.e. in Ω due to L -weak convergence of Jacobians under W , -weak limits [19, Theorem 8.4.2]. The weak convergence also impliesthat R Ω J h | Ω ∗ | . It remains to show that h has an approximating sequenceof homeomorphisms. For this it is enough to prove that h j cδ −→ h in Ω.Indeed, each h j being a cδ -uniform limit of homeomorphisms, the diagonalselection will produce the desired approximating sequence.By Lemma 3.12 the sequence { h j } is equicontinuous on any compactsubset of Ω. With the help of the Arzel`a-Ascoli theorem it is routine toprove that h j → h c -uniformly. In particular, δ h j → δ h pointwise. Theconvergence is uniform because the functions δ h j are equicontinuous in Ω byvirtue of Lemma 3.12. It follows that h j cδ −→ h as claimed. (cid:3) Due to the weak lower semicontinuity of the Dirichlet energy, Lemma 3.13has a useful corollary.
Corollary 3.14.
Under the hypotheses of Lemma 3.13 there exists h ∈ D (Ω , Ω ∗ ) such that E [ h ] = E (Ω , Ω ∗ ).Note that Lemma 3.13 fails for k = 1. Indeed, the M¨obius transformations f a ( z ) = z − a − a ¯ z converge weakly in W , to a constant mapping (not a deformation) as a →
1. We conclude this section with a promised remark on the existenceof homeomorphisms of class H , (Ω , Ω ∗ ). Lemma 3.15.
Let Ω and Ω ∗ be bounded doubly connected domains in C .Then H , (Ω , Ω ∗ ) is nonempty, except for one degenerate case when Mod Ω = ∞ and Mod Ω ∗ < ∞ . In this case there is no homeomorphism h : Ω onto −→ Ω ∗ of Sobolev class W , (Ω) .Proof. Suppose that the degenerate case takes place. Then Ω = V \ { z } where V is a simply connected domain. Since isolated points are removablefor monotone W , functions [17, Theorem 3.1], the mapping h has a con-tinuous extension to V . But then Ω ∗ = h ( V ) \ { h ( z ) } , which contradictsthe finiteness of Mod Ω ∗ . Conversely, suppose that the degenerate case fails. If Mod Ω = Mod Ω ∗ = ∞ , then there exists a conformal mapping h : Ω onto −→ Ω ∗ for which E [ h ] =2 | Ω ∗ | < ∞ by virtue of (1.2). The remaining case is when both domains havefinite modulus. Then we map them conformally onto circular annuli A and A ∗ and compose them with a radial quasiconformal mapping ψ : A onto −→ A ∗ , ψ ( z ) = | z | α − z, α = Mod Ω ∗ Mod Ω . This creates an element in H , (Ω , Ω ∗ ). (cid:3) Harmonic replacement
Let Ω be a domain in C and U ⋐ Ω a bounded simply connected domain.For any continuous function f : Ω → C there exists a unique continuousfunction P U f : Ω → C , called the Poisson modification of f , such that P U f is harmonic in U and agrees with f on Ω \ U . Indeed, the Dirichlet problemwith continuous boundary data has a continuous solution in any simplyconnected domain, e.g., [33, Theorem 4.2.1] or [8, Ch.III]. Furthermore, P U f ∈ W , (Ω) whenever f ∈ W , (Ω). Although the latter fact is surelyknown, we give an explanation. The function P U f can be constructed bythe Wiener method [8, Theorem III.5.1] as a c -uniform limit(4.1) P U f = lim n →∞ P U n f, U ⋐ U ⋐ . . . where { U n } is an exhaustion of U by smooth Jordan domains. Since thedifference P U n f − f vanishes on the smooth boundary ∂U n , it extends by zeroto a function in W , (Ω). Adding f to it, we conclude that P U n f ∈ W , (Ω),with a uniform bound on the W , -norm thanks to Dirichlet’s principle.Thus, { P U n f } contains a subsequence that converges weakly in W , (Ω).Its limit must be P U f since P U n f → P U f uniformly.The following lemma generalizes the well-known Rad´o-Kneser-ChoquetTheorem on the univalence of harmonic extensions. The added generality isin that the domain U is not required to be Jordan. Lemma 4.1 (Modification Lemma) . Let U and D be bounded simply con-nected domains in C with D convex. Suppose that f is a homeomorphismfrom U onto D with continuous extension f : U → D . Then there exists aunique harmonic homeomorphism h : U onto −→ D which agrees with f on theboundary. Specifically, h has a continuous extension to U which coincideswith f on ∂U .Proof. The existence and uniqueness of a continuous harmonic extension h of f (cid:12)(cid:12) ∂U are well known. Also h ( U ) ⊃ D by a straightforward degreeargument and h ( U ) ⊂ D by the maximum principle. Thus the essence ofthe lemma is injectivity of h .Let { D n } be an exhaustion of D by convex domains and define U n = f − ( D n ), which is a Jordan domain. By the Rad´o-Kneser-Choquet Theo-rem [7, p. 29] the mapping h n := P U n f is harmonic homeomorphism of U n XISTENCE OF ENERGY-MINIMAL DIFFEOMORPHISMS 13 onto D n . As n → ∞ , h n → P U f =: h c -uniformly on U , see [8, Ch.III]. Theconvergence of harmonic functions implies the convergence of their deriva-tives. Therefore J h n → J h pointwise, in particular J h >
0. This means thatthe holomorphic functions h z and h ¯ z satisfy the inequality | h ¯ z | | h z | . Thisis only possible when either | h ¯ z | < | h z | in U or | h ¯ z | ≡ | h z | in U . The sec-ond case cannot occur, for it would yield J h ≡
0, contradicting h ( D ) ⊃ U .Therefore J h >
0, so the mapping h is a local diffeomorphism. Being also a c -uniform limit of homeomorphisms, h is a diffeomorphism of U . (cid:3) We are now in the position to apply the Poisson modification to deforma-tions.
Lemma 4.2.
