MMINIMISERS AND KELLOGG’S THEOREM
DAVID KALAJ AND BERNHARD LAMELA
BSTRACT . We extend the celebrated theorem of Kellogg for conformal map-pings to the minimizers of Dirichlet energy. Namely we prove that a diffeo-morphic minimizer of Dirichlet energy of Sobolev mappings between doublyconnected domains D and Ω having C n,α boundary is C n,α up to the boundary,provided Mod( D ) (cid:62) Mod(Ω) . If
Mod( D ) < Mod(Ω) and n = 1 we obtainthat the diffeomorphic minimizer has C ,α (cid:48) extension up to the boundary, for α (cid:48) = α/ (2 + α ) . It is crucial that, every diffeomorphic minimizer of Dirich-let energy has a very special Hopf differential and this fact is used to prove thatevery diffeomorphic minimizer of Dirichlet energy can be locally lifted to a cer-tain minimal surface near an arbitrary point inside and at the boundary. This isa complementary result of an existence results proved by T. Iwaniec, K.-T. Koh,L. Kovalev, J. Onninen (Inventiones 2011). C ONTENTS
1. Introduction and the main results 11.1. Noether harmonic mappings 31.2. Some key properties of Noether harmonic diffeomorphisms 32. Some corollaries and the strategy of the proof 52.1. Minimizing mappings and minimal surfaces 53. H¨older property of minimizers 114. Proof of Theorem 1.1 124.1. Proof of Lipschitz continuity 124.2. The minimizer is C ,α (cid:48) up to the boundary 185. Proof of Theorem 1.2 226. Concluding remark 22Appendix-Proof of Lemma 3.2 23Acknowledgement 25References 251. I NTRODUCTION AND THE MAIN RESULTS
In this paper, we consider two doubly connected domains D and Ω in the com-plex plane C . The Dirichlet energy of a diffeomorphism f : D → Ω is defined Mathematics Subject Classification.
Primary 31A05; Secondary 49Q05.
Key words and phrases.
Minimizers, Kellogg theorem, Minimal surfaces, Annuli. a r X i v : . [ m a t h . C V ] M a r DAVID KALAJ AND BERNHARD LAMEL by(1.1) E [ f ] = (cid:90) D (cid:107) Df (cid:107) dλ = 2 (cid:90) D (cid:0) | ∂f | + | ¯ ∂f | (cid:1) dλ, where (cid:107) Df (cid:107) is the Hilbert-Schmidt norm of the differential matrix of f and λ isstandard Lebesgue measure. The primary goal of this paper is to establish boundaryregularity of a diffeomorphism f : D onto −→ Ω of smallest (finite) Dirichlet energy,provided such an f exists and the boundary is smooth. If we denote by J ( z, f ) theJacobian of f at the point z , then (1.1) yields(1.2) E [ f ] = 2 (cid:90) D J ( z, f ) dλ + 4 (cid:90) D | ¯ ∂f | (cid:62) | Ω | where | Ω | is the measure of Ω . In this paper we will assume that diffeomorphismsas well as Sobolev homeomorphisms are orientation preserving, so that J ( z, f ) > . A conformal mapping of D onto Ω would be an obvious minimizer of (1.2),because ¯ ∂f = 0 , provided it exists. Thus in the special case where D and Ω areconformally equivalent the famous Kellogg theorem yields that the minimizer isas smooth as the boundary in the H¨older category. For an exact statement of theKellogg theorem, we recall that a function ξ : D → C is said to be uniformly α − H¨older continuous and write ξ ∈ C α ( D ) if sup z (cid:54) = w,z,w ∈ D | ξ ( z ) − ξ ( w ) || z − w | α < ∞ . In similar way one defines the class C n,α ( D ) to consist of all functions ξ ∈ C n ( D ) which have their n th derivative ξ ( n ) ∈ C α ( D ) . A rectifiable Jordan curve γ ofthe length l = | γ | is said to be of class C n,α if its arc-length parameterization g : [0 , l ] → γ is in C n,α , n (cid:62) . The theorem of Kellogg (with an extensiondue to Warschawski, see [9, 35, 33, 34, 29]) now states that if D and Ω are Jordandomains having C n,α boundaries and ω is a conformal mapping of D onto Ω , then ω ∈ C n,α .The theorem of Kellogg and of Warshawski has been extended in various direc-tions, see for example the work on conformal minimal parameterization of minimalsurfaces by Nitsche [27] (see also the paper by Kinderlehrer [21] and by F. D. Les-ley [24]), and to quasiconformal harmonic mappings with respect to the hyperbolicmetric by Tam and Wan [30, Theorem 5.5.]. For some other extensions and quan-titative Lipschitz constants we refer to the paper [25].We have the following extension of the Kellogg’s theorem, which is the mainresult of the paper. Theorem 1.1.
Let α ∈ (0 , . Assume that D and Ω are two doubly connecteddomains in the complex plane with C ,α boundaries. Assume that f is a diffeo-morphic minimizer of energy (1.1) throughout the class of all diffeomorphisms be-tween D and Ω . Then f has a C ,α (cid:48) extension up to the boundary, with α (cid:48) = α if Mod( D ) (cid:62) Mod(Ω) and α (cid:48) = α α if Mod( D ) < Mod(Ω) . For higher-degree regularity we will prove the following result:
INIMISERS AND KELLOGG’S THEOREM 3
Theorem 1.2.
Let α ∈ (0 , . Assume that D and Ω are two doubly connecteddomains in the complex plane with C n,α boundaries so that Mod( D ) (cid:62) Mod(Ω) .Assume that f is a diffeomorphic minimizer of energy (1.1) throughout the classof all diffeomorphisms between D and Ω . Then f has a C n,α extension up to theboundary. We will formulate some corollaries of Theorem 1.1 in Section 2, where we willdescribe the key point of the proof. In Section 3 together with the appendix belowwe prove that diffeomorphic minimizers are H¨older continuous at the boundarycomponents. This is needed to prove the global Lipschitz continuity of such dif-feomorphisms, which is done in Subsection 4.1. The proof of the smoothness issueis given in Section 4.2. Section 5 contains the proof of the main results. The lastsection is devoted to an open problem.The following existence result was proved in [12]:
Proposition 1.3.
Suppose that D and Ω are bounded doubly connected domainsin C such that Mod D (cid:54) Mod Ω . Then there exists a diffeomorphism h of finiteDirichlet energy, which minimizes the energy amongst all diffeomorphisms; thatis, E [ h ] = inf {E [ f ] : f is a diffeomorphism between D and Ω } . Moreover, h isharmonic and it is unique up to a conformal automorphism of D . The most important issue in proving Proposition 1.3 was to establish some keyproperties of Noether harmonic which we gather in the next subsections.1.1.
Noether harmonic mappings.
We recall that a mapping g : D → Ω is saidto be Noether harmonic (see [6]) if(1.3) ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 E [ g ◦ φ − t ] = 0 for every family of diffeomorphisms t → φ t : Ω → Ω depending smoothly on thereal parameter t and satisfying φ = id . To be more exact, that means that themapping Ω × [0 , (cid:15) ] (cid:51) ( t, z ) → φ t ( z ) ∈ Ω is a smooth mapping for some (cid:15) > .Not every Noether harmonic mapping h is a harmonic mapping, however if themapping g is a diffeomorphism, then it is harmonic, i.e. it satisfies the equation ∆ g = 0 .1.2. Some key properties of Noether harmonic diffeomorphisms.
