Minimisers and Kellogg’s theorem

We extend the celebrated theorem of Kellogg for conformal mappings to the minimizers of Dirichlet energy. Namely we prove that a diffeomorphic minimizer of Dirichlet energy of Sobolev mappings between doubly connected domains D and Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} having Cn,α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {C}}^{n,\alpha }$$\end{document} boundary is Cn,α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {C}}^{n,\alpha }$$\end{document} up to the boundary, provided Mod(D)⩾Mod(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\mathrm{Mod}\,}}(D)\geqslant {{\,\mathrm{Mod}\,}}(\Omega )$$\end{document}. If Mod(D)<Mod(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\mathrm{Mod}\,}}(D)< {{\,\mathrm{Mod}\,}}(\Omega )$$\end{document} and n=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=1$$\end{document} we obtain that the diffeomorphic minimizer has C1,α′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {C}}^{1,\alpha '}$$\end{document} extension up to the boundary, for α′=α/(2+α)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha '=\alpha /(2+\alpha )$$\end{document}. It is crucial that, every diffeomorphic minimizer of Dirichlet energy has a very special Hopf differential and this fact is used to prove that every diffeomorphic minimizer of Dirichlet energy can be locally lifted to a certain minimal surface near an arbitrary point inside and at the boundary. This is a complementary result of an existence results proved by Iwaniec et al. (Invent Math 186(3):667–707, 2011).


Introduction and the main results
In this paper, we consider two doubly connected domains D and in the complex plane C. The Dirichlet energy of a diffeomorphism f : D → is defined by where D f is the Hilbert-Schmidt norm of the differential matrix of f and λ is standard Lebesgue measure. The primary goal of this paper is to establish boundary regularity of a diffeomorphism f : D onto −→ of smallest (finite) Dirichlet energy, provided such an f exists and the boundary is smooth. If we denote by J (z, f ) the Jacobian of f at the point z, then (1.1) yields where | | is the measure of . In this paper we will assume that diffeomorphisms as well as Sobolev homeomorphisms are orientation preserving, so that J (z, f ) > 0. A conformal mapping of D onto would be an obvious minimizer of (1.2), becausē ∂ f = 0, provided it exists. Thus in the special case where D and are conformally equivalent the famous Kellogg theorem yields that the minimizer is as smooth as the boundary in the Hölder category. For an exact statement of the Kellogg theorem, we recall that a function ξ : D → C is said to be uniformly α−Hölder continuous and write ξ ∈ C α (D) if sup z =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 nth derivative ξ (n) ∈ C α (D). A rectifiable Jordan curve γ of the 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 1. The theorem of Kellogg (with an extension due to Warschawski, see [7,29,[33][34][35]) now states that if D and are Jordan domains having C n,α boundaries and ω is a conformal mapping of D onto , then ω ∈ C n,α .
The theorem of Kellogg and of Warshawski has been extended in various directions, see for example the work on conformal minimal parameterization of minimal surfaces by Nitsche [27] (see also the paper by Kinderlehrer [21] and by Lesley [24]), and to quasiconformal harmonic mappings with respect to the hyperbolic metric by Tam and Wan [30,Theorem 5.5.]. For some other extensions and quantitative Lipschitz constants we refer to the paper [25].
We have the following extension of the Kellogg's theorem, which is the main result of the paper. For higher-degree regularity we will prove the following result: 1). Assume that D and are two doubly connected domains in the complex plane with C n,α boundaries so that Mod(D) Mod( ). Assume that f is a diffeomorphic minimizer of energy (1.1) throughout the class of all diffeomorphisms between D and . Then f has a C n,α extension up to the boundary.
We will formulate some corollaries of Theorem 1.1 in Sect. 2, where we will describe the key point of the proof. In Sect. 3 together with the appendix below we prove that diffeomorphic minimizers are Hölder continuous at the boundary components. This is needed to prove the global Lipschitz continuity of such diffeomorphisms, which is done in Sect. 4.1. The proof of the smoothness issue is given in Sect. 4.2. Section 5 contains the proof of the main results. The last section is devoted to an open problem.
The following existence result was proved in [12]: The most important issue in proving Proposition 1.3 was to establish some key properties of Noether harmonic which we gather in the next subsections.