Let Ω and Ω ∗ be bounded k -connected domains, k < ∞ .Suppose that h ∈ D (Ω , Ω ∗ ) satisfies h (Ω) = Ω ∗ . Let D be a convex domainsuch that D ⊂ Ω ∗ . Denote U = h − ( D ) and g = P U h . Then(i) g ∈ D (Ω , Ω ∗ ) (ii) The restriction of g to U is a harmonic diffeomorphism onto D .(iii) E [ g ] E [ h ] with equality if and only if g ≡ h .Proof. We use the notation of (3.6) and (3.7). By Lemma 3.7 the inducedmapping ⊲⊳ h : S onto −→ S is monotone, so we can apply Theorem II.1.47 in [32].According to which there exists a monotone mapping f : S onto −→ S whichtakes χ ( U ) homeomorphically onto χ ∗ ( D ) and agrees with ⊲⊳ h on S \ χ ( U ).This allows us to apply the Modification Lemma 4.1 to h . Thus the Poissonmodification g = P U h performs a harmonic diffeomorphism of U onto D .Clearly, ⊲⊳ g : S onto −→ S is monotone. Any such mapping can be uniformlyapproximated by homeomorphisms [32, Theorem II.1.57]. We can actuallyalter slightly these homeomorphisms so as to obtain a sequence of homeo-morphisms g j : S onto −→ S that agree with ⊲⊳ g at the punctures P ⊂ S , andstill g j → ⊲⊳ g uniformly on S . Every such homeomorphism g j : S onto −→ S is represented by a homeomorphism h j : Ω onto −→ Ω ∗ by the rule g j = ⊲⊳ h j ,where ⊲⊳ h j is determined from the equation (3.7); ⊲⊳ h j ◦ χ = χ ∗ ◦ h j . Theuniform convergence of g j implies that h j cδ −→ g in Ω. Thus we concludethat g ∈ D (Ω , Ω ∗ ). The inequality E [ P U h ] E [ h ] is merely a restatementof Dirichlet’s principle. (cid:3) Reich-Walczak-type inequalities
The Reich-Walczak inequalities [35] provide the upper and lower boundsfor the conformal modulus of the image of a circular annulus under a qua-siconformal homeomorphism. Propositions 5.1 and 5.2 provide such boundsfor deformations, which are in general neither quasiconformal nor homeo-morphisms. We also treat Sobolev homeomorphisms in W , , for which simi-lar inequalities were established in [30] in the context of self-homeomorphismsof a disk that agree with the identity mapping on the boundary. However, we work with doubly connected domains and do not prescribe boundaryvalues.Let us introduce notation for several quantities associated with the deriva-tives of a mapping f . We make use of polar coordinates ρ and θ and theassociated normal and tangential derivatives f N = f ρ and f T = f θ ρ . In these terms the Wirtinger derivatives f z and f ¯ z are expressed as f z = e − iθ f N − if T ) f ¯ z = e iθ f N + if T )Also, the Jacobian determinant of f is J f = | f z | − | f ¯ z | = Im f N f T . Except for the origin, where polar coordinates collapse, we may define the normal and tangential distortion of f as follows. K fN := | f z + ¯ zz f ¯ z | J f = | f N | J f (5.1) K fT := | f z − ¯ zz f ¯ z | J f = | f T | J f (5.2)By convention these quotients are understood as 0 whenever the numeratorvanishes. Naturally, they assume the value + ∞ if the Jacobian vanishes butthe numerator does not. For a mapping f ∈ W , the quantities f N , f T ,and J f are finite a.e., and therefore K fN and K fT are unambiguously definedat almost every point of the domain of definition of f . Proposition 5.1.
Let Ω and Ω ∗ be bounded doubly connected domains suchthat Ω separates and ∞ . Suppose that either(a) f ∈ D (Ω , Ω ∗ ) or(b) f : Ω onto −→ Ω ∗ is a sense-preserving homeomorphism of class W , (Ω , Ω ∗ ) .Then (5.3) 2 π Mod Ω ∗ Z Ω K fN dz | z | . Proof.
There is nothing to prove if the integral in (5.3) is infinite, so we as-sume K fN < ∞ a.e. There exists a conformal mapping Φ : Ω ∗ → A ( r ∗ ,
1) =: A ∗ where 0 r ∗ < ∗ = log 1 /r ∗ . Let G = { z ∈ Ω : f ( z ) ∈ Ω ∗ } and define g : G → A ∗ by g = Φ ◦ f . Note that G = Ω if weare in the case (b).Fix ǫ >
0. We claim that(5.4) Z G | g N || g | + ǫ dz | z | > π log 1 + ǫr ∗ + ǫ . XISTENCE OF ENERGY-MINIMAL DIFFEOMORPHISMS 15
Let ℓ θ = { ρe iθ ∈ G : ρ > } , θ ∈ [0 , π ]. For almost every θ ∈ [0 , π ] themapping g is locally absolutely continuous on ℓ θ . The image of ℓ θ under g is aunion of curves in A ∗ which approach the boundary of A ∗ in both directions.At least one of them connects two boundary components of A ∗ because G separates the boundary components of Ω by Lemma 3.10. Therefore thefunction | g | attains all values between r ∗ and 1 when restricted to someconnected component of ℓ θ ∩ G . It follows that Z ℓ θ ∩ G | g N || g | + ǫ > log 1 + ǫr ∗ + ǫ . Integration with respect to θ yields (5.4). Using the normal distortion in-equality | g N | K gN J g and the Cauchy-Schwarz inequality we obtain (cid:18)Z G | g N || g | + ǫ dz | z | (cid:19) Z G ( K gN J g ) / | g | + ǫ dz | z | ! Z G J g ( | g | + ǫ ) Z G K gN dz | z | Z G J g ( | g | + ǫ ) Z Ω K gN dz | z | . Since Φ is conformal, K gN = K fN . Thus we infer from (5.4) that(5.5) (cid:18) log 1 + ǫr ∗ + ǫ (cid:19) π ) Z Ω K fN dz | z | Z G J g ( | g | + ǫ ) . From Lemmas 3.3 and 3.8 we obtain(5.6) Z G J g ( | g | + ǫ ) Z A ∗ dw ( | w | + ǫ ) π log 1 + ǫr ∗ + ǫ . It follows from (5.5) and (5.6) thatlog 1 + ǫr ∗ + ǫ π Z Ω K fN dz | z | . Letting ǫ → (cid:3) Unlike Proposition 5.1, our lower bound for the modulus of the imageunder a deformation depends on the rectifiability of the boundary of Ω ∗ .We do not know if this assumption is redundant. Proposition 5.2.
Let A = A ( r, R ) be a circular annulus, r < R < ∞ ,and Ω ∗ a bounded doubly connected domain with finite modulus. Supposethat either(a) f ∈ D ( A , Ω ∗ ) and Ω ∗ is bounded by rectifiable Jordan curves, or(b) f : A onto −→ Ω ∗ is a sense-preserving homeomorphism of class W , ( A , Ω ∗ ) .Then (5.7) Z A K fT dz | z | > π (Mod A ) Mod Ω ∗ . Before proving Proposition 5.2 we collect some results concerning theHardy space H ( A ) on a circular annulus A = A ( r, R ), 0 < r < R < ∞ . Inwhat follows T ρ = { z ∈ C : | z | = ρ } , ρ >
0. By definition, a holomorphicfunction ψ : A → C belongs to H ( A ) if the integrals R T ρ | ψ | are uniformlybounded for r < ρ < R . Such a function ψ has nontangential limits a.e. on ∂ A [36, p.6], and ψ = 0 a.e. on ∂A unless ψ ≡ H ( A ) and domains with rectifiable boundaries is summarized inthe following proposition which is a version of classical theorems due toF. and M. Riesz and V. I. Smirnov. Below H denotes the one-dimensionalHausdorff measure, not to be confused with the Hardy space. Proposition 5.3.