The follow-ing properties of Noether harmonic mappings are derived in the proof of [14,Lemma 1.2.5]. Assume that g : D → Ω is Noether harmonic. Then1. The Hopf differential of g , defined by ϕ := g z g ¯ z , which a priori belongsto L ( D ) , is holomorphic.2. If ∂D is C ,α -smooth then ϕ extends continuously to D , and the quadraticdifferential ϕ dz is real on each boundary curve of D .Using those key properties, [12, Lemma 6.1] (and [16]) show the following: If D = A ( r, R ) , where < r < R < ∞ , is a circular annulus centered at origin, Ω DAVID KALAJ AND BERNHARD LAMEL a doubly connected domain, and g : D → Ω is a stationary deformation, then thereexists a real constant c ∈ R such that(1.4) g z g ¯ z ≡ cz in D We recall that every doubly connected domain D ⊂ C whose inner boundary isnot just one point is conformally equivalent to such an annulus D = A ( r, R ) . Theconformal invariant Mod D := R/r is called the conformal modulus of D .The constant c appearing in (1.4) is related to the conformal modulus by thefollowing proposition. Proposition 1.4. [16, Corollary 5.2] . If g : D → Ω is a Noether harmonic defor-mation, then we have c > if Mod
D <
Mod Ω ,c = 0 if Mod D = Mod Ω ,c < if Mod
D >
Mod Ω . We next recall that sense preserving mapping w of class ACL between two pla-nar domains X and Y is called ( K, K (cid:48) ) -quasi-conformal if(1.5) (cid:107) Dw (cid:107) (cid:54) KJ ( z, w ) + K (cid:48) , for almost every z ∈ X . Here K (cid:62) , K (cid:48) (cid:62) , J ( z, w ) is the Jacobian of w in z and (cid:107) Dw (cid:107) = | w x | + | w y | = 2 | w z | + 2 | w ¯ z | . For a related definition formappings between surfaces the reader is referred to [31].Noether-harmonic maps, and in particular minimizers, belong to the class of ( K, K (cid:48) ) quasiconformal mappings, for a ( K, K (cid:48) ) which is nicely related to thedata c and Mod D : Lemma 1.5. [17]
Every sense-preserving Noether harmonic map g : A ( ρ, → Ω is ( K, K (cid:48) ) quasiconformal, where K = 1 and K (cid:48) = 2 | c | ρ , and c is the constant from (1.4) . The result is sharp and for c = 0 the Noetherharmonic map is (1 , quasiconformal, i.e. it is a conformal mapping. In this case Ω is conformally equivalent to A ( ρ, . Assume that g : [0 , (cid:96) ] → Γ is the arc-length parameterization of a rectifiableJordan curve Γ . Here (cid:96) = | Γ | is the length of Γ . We say that a continuous mapping f : T → Γ of the unit circle onto Γ is monotone if there exists a monotonefunction φ : [0 , π ] → [0 , (cid:96) ] such that f ( e it ) = g ( φ ( s )) . In a similar way wedefine a monotone function between ρ T := { z : | z | = ρ } and Γ . In view of [22,Proposition 5] and Proposition 3.1 below we can formulate the following simplelemma. Lemma 1.6.
Assume that f is a diffeomorphic minimizer of Dirichlet energy be-tween the annuli A ( ρ, and the doubly connected domain Ω , which is bounded bythe outer boundary Γ and inner boundary Γ . Then f has a continuous extensionto the boundary and the boundary mapping is monotone in both boundary curves. INIMISERS AND KELLOGG’S THEOREM 5
2. S
OME COROLLARIES AND THE STRATEGY OF THE PROOF
Theorem 1.1 and Proposition 1.3 imply the following result:
Corollary 2.1.
Assume that D and Ω are two doubly connected domains in C with C ,α boundary. Assume also that Mod( D ) (cid:54) Mod(Ω) . Then there existsa minimizer h of Dirichlet energy E and it has a C ,α/ (2+ α ) extension up to theboundary. Moreover it is unique up to the conformal change of D .In view of Subsection 2.1, we can state the following corollary. Corollary 2.2.
Assume f = u + iv : A ρ → Ω is a diffeomorphic minimizerof Dirichlet energy among the diffeomorphisms, where Ω is a doubly connecteddomain with a C ,α boundary. Then, the mapping F ( z ) = (cid:40) ( u, v, √ c Arg z ) , for c > ; (cid:16) u, v, √− c log | z | (cid:17) , for c (cid:54) ,is a conformal parametrisation of a minimal surface Σ , whose boundary is in C ,α (cid:48) ,where α (cid:48) = α , if c (cid:54) and α (cid:48) = α/ (2 + α ) otherwise. If c (cid:54) , then the surface Σ is a doubly connected catenoidal minimal surface, whose conformal modulus isequal to Mod A ρ (cid:62) Mod Ω . If c > , then the minimal surface Σ is a helicoidalminimal surface.The minimizer of Dirichlet energy is not always a diffeomorphism when Mod( D ) (cid:62) Mod(Ω) . Moreover it fails to be smooth in the domain if the boundary is notsmooth [3]. For more general setting we refer to [10].
Remark . By using Lemma 1.5, the first author in [17] proved that, a minimizerof (cid:37) − energy between doubly connected domains having C boundary is Lipschitzcontinuous. The (cid:37) − energy, is a certain generalization of Euclidean energy, and wewill omit details in this paper.2.1. Minimizing mappings and minimal surfaces.
Since D is conformally equiv-alent to A ρ = { z : ρ < | z | < } , for some ρ ∈ (0 , , we can assume that D = A ρ .Namely, by a Kellogg’s type result of Jost ([15]), a conformal biholomorphism ofa domain D with C n,α boundary onto A ρ is C n,α continuous up to the boundarytogether with its inverse. For every p ∈ ∂ A ρ , there is a Jordan domain A p ⊂ A ρ ,containing a Jordan arc T p in ∂ A ρ , whose interior contains p . Moreover in view ofLemma 1.6, enlarging T p if necessary, we can assume that Γ p := f ( T p ) is a Jordanarc containing q = f ( p ) in its interior in ∂D . Moreover we can assume that A p has a C ∞ boundary. Assume now that Φ p is a conformal mapping of the unit disk D onto A p so that Φ p ( p/ | p | ) = p . Moreover, if p (cid:48) (cid:54) = p , but | p | = | p (cid:48) | we can chosedomains A p (cid:48) to be just rotation of A p . So all those domains A p are isometric to A or A ρ . Moreover we also can assume that Φ p (cid:48) = e iς Φ p . Then f p = f ◦ Φ p has therepresentation(2.1) f p ( z ) = g ( z ) + h ( z ) , DAVID KALAJ AND BERNHARD LAMEL where g ( z ) = g p ( z ) and h ( z ) = h p ( z ) are holomorphic mappings defined on theunit disk. Moreover f p is a sense preserving diffeomorphism and this means that J ( z, f p ) = | g (cid:48) ( z ) | − | h (cid:48) ( z ) | > . From (1.4) we have(2.2) f z f ¯ z = cz , z ∈ A ρ . It follows from (2.2) and (2.1) that(2.3) h (cid:48) p g (cid:48) p = c (Φ (cid:48) p ( z )) Φ p ( z ) . Then it defines locally the minimal surface by its conformal minimal coordinates, ϕ p = ( ϕ , ϕ , ϕ ) , and this is crucial for our approach: ϕ ( z ) = (cid:60) ( g + h ) (2.4) ϕ ( z ) = (cid:61) ( g − h ) (2.5) ϕ ( z ) = (cid:60) (2 i √ c log Φ p ( z )) . (2.6)This can be written ϕ ( z ) = ϕ ( z ) + (cid:60) (cid:90) zz ( g (cid:48) ( z ) + h (cid:48) ( z )) dz (2.7) ϕ ( z ) = ϕ ( z ) + (cid:60) (cid:90) zz i ( h (cid:48) ( z ) − g (cid:48) ( z )) dz (2.8) ϕ ( z ) = ϕ ( z ) + (cid:60) (cid:90) zz i (cid:112) h (cid:48) ( z ) g (cid:48) ( z ) dz. (2.9)Thus the Weierstrass–Enneper parameters are p ( z ) = g (cid:48) ( z ) , q ( z ) = (cid:115) h (cid:48) ( z ) g (cid:48) ( z ) . The first fundamental form is given by ds = λ ( z ) | dz | , where λ ( z ) = 12 (cid:88) j =1 | k j | . Here k ( z ) = g (cid:48) ( z ) + h (cid:48) ( z ) , k ( z ) = i ( h (cid:48) ( z ) − g (cid:48) ( z )) , k ( z ) = 2 i (cid:112) h (cid:48) ( z ) g (cid:48) ( z ) . Then as in [5, Chapter 10], we get λ ( z ) = | p | (1 + | q | ) = | g (cid:48) ( z ) | (cid:18) | g (cid:48) ( z ) || h (cid:48) ( z ) | (cid:19) = ( | g (cid:48) ( z ) | + | h (cid:48) ( z ) | ) . Let us note the following important fact, the boundary curve of the minimalsurface defined in (2.4), (2.5) and (2.6) is ϕ p ( e is ) = ( ϕ ( e is ) , ϕ ( e is ) , ϕ ( e is )) , s ∈ [0 , π ) , INIMISERS AND KELLOGG’S THEOREM 7 p ∈ ∂ A ρ . Its trace is not smooth in general. However the trace of curve z p ( e is ) = ( ϕ ( e is ) , ϕ ( e is )) is smooth as well as the function k is smooth in a small neighborhood of p . Thiswill be crucial in proving our main results.We will prove certain boundary behaviors of f near the boundary by using therepresentation (2.1), and this is why we do not need global representation. The ideais to prove that f is Lipschitz and has smooth extension up to the boundary locally.And this will imply the same behaviour on the whole boundary. The conformalmapping Φ p is a diffeomorphism and it is C ∞ ( D ) , provided the boundary of A p belongs to the same class. So we will go back to the original mapping easily.In the previous part we have showed that every minimizing mapping can belifted locally to a certain minimal surface. In the following part we show that incertain circumstances the lifting is global.Every harmonic mapping f defined on the annulus A ρ can be expressed (see e.g.[1, Theorem 9.1.7]) as(2.10) f ( z ) = a log | z | + b + (cid:88) k (cid:54) =0 ( a k z k + b k ¯ z k ) . Assume now that f is a diffeomorphic minimizer between A ρ and Ω and that c < , i.e. Mod( A ρ ) > Mod(Ω) (see Proposition 1.4). Then we get the followingconformal parameterization of a catenoidal minimal surface Σ , ϕ : A ρ → Σ ,defined by(2.11) ϕ ( z ) = (cid:18) (cid:60) f ( z ) , (cid:61) f ( z ) , √− c log 1 | z | (cid:19) . If a = 0 , then we have the following decomposition f ( z ) = b + g ◦ ( z )+ h ◦ ( z ) ,where g ◦ ( z ) = (cid:88) k (cid:54) =0 a k z k , and h ◦ ( z ) = (cid:88) k (cid:54) =0 b k z k . Then we get the following conformal parameterization of a minimal surface Σ , ϕ : A ρ → Σ , defined by(2.12) ϕ ( z ) = (cid:18) (cid:60) ( g ◦ ( z ) + h ◦ ( z )) , (cid:61) ( g ◦ ( z ) − h ◦ ( z )) , √− c log 1 | z | (cid:19) . The following corollary is a consequence of Theorem 1.1 and (2.11) .