Noether harmonic mappings
We recall that a mapping g : D → is said to be Noether harmonic (see [8] for every family of diffeomorphisms t → φ t : → depending smoothly on the real parameter t and satisfying φ 0 = id. To be more exact, that means that the mapping ∈ is a smooth mapping for some 0 > 0. Not every Noether harmonic mapping h is a harmonic mapping, however if the mapping g is a diffeomorphism, then it is harmonic, i.e. it satisfies the equation g = 0.

Some key properties of Noether harmonic diffeomorphisms
The following properties of Noether harmonic mappings are derived in the proof of [14,Lemma 1.2.5]. Assume that g : D → is Noether harmonic. Then 1. The Hopf differential of g, defined by ϕ := g z gz, which a priori belongs to L 1 (D), is holomorphic. 2. If ∂ D is C 1,α -smooth then ϕ extends continuously to D, and the quadratic differential ϕ dz 2 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 0 < r < R < ∞, is a circular annulus centered at origin, a doubly connected domain, and g : D → is a stationary deformation, then there exists a real constant c ∈ R such that We recall that every doubly connected domain D ⊂ C 2 whose inner boundary is not just one point is conformally equivalent to such an annulus D = A(r , R We next recall that sense preserving mapping w of class ACL between two planar domains X and Y is called for almost every z ∈ X. Here K 1, K 0, J (z, w) is the Jacobian of w in z and For a related definition for mappings between surfaces the reader is referred to [31].
Noether-harmonic maps, and in particular minimizers, belong to the class of (K , K ) quasiconformal mappings, for a (K , K ) which is nicely related to the data c and Mod D: Lemma 1.5 [17] Every sense-preserving Noether harmonic map g : and c is the constant from (1.4). The result is sharp and for c = 0 the Noether harmonic map is (1, 0) quasiconformal, i.e. it is a conformal mapping. In this case is conformally equivalent to A(ρ, 1).
Assume that g : [0, ] → is the arc-length parameterization of a rectifiable Jordan curve . Here = | | is the length of . We say that a continuous mapping f : T → of the unit circle onto is monotone if there exists a monotone function φ : [0, 2π ] → [0, ] such that f (e it ) = g(φ(s)). In a similar way we define a monotone function between ρT := {z : |z| = ρ} and . In view of [22,Proposition 5] and Proposition 3.1 below we can formulate the following simple lemma.

Lemma 1.6
Assume that f is a diffeomorphic minimizer of Dirichlet energy between the annuli A(ρ, 1) and the doubly connected domain , which is bounded by the outer boundary and inner boundary 1 . Then f has a continuous extension to the boundary and the boundary mapping is monotone in both boundary curves.

Some corollaries and the strategy of the proof
is a conformal parametrisation of a minimal surface , whose boundary is in C 1,α , where α = α, if c < 0 and α = α/(2 + α). If c 0, then the surface is a doubly connected catenoidal minimal surface, whose conformal modulus is equal to Mod . If c > 0, then the minimal surface is a helicoidal minimal surface.
The minimizer of Dirichlet energy is not always a diffeomorphism when Mod(D) Mod( ). Moreover it fails to be smooth in the domain if the boundary is not smooth [3]. For more general setting we refer to [10].

Remark 2.3
By using Lemma 1.5, the first author in [17] proved that, a minimizer of −energy between doubly connected domains having C 2 boundary is Lipschitz continuous. The −energy, is a certain generalization of Euclidean energy, and we will omit details in this paper.