Let Ω be a doubly connected domain bounded by rectifiableJordan curves and let Ψ : A = A ( r, R ) → Ω be conformal. Then(i) Ψ ′ ∈ H ( A ) (ii) for any Borel set E ⊂ ∂ A we have H (Ψ( E )) = R E | Ψ ′ | (iii) H (Ψ( E )) = 0 if and only if H ( E ) = 0 .Proof. Part (i) is proved in exactly the same way as the corresponding resultfor the disk [8, p. 200]. Since Ψ ′ ∈ H , the continuous extension of Ψ to ∂ A is absolutely continuous, i.e., (ii) holds. Part (iii) follows from (ii) becauseΨ ′ = 0 a.e. on ∂ A . (cid:3) Proof of Proposition . There is nothing to prove if the integral in (5.3) isinfinite, so we assume K fT < ∞ a.e. Let G = { z ∈ A : f ( z ) ∈ Ω ∗ } . Note that G coincides with A if we are in the case (b). On the set A \ G the Jacobian J f vanishes by Lemma 3.10. Since K fT is finite a.e., it follows that f θ = 0a.e. on A \ G . There exists a conformal mapping Φ : Ω ∗ → A ( r ∗ ,
1) =: A ∗ ,where 0 < r ∗ < ∗ = log 1 /r ∗ . In case (a) Φ extendsto a homeomorphism Φ : Ω ∗ → A ∗ . In either case (a) or (b) we can define g = Φ ◦ f : A → A ∗ .We claim that(5.8) Z G | g T || g | dz | z | > π Mod A . Indeed, for almost every circle T ρ ⊂ A the mapping f is absolutely contin-uous on T ρ and(5.9) f θ = 0 a.e. on T ρ \ G. For any such ρ we are going to prove the inequality(5.10) Z T ρ ∩ G | g T || g | > π, from which (5.8) will follow by integration.In the case (b) the inequality (5.10) is a direct consequence of the factthat the curve g ( T ρ ) separates the boundary components of A ∗ ; indeed, thelength of any such curve in the logarithmic metric | dz | / | z | is at least 2 π . XISTENCE OF ENERGY-MINIMAL DIFFEOMORPHISMS 17
We now turn to the case (a). Let w ◦ be an interior point of the boundedcomponent of C \ Ω ∗ . Choose an approximating sequence { h j } j ∈ N ⊂ H , (Ω , Ω ∗ )that converges to f . Note that for each j the multivalued function arg( h j ( z ) − w ◦ ) increases by 2 π on T ρ . Letting j → ∞ we obtain the same for f ; inparticular, f ( T ρ ) separates w ◦ from ∞ . Since Φ : Ω ∗ → A ∗ is a homeomor-phism, g ( T ρ ) is a closed curve in A ∗ which separates 0 from ∞ . Therefore, itslength in the logarithmic metric | dz | / | z | is at least 2 π . By virtue of (5.9) theintersection of f ( T ρ ) with ∂ Ω ∗ has zero length. By part (iii) of Theorem 5.3we have H ( g ( T ρ ) ∩ ∂ A ∗ ) = 0. Hence, the part of g ( T ρ ) that is contained in A ∗ has logarithmic length at least 2 π . This is exactly what (5.10) claimed.Now that (5.8) is at our disposal, we proceed as in the proof of Proposi-tion 5.1. The Cauchy-Schwarz inequality yields (cid:18)Z G | g T || g | dz | z | (cid:19) Z G ( K gT J g ) / | g | dz | z | ! Z G J g | g | Z G K gT dz | z | π log 1 r ∗ Z A K gT dz | z | = 2 π Mod Ω ∗ Z A K gT dz | z | . (5.11)where the second to last inequality follows from (5.6). It remains to com-bine (5.8) and (5.11). (cid:3) Hopf differentials
We call a deformation h ∈ D (Ω , Ω ∗ ) stationary if(6.1) ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 E [ h ◦ φ − t ] = 0for every family of diffeomorphisms φ t : Ω → Ω which depend smoothly onthe parameter t ∈ R and satisfy φ = id. It should be emphasized thatapart from φ , the diffeomorphisms φ t need not agree with the identity onthe boundary. The derivative in (6.1) exists for any h ∈ W , (Ω), see [37,p. 158]. Every energy-minimal deformation is stationary. Indeed, h ◦ φ − t belongs to D (Ω , Ω ∗ ) by virtue of Lemma 3.2. The minimal property of h implies E [ h ◦ φ − t ] > E [ h ], from where (6.1) follows.The key property of the stationary mapping in (6.1) is that: • The function ϕ := h z h ¯ z , a priori in L (Ω), is actually holomorphic. • If ∂ Ω is C -smooth then ϕ extends continuously to Ω, and the qua-dratic differential ϕ dz is real on each boundary curve of Ω.See [23, Lemma 1.2.5] for the proof of the above facts and [21, Chapter III]for the background on quadratic differentials and their trajectories. Let usconsider the special case Ω = A ( r, R ) with 0 < r < R < ∞ . Since ϕ dz is real on each boundary circle, the function z ϕ ( z ) is real on ∂ Ω. By themaximum principle(6.2) z ϕ ( z ) ≡ c ∈ R . We state this as a lemma for the ease of future references.
Lemma 6.1.
Let
Ω = A ( r, R ) be a circular annulus, < r < R < ∞ , and Ω ∗ a bounded doubly connected domain. If h ∈ D (Ω , Ω ∗ ) is a stationarydeformation, then (6.3) h z h ¯ z ≡ cz in Ω . where c ∈ R is a constant. Furthermore, (6.4) ( | h N | J h , if c | h T | J h , if c > . Proof.
The validity of (6.3) with some c ∈ R was already recognized in (6.2).Separating the real and imaginary parts in (6.3) we arrive at two equations | h N | − | h T | = 4 c | z | ;(6.5) Re( h N h T ) = 0 . (6.6)Recall that J h = Im h N h T >
0, which in view of (6.6) reads as(6.7) J h = | h N || h T | Combining this with (6.5) the claim (6.4) follows. (cid:3)
Lemma 6.1 together with Propositions 5.1 and 5.2 give the following corol-lary.
Corollary 6.2.
Under the hypotheses of Lemma 6.1, we have • if Mod Ω < Mod Ω ∗ , then c > • if Mod Ω > Mod Ω ∗ and Ω ∗ is bounded by rectifiable Jordan curves,then c < Monotonicity of the minimum energy function
Due to the conformal invariance of the Dirichlet integral and of theclass of deformations (Lemma 3.2), the minimal energy level E (Ω , Ω ∗ ), de-fined by (2.1), depends only on the conformal type of Ω as long as Ωis bounded and Ω ∗ is fixed. This leads us to consider a one-parameterfamily of extremal problems for homeomorphisms A ( τ ) onto −→ Ω ∗ of annuli A ( τ ) = A (1 , e τ ), 0 < τ < ∞ . In this section we are concerned with the quan-tity E ( τ, Ω ∗ ) := E ( A ( τ ) , Ω ∗ ) as a function of τ , called the minimum energyfunction . When Ω ∗ has finite conformal modulus, E ( τ, Ω ∗ ) attains its mini-mum at τ = Mod Ω ∗ . Indeed, by (1.2) for every τ we have E ( τ, Ω ∗ ) > | Ω ∗ | ,with equality if and only if Ω and Ω ∗ are conformally equivalent; that is, for τ = Mod Ω = Mod Ω ∗ . The following monotonicity result, which extendsthis observation, will be of crucial importance in the proof of Theorem 2.3. Proposition 7.1.