Corollary 2.4.
Assume that f : A ρ → Ω is a diffeomorphic minimizer of Dirich-let energy, with Mod A ρ (cid:62) Mod Ω and ∂ Ω ∈ C ,α . Then f can be lifted to asmooth doubly connected minimal surface Σ with C ,α boundary, and the liftingis conformal and harmonic. DAVID KALAJ AND BERNHARD LAMEL
Let us continue this subsection with the following explicit example. Let(2.13) f ( z ) = r ( R − r )(1 − r ) ¯ z + (1 − rR ) z − r . Then f ( z ) is a harmonic mapping of the annulus A r onto A R that minimizes theDirichlet energy ([2]). Further, under notation of this subsection we have p ( z ) = 1 − rR − r and q ( z ) = (cid:112) r ( r − R )(1 − rR )(1 − r ) z . Put ϕ = (cid:60) f ( z ) , ϕ ( z ) = (cid:61) f ( z ) and assume that Mod( A r ) > Mod( A R ) , i.e. R > r . Then we have from (2.12) that ϕ ( z ) = (cid:60) (cid:90) iq ( z ) dz = (cid:60) (cid:90) i (cid:112) r ( R − r )(1 − rR )(1 − r ) z dz = 2 (cid:112) r ( R − r )(1 − rR )(1 − r ) log 1 | z | . Here (cid:82) Q ( z ) dz stands for the primitive function of Q ( z ) . It follows that (2.13)defines a global minimal surface by its conformal minimal coordinates ϕ ( z ) =( ϕ ( z ) , ϕ ( z ) , ϕ ( z )) . This minimal graph is a part of the lower slab of catenoid.(see Figure 1).F IGURE
1. A part of catenoid over an annulus. Here R = 2 / and r = 1 / . Remark . It follows from (2.13) that, w ( z ) = √− c log 1 | (cid:112) f − ( z ) | defines the nonparametric minimal surface Σ . This means that Σ = { ( x, y, w ( z )) : z ∈ Ω } . Moreover Mod(Σ) = log ρ (cid:62) Mod Ω . INIMISERS AND KELLOGG’S THEOREM 9
For c > , i.e. for Mod( A ρ ) < Mod(Ω) we get the following counterpart. Thenwe get the following conformal parametrization of a helicoidal minimal surface Σ , ϕ : A ρ → Σ , defined by(2.14) ϕ ( z ) = (cid:0) (cid:60) f ( z ) , (cid:61) f ( z ) , √ c Arg z (cid:1) . If f has not the logarithmic part, then we get the parametrization of a minimalsurface(2.15) ϕ ( z ) = (cid:0) (cid:60) ( g ◦ + h ◦ ) , (cid:61) ( g ◦ − h ◦ ) , √ c Arg z (cid:1) . In particular, if A ρ = A r and Ω = A R , so that R < r , then ϕ ( z ) = − (cid:112) r ( r − R )(1 − rR )(1 − r ) Arg( z ) . In particular, if r = 2 / and R = 1 / then this minimal surface over the annulus A r is shown in Figure 2.We finish this section with a lemma needed in the sequel Lemma 2.6.
Let p ∈ T = ∂ D . a) Assume that Φ is a holomorphic mapping of theunit disk into itself so that Φ( p ) = p and Φ has the derivative at p . Then Φ (cid:48) ( p ) (cid:62) − | Φ(0) | | Φ(0) | > . b) Assume that Φ is a holomorphic mapping of the unit disk into the exterior ofthe disk r D with Φ( p ) = rp . Then Φ (cid:48) ( p ) < r r − | Φ(0) || Φ(0) | + r < . To prove Lemma 2.6 we recall the boundary Schwarz lemma ([8]) which statesthe following.
Lemma 2.7 (Boundary Schwarz lemma) . Let f : D → D be a holomorphicfunction. If f is holomorphic at z = 1 with f (0) = 0 and f (1) = 1 , then f (cid:48) (1) (cid:62) .Moreover, the inequality is sharp.Proof of Lemma . Assume that p = 1 . Otherwise consider the function Φ ( z ) = p Φ( zp ) . Consider F ( z ) = (1 − Φ(0))(Φ( z ) − Φ(0))(1 − Φ(0))(1 − Φ(0)Φ( z )) . Then F (cid:48) (1) = 1 + | Φ(0) | − | Φ(0) | Φ (cid:48) (1) . Since F (0) = 0 , F (1) = 1 , it follows that F satisfies the boundary Schwarzlemma, and therefore F (cid:48) (1) is a real positive number bigger or equal to . Thisimplies a). F IGURE
2. A part of helicoid over an annulus. Here R = 1 / and r = 2 / .In order to prove b), consider the auxiliary function g ( z ) = r Φ( z ) . By applyinga) to g we get g (cid:48) (1) (cid:62) − | g (0) | | g (0) | . Since Φ (cid:48) ( z ) = − rf (cid:48) ( z )Φ ( z ) , we get − r Φ (cid:48) (1) r (cid:62) − | g (0) | | g (0) | and so − Φ (cid:48) ( r ) (cid:62) r − r | Φ(0) | r | Φ(0) | = r | Φ(0) | − r | Φ(0) | + r . INIMISERS AND KELLOGG’S THEOREM 11
This finishes the proof. (cid:3)
3. H ¨
OLDER PROPERTY OF MINIMIZERS
In this section we prove that the minimizers of the energy are global H¨oldercontinuous provided that the boundary is C .We first formulate the following result Proposition 3.1 (Caratheodory’s theorem for ( K, K (cid:48) ) mappings) . [18] Let W bea simply connected domain in C whose boundary has at least two boundary pointssuch that ∞ / ∈ ∂W . Let f : D → W be a continuous mapping of the unit disk D onto W and ( K, K (cid:48) ) quasiconformal near the boundary T .Then f has a continuous extension up to the boundary if and only if ∂W is lo-cally connected. Let Γ ∈ C ,µ , < µ (cid:54) , be a Jordan curve and let g be the arc lengthparameterization of Γ and let l = | Γ | be the length of Γ . Let d Γ be the distancebetween g ( s ) and g ( t ) along the curve Γ , i.e.(3.1) d Γ ( g ( s ) , g ( t )) = min {| s − t | , ( l − | s − t | ) } . A closed rectifiable Jordan curve Γ enjoys a b − chord-arc condition for someconstant b > if for all z , z ∈ Γ there holds the inequality(3.2) d Γ ( z , z ) (cid:54) b | z − z | . It is clear that if Γ ∈ C then Γ enjoys a chord-arc condition for some b = b Γ > .In the following lemma we use the notation Ω(Γ) for a Jordan domain boundedby the Jordan curve Γ . Similarly, Ω(Γ , Γ ) denotes the doubly connected domainbetween two Jordan curves Γ and Γ , such that Γ ⊂ Ω(Γ) .The following lemma is a ( K, K (cid:48) ) -quasiconformal version of [32, Lemma 1]. Lemma 3.2.
Assume that the Jordan curves Γ , Γ are in the class C ,α . Then thereis a constant B > , so that Γ and Γ satisfy B − chord-arc condition and forevery ( K, K (cid:48) ) − q.c. mapping f between the annulus A ρ and the doubly connecteddomain Ω = Ω(Γ , Γ ) there exists a positive constant L = L ( K, K (cid:48) , B, ρ, f ) sothat there holds (3.3) | f ( z ) − f ( z ) | (cid:54) L | z − z | β for z , z ∈ T and z , z ∈ ρ T for β = K (1+2 B ) . See appendix below for the proof of Lemma 3.2. We now can state the followingproposition:
Proposition 3.3.