Let Ω ∗ be a bounded doubly connected domain. The func-tion τ E ( τ, Ω ∗ ) is strictly decreasing for < τ < Mod Ω ∗ . If, in addition, Ω ∗ is bounded by rectifiable Jordan curves, then E ( τ, Ω ∗ ) is strictly increasingfor τ > Mod Ω ∗ . XISTENCE OF ENERGY-MINIMAL DIFFEOMORPHISMS 19
The proof of Proposition 7.1 requires auxiliary results concerning thenormal and tangential energies E N [ h ] = Z Ω | h N | , E T [ h ] = Z Ω | h T | . Clearly E [ h ] = E N [ h ] + E T [ h ]. Unlike E [ h ], both quantities E N [ h ] and E T [ h ]transform in a straighforward way under composition with the power stretchmapping(7.1) ψ α ( z ) := | z | α − z, < α < ∞ . Specifically,(7.2) E N [ h ◦ ψ ] = α E N [ h ] , E T [ h ◦ ψ ] = 1 α E T [ h ] . The direct verification of (7.2) is left to the reader. We only note that thedomain of definition of h here is irrelevant as the computation is local. Lemma 7.2.
Let Ω ∗ be a bounded doubly connected domain, τ ◦ ∈ (0 , ∞ ) .Suppose that h ◦ ∈ D ( A ( τ ◦ ) , Ω ∗ ) is an energy-minimal deformation. Thenfor all < τ < ∞ we have (7.3) E ( τ, Ω ∗ ) τ ◦ τ E N [ h ◦ ] + ττ ◦ E T [ h ◦ ] . Proof.
Let α = τ ◦ τ and note that ψ α defined by (7.1) is a quasiconformalmapping of A ( τ ) onto A ( τ ◦ ). By Lemma 3.2 the composition h ◦ ◦ ψ α belongsto D ( A ( τ ) , Ω ∗ ) and by (7.2) we have E ( τ, Ω ∗ ) E [ h ◦ ◦ ψ α ] = α E N [ h ◦ ] + 1 α E T [ h ◦ ] . (cid:3) Let us apply Lemma 7.2 with τ ◦ = Mod Ω ∗ . In this case h ◦ : Ω onto −→ Ω ∗ isconformal so E N [ h ◦ ] = E T [ h ◦ ] = | Ω ∗ | . We obtain a simple upper bound forthe minimal energy function,(7.4) E ( τ, Ω ∗ ) (cid:18) Mod Ω ∗ τ + τ Mod Ω ∗ (cid:19) | Ω ∗ | , < τ < ∞ . Corollary 7.3.
The function E ( τ, Ω ∗ ) is locally Lipschitz for 0 < τ < ∞ .Indeed the existence of h ◦ in Lemma 7.2 is assured by Corollary 3.14.From Lemma 7.2 for arbitrary 0 < τ ◦ , τ < ∞ we have E ( τ, Ω ∗ ) − E ( τ ◦ , Ω ∗ ) τ ◦ τ E N [ h ◦ ] + ττ ◦ E T [ h ◦ ] − E N [ h ◦ ] − E T [ h ◦ ]= ( τ − τ ◦ ) (cid:26) E T [ h ◦ ] τ ◦ − E N [ h ◦ ] τ (cid:27) (7.5)from where the local Lipschitz property is readily seen. Proof of Proposition . Since E ( τ, Ω ∗ ) is locally Lipschitz, its derivativeexists for almost every τ ∈ (0 , ∞ ). Fix such a point of differentiablity, say0 < τ ◦ < Mod Ω ∗ . Let h ◦ ∈ D ( A ( τ ◦ ) , Ω ∗ ) be an energy-minimal deformation.By Lemma 6.1 | h ◦ N | = | h ◦ T | + 4 c | z | , hence upon integration E N [ h ◦ ] = E T [ h ◦ ] + 8 cπτ ◦ . Now, for any τ ∈ (0 , ∞ ) the estimate (7.5) takes the form(7.6) E ( τ, Ω ∗ ) − E ( τ ◦ , Ω ∗ ) ( τ − τ ◦ ) (cid:8) − cπ + ( τ − ◦ − τ − ) E N [ h ◦ ] (cid:9) Therefore ddt (cid:12)(cid:12)(cid:12)(cid:12) τ = τ ◦ E ( τ, Ω ∗ ) = − πc. Corollary 6.2 completes the proof. (cid:3) Existence: Theorem
Proposition 8.1.
Let Ω and Ω ∗ be bounded doubly connected domains. Sup-pose that h ∈ D (Ω , Ω ∗ ) satisfies E [ h ] = E (Ω , Ω ∗ ) . Let G = { z ∈ Ω : h ( z ) ∈ Ω ∗ } . Then G is a doubly connected domain that separates the boundarycomponents of Ω . The restriction of h to G is a harmonic diffeomorphismonto Ω ∗ .Proof. The fact that G is a domain separating the boundary components ofΩ was established in Lemma 3.10. Each point z ∈ G has a neighborhood inwhich h is a harmonic diffeomorphism. Indeed, otherwise we would be ableto find a deformation with strictly smaller energy by means of Lemma 4.2.Thus, h : G onto −→ Ω ∗ is a local diffeomorphism. On the other hand, for each w ∈ Ω ∗ the preimage h − ( w ) is connected by Lemma 3.7. It follows that h : G onto −→ Ω ∗ is a diffeomorphism. Being a diffeomorphic image of Ω ∗ , thedomain G must be doubly connected. (cid:3) Proof of Theorem . If Mod Ω = Mod Ω ∗ , then the domains are confor-mally equivalent. As observed in §
1, a conformal mapping minimizes theDirichlet energy. Thus we only need to consider the case Mod Ω < Mod Ω ∗ .In particular Mod Ω < ∞ .Let h and G be as in Proposition 8.1. The existence of such h is guaranteedby Corollary 3.14. Since G separates the boundary components of Ω, wehave Mod G Mod Ω with equality if and only if G = Ω [25, Lemma 6.3].If Mod G <
Mod Ω, then by Proposition 7.1 Z G | Dh | > E (Mod G, Ω ∗ ) > E (Mod Ω , Ω ∗ ) = Z Ω | Dh | which is absurd. Thus G = Ω. By Proposition 8.1 the mapping h : Ω → Ω ∗ is a harmonic diffeomorphism. The uniqueness statement will follow fromProposition 10.2. (cid:3) XISTENCE OF ENERGY-MINIMAL DIFFEOMORPHISMS 21
Proof of Theorem . Suppose Mod Ω Mod Ω ∗ and let f ◦ : Ω onto −→ Ω ∗ bean energy-minimal diffeomorphism provided to us by Theorem 2.3. Forevery homeomorphism g : Ω ∗ onto −→ Ω with integrable distortion the inversemap f = g − : Ω onto −→ Ω ∗ belongs to the Sobolev class W , (Ω) and wehave [10, 11, 20](8.1) Z Ω ∗ K g ( w ) dw = Z Ω | Df ( z ) | dz > Z Ω | Df ◦ ( z ) | dz = Z Ω ∗ K g ◦ ( w ) dw where g ◦ = f − ◦ . The latter identity is legitimate because f ◦ is a diffeomor-phism. Thus g ◦ is also a C ∞ -smooth diffeomorphism. It has the smallestpossible L -norm of the distortion. If equality holds in (8.1) then, by The-orem 2.3, the mapping f − ◦ ◦ f is conformal. (cid:3) Nonexistence: Theorem G Ω whenever C \ Ω contains a nondegenerate continuum.Our normalization is G Ω ( z, ζ ) = − log | z − ζ | + O (1) as z → ζ . In particular, G Ω ( z, ζ ) >
0. Green’s function for the unit disk D is G D ( z, ζ ) = log (cid:12)(cid:12)(cid:12)(cid:12) − z ¯ ζz − ζ (cid:12)(cid:12)(cid:12)(cid:12) . If f : Ω → Ω ∗ is a holomorphic function, then the subordination principleholds:(9.1) G Ω ( z, ζ ) G Ω ∗ ( f ( z ) , f ( ζ )) . Proof of Theorem . If Ω is degenerate, so is Ω ∗ because a point is a remov-able singularity for W , -homeomorphisms [17, Theorem 3.1]. Therefore wemay assume, by a conformal change of variables in Ω, that Ω = A ( R − , R ), R >
1. By Lemma 6.1(9.2) h z h ¯ z ≡ cz in Ω . where c is real. If c >
0, then (6.4) yields | h T | J h , hence Mod Ω ∗ > Mod Ωby Proposition 5.2. It remains to consider the case c <
0. Let us write c = − b , b ∈ R . Introduce the so-called second complex dilatation(9.3) ν = h ¯ z h z which is a holomorphic function from Ω into the unit disk D [7, p. 5].Equation (9.2) implies that ν does not vanish and ν = − b z h z Therefore, ν has a single-valued square root, namely(9.4) ω = ibzh z . From (9.3) and (9.4) we have(9.5) h z = ibzω and h ¯ z = − ibω ¯ z . Now we integrate the differential form dh = h z dz + h ¯ z d ¯ z over the unit circle(9.6) 0 = Z T dh = ib Z T (cid:18) dzzω − ω d ¯ z ¯ z (cid:19) = ib Z T (cid:18) ω + ω (cid:19) dzz . The image of T under the map z ω | ω | is an arc Γ ⊂ T . This arc cannot becontained in any open half-circle, for then the values of the function ω − + ω = ω (1 + | ω | − ) on T would lie in an open halfplane, contradicting (9.6).Thus there exist points z , z ∈ T such that(9.7) ω ( z ) ω ( z ) < . We write w j = ω ( z j ), j = 1 ,
2, and invoke a simple lower bound for theGreen function of Ω, derived in [15, (3.9)]:(9.8) G Ω ( z , z ) > log coth π R , z , z ∈ T . By the subordination principle (9.1),(9.9) G Ω ( z , z ) G D ( w , w ) . Because of symmetry we may assume | w | | w | . The right hand sideof (9.9) is estimated from above using (9.7):(9.10) G D ( w , w ) = log 1 + | w w || w | + | w | log 1 + | w | | w | . Combining (9.8)–(9.10) we obtain an upper bound for | ω | on T ,(9.11) | ω ( z ) | tanh π R .
Introduce an auxiliary mapping g = φ ◦ h , where φ is an affine transformationchosen so that g becomes conformal at z ; that is, g ¯ z ( z ) = 0. It was provedin ([15], estimates (3.11) and (3.13)) thatMod g (Ω) > Mod Ω · Λ (cid:18) coth π τ (cid:19) , Λ( t ) = log t − log(1 + log t )2 + log t , t > . From (9.11) we haveMod h (Ω) > − | ω ( z ) | | ω ( z ) | Mod g (Ω) = exp (cid:18) − π R (cid:19) Mod g (Ω) . Combining the last two lines yields (2.3). (cid:3)
XISTENCE OF ENERGY-MINIMAL DIFFEOMORPHISMS 23
We complement Theorem 2.4 with an explicit example of two doublyconnected domains and a harmonic homeomorphism between them, whichdo not admit an energy-minimal homeomorphism.
Example 9.1.
Consider the annulus Ω = A (1 , R ), R >
1. Fix 0 < δ < ∗ be the image of the annulus A ∗ = A (1 , ( R + R − )) under the affinemapping φ ( z ) = z + δ ¯ z . Then there exists no energy-minimal diffeomorphism h : Ω onto −→ Ω ∗ , though there exists a harmonic one. Proof.
The annulus A ∗ is the image of Ω under the extremal Nitsche mapping h ∗ ( z ) = 12 (cid:18) z + 1¯ z (cid:19) which is not only harmonic but also energy-minimal in H , (Ω , A ∗ ) [2, Corol-lary 2]. The uniqueness part of Theorem 1.1 in [14] states that h ∗ is theunique harmonic homeomorphism from Ω onto A ∗ , up to a conformal au-tomorphism of the annulus Ω, rotation or/and inversion. It follows that g := φ ◦ h ∗ is the unique harmonic diffeomorphism of Ω onto Ω ∗ , up toa conformal automorphism of Ω. Thus if H , (Ω , Ω ∗ ) admitted an energyminimizer the mapping g would be one of them. Explicitly, g ( z ) = 12 (cid:18) z + δz + δ ¯ z + 1¯ z (cid:19) . On the other hand, the Hopf differential of g takes the form g z g ¯ z = 14 (cid:18) − δz (cid:19) (cid:18) δ − z (cid:19) cz . By Lemma 6.1 we see that g cannot be stationary in the annulus Ω. Conse-quently, there is no energy-minimal homeomorphism in H , (Ω , Ω ∗ ). (cid:3) Convexity of the minimum energy function In § ∗ thefunction E ( τ, Ω ∗ ) is decreasing for 0 < τ < Mod Ω ∗ . The minimum of thisfunction is attained at τ = Mod Ω ∗ , i.e., in the case of conformal equivalence.In this section we prove: Theorem 10.1.
Let Ω ∗ be a bounded doubly connected domain. The func-tion τ E ( τ, Ω ∗ ) is strictly convex for < τ < Mod Ω ∗ . The main part of the proof of this theorem needs to be stated separately.As a by-product it establishes the uniqueness part of Theorem 1.1.
Proposition 10.2.