Let f be a diffeomorphic minimizer of the Dirichlet energy be-tween the annulus A ρ and the doubly connected domain Ω(Γ , Γ ) , where Γ and Γ are C ,α Jordan curves. Then f is H¨older continuous on A ρ . The proof of Proposition 3.3 follows from Lemma 3.2, Lemma 4.5 below andcompactness property of A ρ .
4. P
ROOF OF T HEOREM
Proposition 4.1.
Assume that Γ is a Jordan curve in R and assume that (cid:126)X ( z ) =( X , X , X ) : D → R is a minimal graph so that (cid:126)X ( T ) = Γ . Assume that (cid:126)X isH¨older continuous in an arc T p ⊂ T containing p in its interior. If the arc T p of T is mapped onto the arc Γ p ⊂ Γ so that Γ p ∈ C ,α , < α < , then (cid:126)X is C ,α ina small neighborhood of p = e it i.e. in a domain D p,δ = { z = re it : 1 / (cid:54) r < , t ∈ ( − δ + t , δ + t ) } . From time to time in the proof we will use the notation D p or D δ instead of D p,δ , but the meaning will be clear from the context.The proof of Proposition 4.1 depends deeply on the proof of a similar statementin [27]. We observe that, almost all results proved in [27] are of local nature (see[27, Lemma 5, Lemma 6, Lemma 7]), thus we will not write the details here.We want to mention that also Lesley in [24, p. 125] have made a similar remark.Further a similar explicit formulation to related to Proposition 4.1 has been statedas Theorem 1 in Section 2.3. of the book of Dierkes, Hildebrandt and Tromba [4].Since the minimising property is preserved under composing by a conformalmapping, in view of the original Kellogg’s theorem [9], we can assume that thedomain is A ρ = { z : ρ < | z | < } .On the other hand, the minimising harmonic mapping has the local representa-tion (2.4). Here Φ p is a C ∞ diffeomorphism, and it does not cause any difficulty.Let p ∈ ∂ A ρ be arbitrary, say | p | = 1 (the other possibility is | p | = ρ ). Be-cause the boundary mapping is continuous and monotone, in view of Lemma 1.6,it follows that, there is a neighborhood T p which is mapped onto the arc Γ p ⊂ ∂ Ω .Therefore by Theorem 4.1, having in mind the notation from subsection 2.1, themapping (cid:126)X ( z ) = (cid:126)X p ( z ) = {(cid:60) f p ( z ) , (cid:61) f p ( z ) , (cid:60) (2 i √ c log Φ p ( z )) } is C ,α in a neighborhood of p , provided the boundary arc is of the same class. Butwe do not know that (cid:126)X ( T p ) ∈ C ,α . We only know that Φ p is a priori in C ∞ ( D ) and Γ p = f p ( T p ) ∈ C ,α . This will be enough for the proof.4.1. Proof of Lipschitz continuity.
We will prove the following lemma neededin the sequel.
Lemma 4.2.
Assume that f = u + iv : A ρ → Ω is a diffeomorphic minimizer,where A ρ = { z : ρ < | z | < } and assume that ∂ Ω ∈ C ,α . Then f is Lipschitzcontinuous.Proof. We use the notation from Subsection 2.1. The constant C that appear in theproof is not the same and its value can vary from one to the another appearance.Assume also q ∈ Γ = ∂ Ω , and, by using a rotation and a translation (if it isnecessary) we can assume that q = 0 , and the tangent line of Γ at q is the realaxis. Post-composing by a such Euclidean isometry, the Euclidean harmonicity is INIMISERS AND KELLOGG’S THEOREM 13 preserved. Then in a small neighborhood of q , Γ has the following parameterization γ ( x ) = ( x, φ ( x )) , x ∈ ( − x , x ) , so that φ (0) = φ (cid:48) (0) = 0 . Assume also that, p = 1 and f (1) = q = 0 . And assume that for a small angle Λ = Λ p = { e iθ : | θ | (cid:54) (cid:15) } we have f (Λ) ⊂ γ ( − x , x ) . We can assume also that x is a smallenough positive constant global for all points q ∈ ∂ Ω . We want to localize theproblem. We only need to prove that f is C ,α (cid:48) in a small neighborhood of . Wealso work with f p = f ◦ Φ p : D → f ( A p ) instead of f , where Φ p ( p/ | p | ) = p ,and assume that γ ( − x , x ) ⊂ ∂A p for every p ∈ ∂ A ρ . We will from time totime use notation f instead of f p , since they behave in the same way in a smallneighborhood of p , because Φ p is a priori in C ∞ Thus, there exists a function x : Λ → R so that f ( e it ) = ( u ( e it ) , v ( e it )) = γ ( x ( e it )) = ( x ( e it ) , φ ( x ( e it ))) . We will also from time to time use notation x ( t ) instead of x ( e it ) . Similarly v ( t ) instead of v ( e it ) .Now we have v = (cid:61) ( f ) = (cid:61) ( g + h ) = (cid:61) ( g − h ) = (cid:60) ( i ( h − g )) and therefore,(4.1) v θ = (cid:60) ( z ( g (cid:48) − h (cid:48) )) . Because Γ p ∈ C ,α we have as in [27, eq. 3], the following relation(4.2) | φ ( s ) − φ ( t ) | (cid:54) C | s − t |{ min {| s | α , | t | α } + | t − s | α } , | t | < t , | s | < t . The constant C and t are the same for all points p ∈ ∂ A ρ . Recall that p = 1 and f (1) = 0 . By using translations and rotations in the domain and image domain,we will obtain this property, and therefore we do not loos the generality.Further | φ ( x ) − φ (0) − φ (cid:48) (0) x || x | α = | φ (cid:48) ( θx ) − φ (cid:48) (0) || x | α (cid:54) C, where θ ∈ (0 , .Since(4.3) v ( e it ) = φ ( x ( e it )) we get(4.4) | v ( e it ) − v (1) | = | φ ( x ( e it )) | (cid:54) C | x ( e it ) | α . Now, the following sequence of the inequalities follow from (4.2), (4.4) and Lemma 3.2.(4.5) | v ( e it ) − v (1) | (cid:54) C | t | β (1+ α ) , and | v ( e it ) − v ( e is ) | = | φ ( x ( t )) − φ ( x ( s )) | (cid:54) C | x ( s ) − x ( t ) |{ min {| x ( s ) | α , | x ( t ) | α } + | x ( t ) − x ( s ) | α } (4.6)and so | v ( e it ) − v ( e is ) | (cid:54) CL α | s − t | β { min {| s | αβ , | t | βα } + | t − s | βα } . (4.7) Here(4.8) L = L Φ , where Φ = sup | z | =1 ,p ∈ ∂ A ρ | Φ (cid:48) p ( z ) | , where L is defined in Lemma 3.2.In order to continue we collect some results from [27] and [9].First we formulate [27, Lemma 7] and a relation from its proof: Lemma 4.3.
Assume that F is a bounded holomorphic mapping defined in theunit disk, so that | F | (cid:54) M in D . Further assume that for a constants (cid:54) δ , (cid:54) η, µ (cid:54) π/ so that for almost every − δ (cid:54) t, s (cid:54) δ we have |(cid:60) F ( t ) − (cid:60) F ( s ) | (cid:54) M | t − s | µ { min {| t | η , | s | η } + | t − s | η } . Then for ζ = τ e is , with | s | (cid:54) δ/ , / (cid:54) τ (cid:54) we have the estimates (4.9) | F (cid:48) ( ζ ) | (cid:54) M | s | η (1 − τ ) µ − + M (1 − τ ) µ + η − + M , if µ + η < ; M | s | η (1 − τ ) µ − + M log − τ + M , if µ + η = 1 ; M | s | η (1 − τ ) µ − + M , if µ < ∧ µ + η > ; M | s | η · log − τ + M , if µ = 1 ; M , if µ > ;and (4.10) | F ( τ ) − F (1) | (cid:54) N (1 − τ ) µ + η , if µ + η < ; N (1 − τ ) log − τ , if µ + η = 1 ; N (1 − τ ) , if µ + η > ,and (4.11) | F ( e is ) − F (1) | (cid:54) N | s | µ + η , if µ + η < ; N | s | log | s | , if µ + η = 1 ; N | s | , if µ + η > .Here N , M , M , M depends on M, η, µ and δ . By repeating the proof of the theorem of Hardy and Littlewood, [9, Theorem 3,p. 411] and [9, Theorem 4, p. 414], we can state the following two theorems.
Lemma 4.4.