Let Ω ∗ be a bounded doubly connected domain. Supposethat h ∈ D ( A ( τ ◦ ) , Ω ∗ ) is an energy-minimal deformation. In particular, byLemma , (10.1) h z h ¯ z ≡ cz in A ( τ ◦ ) . Then for any diffeomorphism g : A ( τ ) → Ω ∗ we have (10.2) E [ g ] − E [ h ] > πc ( τ ◦ − τ ) . If, in addition, h is a diffeomorphism, then equality holds in (10.2) if andonly if τ = τ ◦ and g − ◦ h is a conformal mapping of A ( τ ◦ ) onto itself.Proof. First we dispose of the easy case c = 0. In this case h is conformal,which implies E [ h ] = 2 | Ω ∗ | . On the other hand, E [ g ] > | Ω ∗ | with equalityif and only if g is conformal, see (1.2).It remains to deal with c = 0. The composition f = g − ◦ h : A ( τ ◦ ) onto −→ A ( τ )lies in W , ( A ( τ ◦ )) and is not homotopic to a constant mapping. Moreover,the restriction of f to the domain G := { z ∈ A ( τ ◦ ) : h ( z ) ∈ Ω ∗ } is a harmonicdiffeomorphism onto A ( τ ), by virtue of Proposition 8.1. Thus, f possesses aright inverse f − : A ( τ ) onto −→ G which is also a diffeomorphism. We estimate E [ g ] − E [ h ] in several steps. The first step is to apply the chain rule to thederivatives of g = h ◦ f − ( w ) at w = f ( z ). ∂g∂w = h z f z − h ¯ z f ¯ z J f ∂g∂ ¯ w = h ¯ z f z − h ¯ z f ¯ z J f (10.3)Then by change of variables the Dirichlet energy of g in A ( τ ) reduces to anintegral over G . E [ g ] = 2 Z A ( τ ) (cid:0) | g w | + | g ¯ w | (cid:1) dw = 2 Z G | h z f z − h ¯ z f ¯ z | + | h ¯ z f z − h z f z | J f dz Next, subtract R G | Dh | from E [ g ], use the inequality | h z | + | h ¯ z | > | h z h ¯ z | ,and recall (10.1) to obtain E [ g ] − Z G | Dh | = 4 Z G (cid:0) | h z | + | h ¯ z | (cid:1) | f ¯ z | − (cid:2) h z h ¯ z f z f ¯ z (cid:3) J f dz > Z G | h z h ¯ z | | f ¯ z | − (cid:2) h z h ¯ z f z f ¯ z (cid:3) J f dz (10.4) = 4 | c | Z G (cid:20) | f z − σf ¯ z | J f − (cid:21) dz | z | , where σ = σ ( z ) = c ¯ z | c | z We must also account for the integral of | Dh | over A ( τ ◦ ) \ G . On this set J h = 0 a.e. by Lemma 3.10, which in view of (10.1) implies | h z | + | h ¯ z | = 2 | h z | = 2 | c || z | . XISTENCE OF ENERGY-MINIMAL DIFFEOMORPHISMS 25
Hence(10.5) Z A ( τ ◦ ) \ G | Dh | = 4 | c | Z A ( τ ◦ ) \ G dz | z | . Combining (10.4) and (10.5) we arrive at E [ g ] − E [ h ] > | c | Z G | f z − σf ¯ z | J f dz | z | − | c | Z A ( τ ◦ ) dz | z | = 4 | c | Z G | f z − σf ¯ z | J f dz | z | − π | c | τ ◦ . (10.6)At this stage the sign of c comes into play. Note that | f z − σf ¯ z | J f = ( K fT if c > K fN if c < . Lemma 3.10 tells us that the Jacobian J h vanishes almost everywhere on A ( τ ◦ ) \ G . This together with (6.4) imply that one of directional derivativesof h must vanish a.e. on A ( τ ◦ ) \ G : h T = 0 if c > h N = 0 if c <
0. Since f = g − ◦ h , the same alternative applies to the directional derivatives f T and f N . In summary, the last integral in (10.6) may as well be taken over A ( τ ◦ ) instead of G .(10.7) E [ g ] − E [ h ] > | c | Z A ( τ ◦ ) K fT dz | z | − π | c | τ ◦ if c > | c | Z A ( τ ◦ ) K fN dz | z | − π | c | τ ◦ if c < . In the case c > f and obtain the estimate(10.8) Z A ( τ ◦ ) K fT dz | z | > π τ ◦ τ which together with (10.7) yield(10.9) E [ g ] − E [ h ] > πc τ ◦ τ ( τ ◦ − τ ) > πc ( τ ◦ − τ ) , < τ ◦ , τ < ∞ . If c <
0, then Proposition 5.1 (b) applies to the restriction of f to G ,yielding(10.10) Z G K fN dz | z | > πτ which together with (10.6) imply (10.2).It remains to prove the equality statement. Since h is a sense-preservingdiffeomorphism, we have G = A ( τ ◦ ) and | h z | > | h ¯ z | everywhere in A ( τ ◦ ).If equality holds in (10.2), then it also holds in (10.4). The latter is onlypossible if f ¯ z ≡ A ( τ ◦ ). Thus f : A ( τ ◦ ) onto −→ A ( τ ) is a conformal mapping.This implies τ ◦ = τ , as desired. (cid:3) Proof of Theorem . Pick τ ◦ ∈ (0 , Mod Ω ∗ ). By Theorem 2.4 there exists h ∈ H , ( A ( τ ◦ ) , Ω ∗ ) such that E [ h ] = E ( τ ◦ , Ω ∗ ). Consequently, (10.1) holds.That c > E ( τ, Ω ∗ ) − E ( τ ◦ , Ω ∗ ) > − πc ( τ − τ ◦ ) , τ ∈ (0 , Mod Ω ∗ ) , τ = τ ◦ Indeed, by Theorem 2.4 there exists g ∈ H , ( A ( τ ) , Ω ∗ ) such that E [ g ] = E ( τ, Ω ∗ ). Proposition 10.2 is exactly what we need for (10.11).Inequality (10.11) tells us that E ( τ, Ω ∗ ) is strictly convex. Together with (7.5)it yields the existence of the derivative ddτ (cid:12)(cid:12)(cid:12)(cid:12) τ = τ ◦ E ( τ, Ω ∗ ) = − πc, Incidentally or not, this shows that c depends only on τ ◦ and Ω ∗ , but not on h . Every convex function, once differentiable everywhere, is automatically C -smooth; the theorem is fully established. (cid:3) The strict convexity part of Theorem 10.1 fails for τ >
Mod Ω ∗ . Wedemonstrate this with an example based on the results of [2]. Althoughthe paper [2] is concerned with the minimization of energy in a somewhatdifferent class of Sobolev mappings, its approach carries over to our settingwith no changes. Example 10.3.
Let Ω ∗ = A (1 , R ∗ ) where 1 < R ∗ < ∞ . The function τ E ( τ, Ω ∗ ) is C -smooth on (0 , ∞ ), strictly convex for 0 < τ < log cosh Mod Ω ∗ and affine for τ > log cosh Mod Ω ∗ . Proof.
Let Ω = A (1 , R ) where R = e τ . We begin with the case 0 < τ < log cosh Mod Ω ∗ . In terms of R this condition reads as(10.12) R ∗ > (cid:18) R + 1 R (cid:19) , equivalently, R R ∗ + p R ∗ − . Let λ ∈ ( − ,
1] be determined by the equation(10.13) R ∗ = 11 + λ (cid:18) R + λR (cid:19) ; that is, λ = R ( R − R ∗ ) RR ∗ − . By [2, Corollary 2] the infimum of energy E (Ω , Ω ∗ ) is achieved by the map-ping(10.14) h λ ( z ) = 11 + λ (cid:18) z + λ ¯ z (cid:19) , for which we compute(10.15) E [ h λ ] = 2 π ( R − R + λ ) R (1 + λ ) . which yields(10.16) E (log R, Ω ∗ ) = 2 π ( R + 1)[( R ∗ + 1) − RR ∗ ] R − . XISTENCE OF ENERGY-MINIMAL DIFFEOMORPHISMS 27
A straightforward computation reveals that the righthand side of (10.16) is aconvex function of log R in the range given by (10.12). Indeed, its derivativewith respect to log R is equal to(10.17) R ddR E (log R, Ω ∗ ) = 8 πR ( R − R ∗ )( RR ∗ − R − . Differentiating (10.17) once again, we find
R ddR (cid:18)
R ddR E (log R, Ω ∗ ) (cid:19) = 8 πR ( R − (cid:8) ( R ∗ R − R + R ∗ )(2 RR ∗ − R − (cid:9) . (10.18)The right hand side of (10.18) has the same sign as (2 RR ∗ − R − R = R ∗ + p R ∗ − E ( τ, Ω ∗ ), namely(10.19) ddτ E ( τ, Ω ∗ ) = 2 π, d dτ E ( τ, Ω ∗ ) = 0 . It remains to consider the case τ > log cosh Mod Ω ∗ . Now it is moreconvenient to work with Ω = A ( r, R ) where R = R ∗ + p R ∗ − r < E (Ω , Ω ∗ ) is realized by a non-injectivedeformation h : Ω onto −→ Ω ∗ . h = ( z | z | for r < | z | (cid:0) z + z (cid:1) for 1 | z | < R Here the radial projection z z/ | z | hammers A ( r,
1) onto the unit circlewhile the Nitsche mapping (cid:0) z + z (cid:1) takes A (1 , R ) homeomorphically ontoΩ ∗ . The contribution of the radial projection to the energy of h is equal to(10.20) 2 π log 1 r = 2 π ( τ − log cosh Mod Ω ∗ ) . This is an affine function of log R whose first derivative equals 2 π and thesecond derivative vanishes. This result remains in agreement with formu-las (10.19). Thus E ( τ, Ω ∗ ) is a C -smooth function. (cid:3) Open questions and conjectures
In (1.5) and (2.1) we defined two infima of energy; the one denoted E H (Ω , Ω ∗ ) runs over homeomorphisms and the other, E (Ω , Ω ∗ ), over de-formations in the sense of Definition 2.2). Clearly E H (Ω , Ω ∗ ) > E (Ω , Ω ∗ ).Under the hypotheses of Theorem 2.3 E H (Ω , Ω ∗ ) = E (Ω , Ω ∗ ). Question 11.1.
For k >
2, is E H (Ω , Ω ∗ ) = E (Ω , Ω ∗ ) for all k -connectedbounded domains Ω and Ω ∗ in C ? This question has the affirmative answer in the case k = 1 thanks to theRiemann mapping theorem. Indeed, due to Corollary 3.9 the formula (1.2)remains valid for all deformations. Therefore, the conformal mapping mini-mizes the energy.Theorem 10.1 and Example 10.3 motivate the following conjecture. Conjecture 11.2.
The function τ E ( τ, Ω ∗ ) is convex for < τ < ∞ . Note that it would follow from the positive answer to Question 11.1, bymeans of Proposition 10.2.We expect that Theorem 2.4 can be given the following sharp form.
Conjecture 11.3.
If two bounded doubly connected domains Ω and Ω ∗ in C admit an energy-minimal diffeomorphism h : Ω onto −→ Ω ∗ , then Mod Ω ∗ > log cosh Mod Ω . Moreover, if both sides are finite and equal, then Ω ∗ is a circular annulus. Concerning the existence of energy-minimal diffeomorphisms between do-mains of higher connectivity, we propose a generalization of Theorem 1.1.
Conjecture 11.4.
Let Ω and Ω ∗ be bounded k -connected domains in C ,where k > . Suppose that Ω ⊂ Ω ∗ where the inclusion is a homotopyequivalence. Then there exists an energy-minimal diffeomorphism of Ω onto Ω ∗ . For k = 2 Conjecture 11.4 is true, by virtue of Theorem 1.1. In theconverse direction, we propose a qualitative version of Theorem 2.4 for k -connected domains. Conjecture 11.5.
Let Ω and Ω ∗ be bounded k -connected domains in C ,where k > . If ǫ > is sufficiently small (depending on both Ω and Ω ∗ ),then there is no energy-minimal homeomorphism of Ω onto φ (Ω ∗ ) , where φ ( x + iy ) = ǫx + iy . In other words, if we flatten Ω ∗ too much in one direction the injectivityof energy-minimal deformations f ∈ D (Ω , Ω ∗ ) will be lost.12. Appendix: Monotone Sobolev mappings
Throughout this section X will be a bounded domain in C whose comple-ment consists of k mutually disjoint closed connected sets denoted by C \ X = X ∪ · · · ∪ X k =: X , k > . It then follows that to every X i there corresponds one and only one compo-nent of ∂ X , precisely equal to ∂ X i , ∂ X = ∂ X = ∂ X ∪ · · · ∪ ∂ X k . XISTENCE OF ENERGY-MINIMAL DIFFEOMORPHISMS 29
Among those components there is exactly one unbounded. Similarly to X ,we consider a bounded domain in Y ⊂ C whose complement consists of k -mutually disjoint closed connected sets denoted by C \ Y = Υ ∪ · · · ∪ Υ k =: Υ . We make one standing assumption on X ; namely, none of the components X , . . . , X k degenerates to a single point.(12.1) min i k diam X i = d > Y will not be required. Let us denote(12.2) ρ Y = inf α = β dist(Υ α , Υ β ) > . We shall examine the class F Y ( X ) of mappings h : X → C such that(i) h ∈ C ( X ) ∩ W , ( X );(ii) h ( X ) ⊃ Y ;(iii) h { ∂ X i } ⊂ ∂ Υ i , i = 1 , . . . , k , in the sense of cluster sets;(iv) the restriction of h to h − ( Y ) is monotone.It follows from (iii) that h − (Γ) is compact for any compact set Γ ⊂ Y .If in addition Γ is connected, then h − (Γ) is connected by Proposition 3.6. Lemma 12.1.
There is a constant c = c ( X , Y ) > such that (12.3) E [ h ] > c ( X , Y ) , for every h ∈ F Y ( X ) . In fact, we have the following explicit bound. Z X | Dh | > ρ Y d diam X . Proof.
Choose a bounded component X i . Let a line segment I with theend-points in X i represent the diameter of X i ; thus | I | = diam X i . Throughevery point t ∈ I there passes a straight line L t perpendicular to I . One ofthe components of X ∩ L t , say an open interval γ , connects ∂ X i with ∂ X α ,for some α = i . Thus, by condition (iii), Z X ∩ L t | Dh | > Z γ | Dh | > dist(Υ α , Υ i ) > ρ Y . This is true for almost every t ∈ I , as long as h is locally absolutely contin-uous on X ∩ L t . By H¨older’s inequality Z X ∩ L t | Dh | > | X ∩ L t | (cid:18)Z X ∩ L t | Dh | (cid:19) > ρ Y diam X . Integrating with respect to t ∈ I , by Fubini’s theorem, we conclude that Z X | Dh | > Z I (cid:18)Z X ∩ L t | Dh | (cid:19) dt > diam X i diam X ρ Y > ρ Y d diam X as desired. (cid:3) Theorem 12.2.
For each i = 1 , . . . , k there exists a continuous function η i = η i X , Y ( z ) on C , vanishing on X i , such that for every h ∈ F Y ( X ) we have (12.4) dist (cid:0) h ( z ) , Υ i (cid:1) η i ( z ) p E [ h ] , z ∈ X . Proof.