Let µ ∈ (0 , and let D δ = { z = re i ( s + s ) : 1 / (cid:54) r (cid:54) , s ∈ ( − δ, δ ) } . Assume that f is a holomorphic mapping defined in the unit disk so that | f (cid:48) ( z ) | (cid:54) M (1 − | z | ) µ − , where < µ < and z ∈ D δ . Then the radial limit lim τ → − f ( τ e iθ ) = f ( e iθ ) exists for every θ ∈ ( − δ + s , δ + s ) and we have there the inequality | f ( w ) − f ( w (cid:48) ) | (cid:54) N | w − w (cid:48) | µ , w, w (cid:48) ∈ D δ , where N depends on M and µ . The converse is also true. INIMISERS AND KELLOGG’S THEOREM 15
Lemma 4.5.
Let µ ∈ (0 , . Assume that f is continuous harmonic mapping onthe closed unit disk and satisfies on a small arc Λ = { e iθ : | θ − s | < δ } thecondition: | f ( e is ) − f ( e it ) | (cid:54) A | t − s | µ , e it , e is ∈ Λ , for almost every points s and t . Then f satisfies the H¨older condition | f ( z ) − f ( w ) | (cid:54) B | z − w | µ for z, w ∈ D δ = { z = re is : 1 / (cid:54) r (cid:54) , s ∈ ( − δ + s , δ + s ) } . We now reformulate a result of Privalov [9, p. 414, Theorem 5] in its local form(w.r.t. the boundary).
Lemma 4.6.
Let µ ∈ (0 , . Assume that f = u + iv is a holomorphic boundedfunction defined on the unit disk D and assume that u satisfies the condition | u ( e it ) − u ( e is ) | (cid:54) M | e it − e is | µ , for almost every s and t so that | s − s | < δ and | t − s | < δ . Then there is a constant N depending on M and µ so that | f ( z ) − f ( w ) | (cid:54) N | z − w | µ for z, w ∈ D δ , where D δ = { z = re is : 1 / (cid:54) r (cid:54) , | s − s | (cid:54) δ } . Proof of Lemma . From Schwarz formula we have f ( ζ ) = 12 π (cid:90) π u ( e it ) e it + ζe it − ζ dt + iC. Thus f (cid:48) ( ζ ) = 22 π (cid:90) π u ( e it ) e it dt ( e it − ζ ) = 1 π (cid:90) π u ( e it ) − u ( e is )( e it − ζ ) e it dt, ζ = re is . Let ζ = re is ∈ D δ . Then we get | f (cid:48) ( ζ ) | (cid:54) π (cid:90) π − π | u ( e i ( s + t ) ) − u ( e is ) | − r cos t + r dt. If t ∈ [ − π, π ] , then − r cos t + r (cid:62) (1 − r ) + 4 rπ t . Further, if s ∈ ( s − δ, s + δ ) , t ∈ ( − δ, δ ) then we get | u ( e i ( s + t ) ) − u ( e is ) | (cid:54) K | t | µ . If t ∈ [ − π, π ] \ ( − δ, δ ) , then | u ( e i ( s + t ) ) − u ( e is ) | (cid:54) M ≤ Mδ µ | t | µ . The conclusion is that | f (cid:48) ( ζ ) | (cid:54) N (1 − | ζ | ) − µ , for ζ ∈ D δ . Then from Lemma 4.4 we get the desired result. (cid:3)
Repeating the proof of the preceding lemma, we also obtain the following point-wise statement.
Lemma 4.7.
Let µ ∈ (0 , and δ > . Assume that f = u + iv is a holomorphicbounded function defined on the unit disk D and assume that u satisfies the condi-tion | u ( e it ) − u ( e is ) | (cid:54) M | e it − e is | µ , for almost every t : | t − s | < δ . Thenthere is a constant N so that | f (cid:48) ( re is ) | (cid:54) N (1 − r ) − µ for < r < . Now we continue the proof of Lemma 4.2. Observe that β < / and so β (1 + α ) < . For F p ( z ) = i ( h p ( z ) − g p ( z )) we get from (4.9)(4.12) | F (cid:48) p ( τ ) | (cid:54) C (1 − τ ) (1+ α ) β − , for / (cid:54) τ < . Since(4.13) F p ◦ Φ − p ( z ) = F ◦ Φ − ( z ) , for z ∈ Φ ( D ) ∩ Φ p ( D ) = A ∩ A p . (and for | p | = ρ , F p ◦ Φ − p ( z ) = F ρ ◦ Φ − ρ ( z ) ,for z ∈ A ρ ∩ A p ) ), we get that(4.14) | F (cid:48) p ( z ) | (cid:54) C (1 − | z | ) (1+ α ) β − , for all z ∈ D p,(cid:15) := { z = re it : 1 / (cid:54) | z | < , t ∈ ( − (cid:15), (cid:15) ) } . Further since p = 1 is not a special point we get that (4.14), is valid for all z ∈ D p,(cid:15) := { z = re it + it :1 / (cid:54) | z | < , t ∈ ( − (cid:15), (cid:15) ) } . Here p = e it or p = ρe it . Recall that (cid:15) > issmall enough so that, for every point q ∈ ∂ Ω , the graph of ∂ Ω after a rotation andtranslation has the form ( x ( e it ) , φ ( x ( e it ))) , t ∈ ( − (cid:15), (cid:15) ) .Then from (4.14) and Lemma 4.4 we get that F p is C , (1+ α ) β in D p,(cid:15) .Let(4.15) G p ( z ) = g p ( z ) + h p ( z ) . Then we also have(4.16) G p ◦ Φ − p ( z ) = G ◦ Φ − ( z ) , for z ∈ Φ ( D ) ∩ Φ p ( D ) = A ∩ A p . Then we have(4.17) (cid:0) G (cid:48) p ( z ) (cid:1) + (cid:0) F (cid:48) p ( z ) (cid:1) = 4 g (cid:48) p ( z ) h (cid:48) p ( z ) = 4 (cid:18) Φ (cid:48) p ( z )Φ p ( z ) (cid:19) . Since the right hand side of (4.17) is bounded, it follows that G p is (1 + α ) β H¨older continuous. Namely (cid:12)(cid:12)(cid:12) G (cid:48) p ( z )(1 − | z | ) − (1+ α ) β (cid:12)(cid:12)(cid:12) (cid:54) (cid:12)(cid:12)(cid:12)(cid:12) (cid:48) ( z )Φ p ( z ) (1 − | z | ) − (1+ α ) β (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) F (cid:48) p ( z )(1 − | z | ) − (1+ α ) β (cid:12)(cid:12)(cid:12) (cid:54) N . INIMISERS AND KELLOGG’S THEOREM 17
Now we have(4.18) h p = 12 ( iF p + G p ) , g p = 12 ( − iF p + G p ) . Since f p = g p + h p and f p ( e it ) = γ ( x ( e it )) , where γ ( x ) = ( x, φ ( x )) ∈ C ,α , itfollows that x ( e it ) = (cid:60) G ( e it ) and therefore(4.19) x ∈ C , (1+ α ) β ( ∂D p,(cid:15) ∩ ∂ D ) . Chose β < / so that none of numbers (1 + α ) k β is equal to for every k . Let n be so that (1 + α ) n β < < (1 + α ) n +1 β . Then by successive application of theprevious procedure we get | F (cid:48) p ( z ) | (cid:54) M (1 − | z | ) (1+ α ) n β − , z = ρe is , / < ρ < , s ∈ ( − (cid:15), (cid:15) ) , and | G (cid:48) p ( z ) | (cid:54) M (1 − | z | ) (1+ α ) n β − , z = ρe is , / < ρ < , s ∈ ( − (cid:15), (cid:15) ) . Then we get | F p ( w ) − F p ( w (cid:48) ) | (cid:54) N | w − w (cid:48) | (1+ α ) n β , w, w (cid:48) ∈ D (cid:15) , | G p ( w ) − G p ( w (cid:48) ) | (cid:54) N | w − w (cid:48) | (1+ α ) n β , w, w (cid:48) ∈ D (cid:15) , where N depends on M and µ and so | F p ( e it ) − F p ( e is ) | (cid:54) N | s − t | (1+ α ) n β , and | G p ( e it ) − G p ( e is ) | (cid:54) N | s − t | (1+ α ) n β , for | s | < (cid:15), | s | < (cid:15) . Thus x ∈ C , (1+ α ) n β ( ∂D p,(cid:15) ∩ D ) , and, as in (4.6) and (4.32) we get | f p ( e it ) − f p ( e is ) | (cid:54) CL α | s − t | (1+ α ) n β { min {| s | (1+ α ) n αβ , | t | (1+ α ) n αβ } + | t − s | (1+ α ) n αβ } . From Lemma 4.3, for µ = (1 + α ) n β and η = (1 + α ) n αβ , by choosing s = 0 ,we get(4.20) | F p (cid:48) ( τ ) | (cid:54) M , τ ∈ (1 / , . Since the functions F (cid:48) p ( z ) and (Φ (cid:48) ( z )) Φ p ( z ) are bounded in [1 / , , it follows that G (cid:48) p ( z ) is also bounded in [1 / , . Let M > , so that | F (cid:48) p ( τ ) | (cid:54) M , | G (cid:48) p ( τ ) | (cid:54) M , τ ∈ (1 / , . Recall that h p = 12 ( iF p + G p ) , g p = 12 ( − iF p + G p ) . Thus we get | Df p ( τ ) | = | g (cid:48) p ( τ ) | + | h (cid:48) p ( τ ) | (cid:54) M . Then we get(4.21) | Df ( τ ) | = | f ζ ( τ ) | + | f ¯ ζ ( τ ) | (cid:54) L Φ , ρ (cid:54) τ (cid:54) ρ ∨ ρ < τ < , where ρ < ρ are certain positive constants and Φ = max z ∈ D | Φ (cid:48) ( z ) | . Since the real interval [ ρ, has not a special geometric character for A ρ , we getthat(4.22) | Df ( z ) | = | f ζ ( z ) | + | f ¯ ζ ( z ) | (cid:54) M , z ∈ B ρ ( ρ , ρ ) , where B ρ ( ρ , ρ ) = { z = τ e is : ρ < τ (cid:54) ρ ∨ ρ < τ < , s ∈ [0 , π ) } . Since f ∈ C ∞ ( A ρ ) we get f ∈ C , ( A ρ ) as claimed, and thus the proof ofLemma 4.2 is finished. (cid:3) The minimizer is C ,α (cid:48) up to the boundary. We continue to use the notationfrom Subsections 4.1 and 2.1. The constant C that appear in the proof is not thesame and its value can vary from one to the another appearance, but it is globaland the same for all points of ∂ A ρ . Assume that f = u + iv : A ρ → Ω is adiffeomorphic minimizer, where A ρ = { z : ρ < | z | < } , we need to show thatis C ,α (cid:48) ( A ρ ) , provided that ∂ Ω ∈ C ,α . We only need to prove that f is C ,α (cid:48) insmall neighborhood of p ∈ ∂ A ρ . We also work with f p = f ◦ Φ p : D → f ( A p ) instead of f , where Φ p (1) = 1 as in the previous part of the paper. We will showthat f p ∈ C ,α (cid:48) ( D p ) , where D p = { z = re is + is : 1 / (cid:54) r < , s ∈ ( − (cid:15), (cid:15) ) } ,where p/ | p | = e is , and (cid:15) > is a small enough positive constant valid for allpoints p ∈ ∂ A ρ .Assume as before that p = 1 and f ( p ) = 0 = q ∈ ∂ Ω . Recall that Λ = Λ p = { e iθ : | θ | (cid:54) (cid:15) } . We already proved that f is Lipschitz continuous. We know as wellthat γ ( x ) = ( x, φ ( x )) ∈ C ,α . Since f ( e iθ ) = γ ( x ( e iθ )) , we have that x = x ( e iθ ) is Lipschitz continuous.Since f is a diffeomorphism, there exists a non-decreasing continuous function x : Λ → R so that f ( e it ) = u ( e it ) + iv ( e it ) = γ ( x ( e it )) . We can also assume that(4.23) ∂ t x ( e it ) (cid:62) for almost every t , because f is a restriction of a harmonic diffeomorphism betweendomains and by Proposition 1.6 it is monotone at the boundary.It follows from (4.4) and the fact that x is Lipschitz and x (1) = 0 that v isdifferentiable with respect to θ for θ = 0 , i.e. in and(4.24) ∂ θ v (1) = ∂ θ v ( e iθ ) | θ =0 = 0 . INIMISERS AND KELLOGG’S THEOREM 19
Therefore | ∂ θ v ( e iθ ) − ∂ θ v (1) | = | ∂ θ v ( e iθ ) | = | φ (cid:48) ( x ( e iθ )) | · | ∂ θ x ( e iθ ) | (cid:54) C | x ( e iθ ) | α (cid:54) C | θ | α , (4.25)holds for a.e. θ in a certain interval. Recall from (4.1) that v θ = (cid:60) ( z ( g (cid:48) − h (cid:48) )) , sofrom Lemma 4.7 we conclude that(4.26) | ( z ( g (cid:48) − h (cid:48) )) (cid:48) ( τ ) | (cid:54) C (1 − τ ) α − , / (cid:54) τ (cid:54) . We also recall that we defined(4.27) k ( z ) = i ( g (cid:48) ( z ) + h (cid:48) ( z )) , k ( z ) = h (cid:48) ( z ) − g (cid:48) ( z ) . In view of (2.3)(4.28) k ( z ) = (cid:112) h (cid:48) ( z ) g (cid:48) ( z ) = (cid:115) c (Φ (cid:48) p ( z )) Φ p ( z ) . Then from (4.26) we have that the following limit k (1) = i∂ t ( h ( e it ) − g ( e it )) | t =0 exists. Moreover, it can be assumed that k (1) = i∂ t ( h ( e it ) − g ( e it )) , because lim τ → h (cid:48) ( τ e is ) = − ie − is ∂ t h ( e it ) | t = s lim τ → g (cid:48) ( τ e is ) = − ie − is ∂ t g ( e it ) | t = s for almost every s ∈ ( − π, π ) . This follows from the F. Riesz theorem [9, Theo-rem 1, p. 409], because | h (cid:48) | and | g (cid:48) | are bounded. Moreover we have from (4.26)(4.29) | k (1) − k ( τ ) | (cid:54) C (1 − τ ) α , / (cid:54) τ (cid:54) . We conclude that k (1) = 2 lim r → (cid:112) h (cid:48) ( r ) g (cid:48) ( r ) exists and(4.30) | k (1) − k ( τ ) | (cid:54) C | − τ | α , / (cid:54) τ < . Further since x ( e iθ ) = u ( e iθ ) = (cid:60) ( f ( e iθ )) = (cid:60) ( g + h ) , we get(4.31) ∂ θ x ( e iθ ) = (cid:60) (cid:104) ie iθ ( g (cid:48) ( e iθ ) + h (cid:48) ( e iθ )) (cid:105) = (cid:60) (cid:104) e iθ k ( e iθ ) (cid:105) (cid:62) . Then the following equality is important in our approach(4.32) k + k + k = 0 . We now proceed as J. C. C. Nitsche did in [27]. So k ( τ ) = − k ( τ ) − k ( τ ) . It follows that the following limit k (1) := lim τ → k ( τ ) = − k (1) − k (1) , exists. Therefore we get | k (1) − k ( τ ) | = | k (1) − k ( τ ) + k (1) − k ( τ ) | . Then from (4.29) and (4.30) we get(4.33) | k ( τ ) − k (1) | (cid:54) C | τ − | α ≡ ε, / (cid:54) τ < . From (4.24), in view of (4.1), we get(4.34) (cid:60) ( k (1)) = 0 . Further, from (4.32), we have(4.35) (cid:60) ( k (1)) (cid:61) ( k (1)) + (cid:60) ( k (1)) (cid:61) ( k (1)) + (cid:60) ( k (1)) (cid:61) ( k (1)) = 0 and (cid:60) ( k (1)) + (cid:60) ( k (1)) + (cid:60) ( k (1))= (cid:61) ( k (1)) + (cid:61) ( k (1)) + (cid:61) ( k (1)) . (4.36)Notice the following important fact, the relations (4.35) and (4.36) make sense foralmost every p ∈ ∂ A ρ . Namely k ( z ) = P [ k i | T ]( z ) , i = 1 , , . We assume that p = 1 is one of such points. From Lemma 2.6 it follows that k (1) is a real or animaginary number. Therefore we have (cid:60) ( k (1)) (cid:61) ( k (1)) = 0 . Thus(4.37) (cid:60) ( k (1)) (cid:61) ( k (1)) = 0 . Now we divide the proof into two cases, and remember that the case c = 0 coincides with the the case when the minimizer is a conformal biholomorphism:(1) We first consider the case c < and put ξ = 1 . In this case (cid:60) k (1) = 0 . Butthen cannot be (cid:61) ( k (1)) (cid:54) = 0 , because in that case (cid:60) ( k (1)) = 0 , and therefore by(4.36), we get (cid:61) ( k (1)) + (cid:61) ( k (1)) + (cid:61) ( k (1)) = 0 . The conclusion is that (cid:61) ( k (1)) = 0 . Observe also that(4.38) k (1) = (cid:112) −(cid:61) ( k (1)) − c (cid:62) √− c > . (2) Then we consider the case c > and put ξ = − i sign (cid:61) k (1) if (cid:61) k (1) (cid:54) = 0 and ξ = 1 for the case (cid:61) k (1) = 0 .Then we apply the following lemma for w = ξk ( τ ) and w = ξk (1) and for ε defined in (4.33) Lemma 4.8. [27]
Let w = a + ib and w = ω be complex numbers satisfying theinequalities ω (cid:62) and | w − w | (cid:54) ε for some ε > . Then either | w − w | (cid:54) √ ε or | w | (cid:62) √ ε and a < , ω > . Then as in [27] we get(4.39) | k ( τ ) − k (1) | (cid:54) C (1 − τ ) α/ , / (cid:54) τ < . INIMISERS AND KELLOGG’S THEOREM 21
Recall now that(4.40) k ( z ) = G (cid:48) p ( z ) k ( z ) = F (cid:48) p ( z ) . Then from (4.13) and (4.16) we get(4.41) Φ (cid:48) p ( F ( z )) F (cid:48) p ( z ) = Φ (cid:48) ( F p ( z )) F (cid:48) ( z ) and(4.42) Φ (cid:48) p ( G ( z )) G (cid:48) p ( z ) = Φ (cid:48) ( G p ( z )) G (cid:48) ( z ) for z ∈ A p ∩ A . Therefore from (4.29), (4.30) and (4.39), in a small (cid:15) − neighborhoodof p = 1 , we get the inequalities(4.43) | k j ( τ e it ) − k j ( e it ) | (cid:54) C (1 − τ ) α/ , / (cid:54) τ < , j = 1 , for almost every t ∈ ( − (cid:15), (cid:15) ) .Further as in [27, p. 325-326] we obtain that | k j ( e it ) − k j ( e is ) | (cid:54) C | s − t | αα +2 , j = 1 , and almost every t, s ∈ ( − (cid:15), (cid:15) ) . The function k has the same behavior a priori.From this it follows that(4.44) k j ∈ C , αα +2 ( D p,(cid:15) ) , j = 1 , , . This concludes the case c (cid:62) .Now we continue to prove the case c (cid:54) . This case we use of the parameteri-zation(4.45) ϕ ( z ) = (cid:18) (cid:60) f ( z ) , (cid:61) f ( z ) , √− c log 1 | z | (cid:19) . By using this we get that ϕ ( A ρ ) = Σ is a doubly connected minimal surfacebounded by two Jordan curves Υ = { ( x, y, − √− c log r ) , ( x, y ) ∈ Γ } and Υ = { ( x, y, − √− c log R ) , ( x, y ) ∈ Γ } , where Γ and Γ are the inner and outer boundaries of Ω . Moreover ∂ Ω ∈ C ,α ifand only if ∂ Σ ∈ C ,α .This time Proposition 4.1 will imply the result.From Proposition 4.1 we obtain that f ∈ C ,α (Φ − p ( A p )) , where A p is a smallneighborhood of a fixed point p . Notice that A p = p/ | p | A or A p = p/ | p | A ρ ,where A and A ρ are fixed domains, whose boundary contains a Jordan arc Λ ,whose interior contains respectively ρ . Since A ρ = Φ − p ( A p ) , we get that f ∈ C ,α ( A ρ ) as claimed.Thus we have finished the proof of Theorem 1.1.
5. P
ROOF OF T HEOREM ϕ defined (4.45), in order toget that ϕ ∈ C n,α ( A ρ ) , provided that ∂ Ω ∈ C n,α ( A ρ ) , which is equivalent withthe condition ∂ Σ ∈ C n,α ( A ρ ) .
6. C
ONCLUDING REMARK
We expect that the following statement is true:
Conjecture 6.1.
Assume that f : D → Ω is a energy minimal diffeomorphism ofthe energy between two domains with C ,α boundaries. If Mod( D ) (cid:54) Mod(Ω) then the diffeomorphic minimizer of Dirichlet energy, which is shown to have a C ,α (cid:48) extension up to the boundary is diffeomorphic on the boundary also and theextension is C ,α . This conjecture is motivated by the existing result described in Proposition 1.3and the example presented in (2.13) of the unique minimizer (up to the rotation) ofDirichlet energy between annuli A r and A R , that maps the outer boundary onto theouter boundary (see [2] for details). The mapping is a a diffeomorphism between A r and A R , provided that(6.1) R < r r . If R = r r , and < r < , then the mapping w ( z ) = r + | z | ¯ z (1 + r ) is a harmonic minimizer (see [2]) of the Euclidean energy of mappings between A ( r, and A ( r r , , however | w z | = | w ¯ z | = 11 + r for | z | = r , and so w is not bi-Lipschitz.Note that (6.1) is satisfied provided that Mod A r (cid:54) Mod A R . The inequality(6.1) (with (cid:54) instead of < ) is necessary and sufficient for the existence of a har-monic diffeomorphism between A r and A R a conjecture raised by J. C. C. Nitschein [28] and proved by Iwaniec, Kovalev and Onninen in [11], after some partialresults given by Lyzzaik [26], Weitsman [36] and Kalaj [19]. If R > r r , then the minimizer of Dirichlet energy throughout the deformations D ( A r , A R ) isnot a diffeomorphism ( see [2] and [3, Example 1.2]).We want to refer to one more interesting behavior that minimizers of energyshare with conformal mappings. Namely, if f is a diffeomorphic minimizer of INIMISERS AND KELLOGG’S THEOREM 23
Dirichlet energy between the domains A ρ and Ω(Γ , Γ ) so that Γ and Γ are con-vex, then f ( t T ) is convex for t ∈ ( ρ, [22]. Further if Γ and Γ are circles, then f ( t T ) is a circle [23]. A PPENDIX -P ROOF OF L EMMA a ∈ C and r > , put D ( a, r ) := { z : | z − a | < r } and define ∆ r =∆ r ( z ) = D ∩ D ( z , r ) . Denote by k τ the circular arc whose trace is { ζ ∈ D : | ζ − z | = τ } . Lemma 6.2 (The length-area principle) . [18] Assume that f is a ( K, K (cid:48) ) − q.c. on ∆ r , < r < r (cid:54) , z ∈ T . Then (6.2) F ( r ) := (cid:90) r l τ τ dτ ≤ πKA ( r ) + π K (cid:48) r , where l τ = | f ( k τ ) | denote the length of f ( k τ ) and A ( r ) is the area of f (∆ r ) .Proof of Lemma . Let Φ be a conformal mapping of Ω(Γ) onto the unit disk ,where
Ω(Γ) is the Jordan domain bounded by Γ , so that Φ( f (1)) = 1 , Φ( f ( e ± i π )) = e ± i π . Then Φ ◦ f is a normalized ( K , K (cid:48) ) quasiconformal mapping near T ⊂ ∂ A ρ .For a ∈ C and r > , put D ( a, r ) := { z : | z − a | < r } . Since Φ is a dif-feomorphism near T , the inequality (3.3) will be proved for f if we prove it for Φ ◦ f .It is clear that if z ∈ T , then, because of normalization, f ( T ∩ D ( z , hascommon points with at most two of three arcs w w , w w and w w . (Here w , w , w ∈ Γ divide Γ into three arcs with the same length such that f (1) = w , f ( e πi/ ) = w , f ( e πi/ ) = w , and T ∩ D ( z , do not intersect at least one ofthree arcs defined by , e πi/ and e πi/ ).Let κ τ = { t ∈ [0 , π ] : z + τ e it ∈ k τ } . Let l τ = | f ( k τ ) | denotes the lengthof f ( k τ ) . Let Γ τ := f ( T ∩ D ( z , τ )) and let | Γ τ | be its length. Assume w and w (cid:48) are the endpoints of Γ τ , i.e. of f ( k τ ) . Then | Γ τ | = d Γ ( w, w (cid:48) ) or | Γ τ | = | Γ | − d Γ ( w, w (cid:48) ) . If the first case holds, then since Γ enjoys the B − chord-arc condition,it follows | Γ τ | (cid:54) B | w − w (cid:48) | (cid:54) Bl τ . Consider now the last case. Let Γ (cid:48) τ = Γ \ Γ τ .Then Γ (cid:48) τ contains one of the arcs w w , w w , w w . Thus | Γ τ | (cid:54) | Γ (cid:48) τ | , andtherefore | Γ τ | (cid:54) Bl τ . Using the first part of the proof, it follows that the length of boundary arc Γ r of f (∆ r ) does not exceed Bl r which, according to the fact that ∂f (∆ r ) = Γ r ∪ f ( k r ) , implies(6.3) | ∂f (∆ r ) | (cid:54) l r + 2 Bl r . Therefore, by the isoperimetric inequality A ( r ) (cid:54) | ∂f (∆ r ) | π (cid:54) ( l r + 2 Bl r ) π = l r (1 + 2 B ) π . Employing now (6.2) we obtain F ( r ) := (cid:90) r l τ τ dτ (cid:54) Kl r (1 + 2 B ) πK (cid:48) r . Observe that for < r (cid:54) − ρ there holds rF (cid:48) ( r ) = l r . Thus F ( r ) (cid:54) KrF (cid:48) ( r ) (1 + 2 B ) πK (cid:48) r . Let G be the solution of the equation G ( r ) = KrG (cid:48) ( r ) (1 + 2 B ) πK (cid:48) r , G (0) = 0 , defined by G ( r ) = πK (cid:48) K (1+2 B ) + 1 r = 2 πK (cid:48) K (1 + 2 B ) + 4 r . It follows that for β = 2 K (1 + 2 B ) there holds ddr log([ F ( r ) − G ( r )] · r − β ) (cid:62) , i.e. the function [ F ( r ) − G ( r )] · r − β is increasing. This yields [ F ( r ) − G ( r )] (cid:54) [ F (1 − ρ ) − G (1 − ρ )]( r/ (1 − ρ )) β (cid:54) C ( K, K (cid:48) , B, ρ, f ) r β . Now for every r (cid:54) − ρ there exists an r ∈ [ r/ √ , r ] such that F ( r ) = (cid:90) r l τ τ dτ (cid:62) (cid:90) rr/ √ l τ τ dτ = l r log √ . Hence, l r (cid:54) C ( K, K (cid:48) , B, ρ, f )log 2 r β . If z is a point with | z | (cid:54) and | z − z | = r/ √ , then by (6.3) | f ( z ) − f ( z ) | (cid:54) (1 + 2 B ) l r . Therefore | f ( z ) − f ( z ) | (cid:54) H | z − z | β , where H = H ( K, K (cid:48) , B, ρ, f ) . Now for z , z ∈ T , then the arch ( z , z ) can be divided into Q = Q ( ρ ) equalarcs by points w , . . . , w Q , so that | w i − w i +1 | (cid:54) − ρ √ . Then we get the inequality | f ( z ) − f ( z ) | (cid:54) Q (cid:88) j =1 | f ( w j ) − f ( w j − ) | (cid:54) QH | w − w | β (cid:54) QH | z − z | β . INIMISERS AND KELLOGG’S THEOREM 25