It suffices to construct for each i = 1 , . . . , k , a function η i = η i ( z ) in X which is bounded and satisfies the conditions lim z → X i η i ( z ) = 0 and (12.4)for all h ∈ F Y ( X ). Continuity of η i can easily be accomplished by taking acontinuous majorant. The obvious choice for η i is:(12.5) η i ( z ) = sup h ∈F ( X , Y ) dist (cid:0) h ( z ) , Υ i (cid:1)p E [ h ] , i = 0 , , . . . , k. By Lemma 12.1 we see that η i ( z ) diam Y √ c ( X , Y ) . Fix an index i and suppose,to the contrary, that lim z → X i η i ( z ) = 0. Then η i ( z ν ) > ǫ > { z ν } ⊂ X converging to a point z ◦ ∈ X i . This means that there is a sequence { h ν } of functions in F Y ( X ) such that(12.6) dist (cid:0) h ν ( z ν ) , Υ i (cid:1) > ǫ p E [ h ν ] > ǫ p c ( X , Y ) , by Lemma 12.1. Obviously, we have(12.7) E [ h ν ] (cid:18) diam Y ǫ (cid:19) , ν = 1 , , . . . Choose and fix a doubly connected domain G ⊂ Y so that one of the con-nected components of C \ G is Υ i . The following lemma provides us withwhat we call a potential function for Υ i . Claim A.
There exists a C -smooth function U : C → [0 , such that U − { } = Υ i and U − { } is precisely the other connected component of C \ G . Moreover, for each < t < the set Γ t = U − { t } is a Jordan curveseparating the boundary components of G .Proof. Let Φ : G → A ( r, R ) be a conformal mapping of G onto a circular an-nulus A ( r, R ) or a punctured disk. The function | Φ | has the desired structureof level sets but may lack smoothness on the boundary. The latter is reme-died with the help of a smooth strictly increasing function ψ : ( r, R ) → (0 , ψ ′ → r and R . We define U as thecomposition ψ ( | Φ | ), extended by 0 and 1 to the entire plane C . (cid:3) For h ∈ F Y ( X ) we consider the continuous function(12.8) V ( z ) = V h ( z ) = U ( h ( z )) if z ∈ X z ∈ X i z / ∈ X ∪ X i . The continuity of V follows from the condition (iii) after taking into accountthat U (Υ i ) = { } while U (Υ α ) = { } for α = i . XISTENCE OF ENERGY-MINIMAL DIFFEOMORPHISMS 31
Recall the constant d that was defined in (12.1) as the smallest of thenumbers diam X i , i = 1 , . . . , k . Claim B.
For any h ∈ F Y ( X ) and < t < the level set V − h { t } is acontinuum of diameter at least d .Proof. That V − h { t } is a continuum follows from the monotonicity assump-tion (iv). Choose α such that: α = i if X i is bounded and α = i otherwise.In either case the component X α is bounded. Consider a straight line L passing through two points a, b ∈ ∂ X α such that | a − b | = diam X α . The set L \ ( a, b ) consists of two closed half-lines L a and L b . We will show that oneach of them V h attains the value t , which yieldsdiam V − h { t } > | a − b | = diam X α > d. The half-line L a meets a bounded component X α at the point a , and mustalso intersect the unbounded component of C \ X . Considering our choice of α we find that L a meets both X i and some other component of C \ X . Thus V h attains the values 0 and 1 on L a . Being continuous, it also attains thevalue t . Similarly we argue with the half-line L b . (cid:3) Claim C.
We have V ∈ W , ( C ) . Moreover, V has the oscillation propertyon every open disk B ⊂ C of diameter not greater than d = min α k diam X α .Proof. First note that V ∈ W , ( X ) and we have the pointwise estimate |∇ V ( z ) | k∇ U k L ∞ ( C ) | Dh ( z ) | , z ∈ X . Recall that V is continuous on C and is constant on each component of C \ X .The classical Sobolev theory tells us that such function belongs to W , ( C )with the energy bound(12.9) Z C |∇ V ( z ) | dz k∇ U k ∞ Z X | Dh | . We now proceed to check the oscillation property of V on a disk B ⊂ C .For this we choose a compact set F ⊂ B . Consider an arbitrary componentof F , denoted F ◦ . Note that diam F ◦ < diam B d . The set V ( F ◦ ) is acompact subinterval of [0 ,
1] which we denote by [ a, b ]. We will show that(12.10) [ a, b ] ⊂ V ( ∂ F ◦ ) . This is obvious when a = b , for then V is constant on F ◦ . When a < b ,the inclusion (12.10) will follow once we prove ( a, b ) ⊂ V ( ∂ F ◦ ) since thelatter set is compact.Suppose that t ∈ ( a, b ) but t / ∈ V ( ∂ F ◦ ). Then V − { t } ⊂ (Int F ◦ ) ∪ ( C \ F ◦ )where Int F ◦ stands for the interior of F ◦ . By Claim B the set V − { t } is a continuum of diameter at least d and therefore cannot be a subsetof Int F ◦ . Hence V − { t } ⊂ C \ F ◦ , but this contradicts the assumption t ∈ ( a, b ) ⊂ V ( F ◦ ). Completing the proof of (12.10). From ∂ F ◦ ⊂ ∂ F it follows that V ( F ◦ ) ⊂ V ( ∂ F ◦ ) ⊂ V ( ∂ F ). Since F ◦ wasan arbitrary component of F , the lemma is proved. (cid:3) We now return to the sequence { h ν } ⊂ F Y ( X ) defined in (12.6) and theassociated functions(12.11) V ν ( z ) = V h ν ( z ) in C ( C ) ∩ W , ( C ) . In view of (12.9) and (12.7) we have the uniform bound on the Dirichletintegrals Z C |∇ V ν ( z ) | dz k∇ U k ∞ diam Y ǫ , ν = 1 , , . . . Since V ν have the oscillation property on every disk B of diameter d , theestimate (3.2) applies, yielding | V ν ( a ) − V ν ( b ) | C k∇ U k ∞ diam Y ǫ log (cid:0) e + d | a − b | (cid:1) whenever a, b ∈ C and | a − b | d .This shows that the functions V ν are equicontinuous on C . By the Arzel`a-Ascoli theorem there is a subsequence, again denoted { V ν } , that convergesuniformly on C to a continuous function V = V ( z ). In particular,(12.12) V ν ( z ν ) − V ( z ν ) → ν → ∞ . Also note that V ν ≡ X i so V ≡ X i as well. On the other hand, itfollows from the definition of V ν that V ν ( z ν ) = U (cid:0) h ν ( z ν ) (cid:1) , and from (12.6)we know that h ν ( z ν ) stay away from Υ i , preciselydist (cid:0) h ν ( z ν ) , Υ i (cid:1) > ǫ p E [ h ν ] > ǫ p c ( X , Y ) . Hence there is t ◦ > V ν ( z ν ) = U ( h ν ( z ν )) > t ◦ for all ν = 1 , , . . . .Passing to the limit in (12.12) as z ν → z ◦ ∈ X i , we obtain a contradiction t ◦ = t ◦ − V ( z ◦ ) lim ν →∞ (cid:2) V ν ( z ν ) − V ( z ν ) (cid:3) = 0thus completing the proof of Theorem 12.2. (cid:3) References
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Department of Mathematics, Syracuse University, Syracuse, NY 13244, USA
E-mail address : [email protected] Department of Mathematics, Syracuse University, Syracuse, NY 13244, USA
E-mail address : [email protected] Department of Mathematics, Syracuse University, Syracuse, NY 13244, USA
E-mail address : [email protected] Department of Mathematics, Syracuse University, Syracuse, NY 13244, USA
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