Thus(6.4) | f ( z ) − f ( z ) | (cid:54) L ( K, K (cid:48) , B, ρ, f ) | z − z | β . In order to deal with the inner boundary, we take the composition F ( z ) = 1 f ( ρ/z ) − a , which maps the annulus A ρ into Ω (cid:48) = { / ( z − a ) : z ∈ Ω } . Here a is a pointinside the inner Jordan curve. Then Ω (cid:48) = Ω (cid:48) (Γ (cid:48) , Γ (cid:48) ) is a doubly connected domainwith C ,α boundary.Now we construct a conformal mapping Φ between the domain Ω(Γ (cid:48) ) and theunit disk and repeat the previous case in order to get that the inequality (3.3) doeshold in both boundary components. (cid:3) A CKNOWLEDGEMENT
We would like to thank the anonymous referee for a large number of remarksthat helped to improve this paper. His/her idea is used to shorten the proof of thecase c < . R EFERENCES [1] S. Axler, P. Bourdon and W. Ramey:
Harmonic function theory , Springer-Verlag, New York1992.[2] K. A
STALA , T. I
WANIEC , AND
G. J. M
ARTIN , Deformations of annuli with smallest meandistortion . Arch. Ration. Mech. Anal. , 899–921 (2010).[3] J. C
RISTINA , T. I
WANIEC , L. V. K
OVALEV , J. O
NNINEN , The Hopf-Laplace equation: har-monicity and regularity.
Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) , No. 4, 1145–1187 (2014).[4] U. D IERKES , S. H
ILDEBRANDT , A. J. T
ROMBA
Regularity of minimal surfaces,
Grundlehrender mathematischen Wissenschaften 340 Springer-Verlag Berlin Heidelberg, 2010.[5] P. D
UREN : Harmonic mappings in the plane.
Cambridge University Press, 2004.[6] F. H ´e LEIN , On weakly harmonic maps and Noether harmonic maps from a Riemann surface intoa Riemannian manifold
Banach Center Publications , 175–181 (1992).[7] F. H ´e LEIN , Regularity of weakly harmonic maps between a surface and a Riemannian manifold .(French) C. R. Acad. Sci., Paris, S ´e r. I , 591–596 (1991).[8] J. G ARNETT : Bounded Analytic Functions.
Academic Press, New York (1981)[9] G. M. G
OLUZIN : Geometric function theory of a Complex Variable , –Transl. Of Math. Mono-graphs 26. - Providence: AMS, 1969.[10] T. I
WANIEC , L. V. K
OVALEV AND
J. O
NNINEN : Lipschitz regularity for inner-variationalequations.
Duke Math. J. 162, No. 4, 643-672 (2013).[11] T. I
WANIEC , L. V. K
OVALEV AND
J. O
NNINEN : The Nitsche conjecture , J. Amer. Math. Soc. , 345-373 (2011).[12] T. I WANIEC , K.-T. K OH , L. K OVALEV , J. O
NNINEN , Existence of energy-minimal diffeomor-phisms between doubly connected domains.
Invent. Math. , No. 3, 667-707 (2011).[13] J. J
OST : Minimal surfaces and Teichm¨uller theory . Yau, Shing-Tung (ed.), Tsing Hua lec-tures on geometry and analysis, Taiwan, 1990-91. Cambridge, MA: International Press. 149-211(1997).[14] J. J
OST : Two-dimensional geometric variational problems , John Wiley and Sons, Ltd., Chich-ester, 1991. [15] J. J
OST : Harmonic maps between surfaces (with a special chapter on conformal mappings).
Lecture Notes in Mathematics. 1062. Berlin etc.: Springer-Verlag. X, 133 p.; (1984).[16] D. K
ALAJ , Energy-minimal diffeomorphisms between doubly connected Riemann surfaces.
Calc. Var. Partial Differ. Equ. , No. 1-2, 465-494 (2014).[17] D. K ALAJ , Lipschitz Property of Minimisers Between Double Connected Surfaces
J GeomAnal (2019). https://doi.org/10.1007/s12220-019-00235-x[18] D. K
ALAJ , M. M
ATELJEVIC : ( K, K (cid:48) ) -quasiconformal harmonic mappings. Potential Anal. , No. 1, 117–135 (2012).[19] D. K ALAJ : On the Nitsche conjecture for harmonic mappings in R and R . Israel J. Math. , 241–251 (2005).[20] O. K
ELLOGG : Harmonic functions and Green’s integral,
Trans. Amer. Math. soc. vol. (1912) pp. 109-132.[21] D. K INDERLEHRER : The boundary regularity of minimal surfaces.
Ann. Sc. Norm. Super.Pisa, Sci. Fis. Mat., III. Ser. 23, 711–744 (1969).[22] N.-T. K OH : Hereditary convexity for harmonic homeomorphisms.
Indiana Univ. Math. J. ,231–243 (2015).[23] N.-T. K OH : Hereditary circularity for energy minimal diffeomorphisms.
Conform. Geom.Dyn. , 369–377 (2017).[24] F. D. L ESLEY
Differentiability of minimal surfaces at the boundary.
Pac. J. Math. 37, 123-139(1971).[25] F. D. L
ESLEY , S. E. W
ARSCHAWSKI : Boundary behavior of the Riemann mapping functionof asymptotically conformal curves
Math. Z. (1982), 299–323.[26] A. L
YZZAIK : The modulus of the image annuli under univalent harmonic mappings and aconjecture of J.C.C. Nitsche , J. London Math. Soc., , 369–384 (2001).[27] J. C. C. N ITSCHE : The boundary behavior of minimal surfaces. Kellogg ’s theorem and branchpoints on the boundary.
Invent. Math. 8, 313-333 (1969).[28] J.C.C. N
ITSCHE : On the modulus of doubly connected regions under harmonic mappings ,Amer. Math. Monthly, , 781–782 (1962).[29] C H . P OMMERENKE : Boundary behaviour of conformal maps.
Grundlehren der Mathematis-chen Wissenschaften. 299. Berlin: Springer- Verlag. ix, 300 p. (1992).[30] L. T
AM AND
T. W AN : Quasiconformal harmonic diffeomorphism and universal Teichm¨ulerspace , J. Diff. Geom. (1995) 368-410.[31] L. S IMON
A H¨older estimate for quasiconformal maps between surfaces in Euclidean space .Acta Math. , 19–51 (1977).[32] S. E. W
ARSCHAWSKI : On Differentiability at the Boundary in Conformal Mapping,
Proc.Amer. Math. Soc., 12:4 (1961), 614-620.[33] S. E. W
ARSCHAWSKI : On differentiability at the boundary in conformal mapping,
Proc. Amer.Math. Soc, (1961), 614-620.[34] : On the higher derivatives at the boundary in conformal mapping,
Trans. Amer. Math.Soc. , No. 2 (1935), 310–340.[35] S. E. W ARSCHAWSKI : ¨Uber das Randverhalten der Ableitung der Abbildungsfunktion beikonformer Abbildung. Math. Z. 35, 321–456 (1932).[36] A. W
EITSMAN : Univalent harmonic mappings of annuli and a conjecture of J.C.C. Nitsche ,Israel J. Math., , (2001), 327–331.U
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