Minimizing mappings and minimal surfaces
Since D is conformally equivalent to A ρ = {z : ρ < |z| < 1}, for some ρ ∈ (0, 1), we can assume that D = A ρ . Namely, by a Kellogg's type result of Jost [15], a conformal biholomorphism of a domain D with C n,α boundary onto A ρ is C n,α continuous up to the boundary together 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 of Lemma 1.6, enlarging T p if necessary, we can assume that p := f (T p ) is a Jordan arc containing q = f ( p) in its interior in ∂ D. Moreover we can assume that A p has a C ∞ boundary. Assume now that p is a conformal mapping of the unit disk D onto A p so that p ( p/| p|) = p. Moreover, if p = p, but | p| = |p | we can chose domains A p to be just rotation of A p . So all those domains A p are isometric to A 1 or A ρ . Moreover we also can assume that p = e iς p . Then f p = f • p has the representation where g(z) = g p (z) and h(z) = h p (z) are holomorphic mappings defined on the unit disk. Moreover f p is a sense preserving diffeomorphism and this means that It follows from (2.2) and (2.1) that Then it defines locally the minimal surface by its conformal minimal coordinates, , and this is crucial for our approach: This can be written Thus the Weierstrass-Enneper parameters are The first fundamental form is given by Here Then as in [5, Chapter 10], we get Let us note the following important fact, the boundary curve of the minimal surface defined in (2.4), (2.5) and (2.6) is p ∈ ∂A ρ . Its trace is not smooth in general. However the trace of curve is smooth as well as the function k 3 is smooth in a small neighborhood of p. This will be crucial in proving our main results.
We will prove certain boundary behaviors of f near the boundary by using the representation (2.1), and this is why we do not need global representation. The idea is 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 conformal mapping 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 be lifted locally to a certain minimal surface. In the following part we show that in certain circumstances the lifting is global.
Every harmonic mapping f defined on the annulus A ρ can be expressed (see e.g. [2, Theorem 9.1.7]) as Assume now that f is a diffeomorphic minimizer between A ρ and and that c < 0, i.e. Mod(A ρ ) > Mod( ) (see Proposition 1.4). Then we get the following conformal parameterization of a catenoidal minimal surface , ϕ : A ρ → , defined by If a 0 = 0, then we have the following decomposition f (z) Then we get the following conformal parameterization of a minimal surface , ϕ : The following corollary is a consequence of Theorem 1.1 and (2.11).

Corollary 2.4 Assume that f : A ρ → is a diffeomorphic minimizer of Dirichlet energy, with Mod A ρ
Mod and ∂ ∈ C 1,α . Then f can be lifted to a smooth doubly connected minimal surface with C 1,α boundary, and the lifting is conformal and harmonic.
Let us continue this subsection with the following explicit example. Let Then f (z) is a harmonic mapping of the annulus A r onto A R that minimizes the Dirichlet energy [1]. Further, under notation of this subsection we have Put Then we have from (2.12) that Here 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 1 (z), ϕ 2 (z), ϕ 3 (z)). This minimal graph is a part of the lower slab of catenoid (see Fig. 1).

Remark 2.5
It follows from (2.13) that, For c > 0, i.e. for Mod(A ρ ) < Mod( ) we get the following counterpart. Then we get the following conformal parametrization of a helicoidal minimal surface , ϕ : A ρ → , defined by If f has not the logarithmic part, then we get the parametrization of a minimal surface In particular, if A ρ = A r and = A R , so that R < r , then In particular, if r = 2/3 and R = 1/3 then this minimal surface over the annulus A r is shown in Fig. 2. We finish this section with a lemma needed in the sequel Lemma 2.6 Let p ∈ T = ∂D.
(a) Assume that is a holomorphic mapping of the unit disk into itself so that ( p) = p and has the derivative at p. Then (b) Assume that is a holomorphic mapping of the unit disk into the exterior of the disk r D with ( p) = r p. Then To prove Lemma 2.6 we recall the boundary Schwarz lemma [6] which states the following.
Proof of Lemma 2.6 Assume that p = 1. Otherwise consider the function 1 Then Since F(0) = 0, F(1) = 1, it follows that F satisfies the boundary Schwarz lemma, and therefore F (1) is a real positive number bigger or equal to 1. This implies a). In order to prove b), consider the auxiliary function g(z) = r (z) . By applying a) to g we get This finishes the proof.

Hölder property of minimizers
In this section we prove that the minimizers of the energy are global Hölder continuous provided that the boundary is C 1 . We first formulate the following result Let ∈ C 1,μ , 0 < μ 1, be a Jordan curve and let g be the arc length parameterization of and let l = | | be the length of . Let d be the distance between g(s) and g(t) along the curve , i.e.
for z 1 , z 2 ∈ T and z 1 , See appendix below for the proof of Lemma 3.2. We now can state the following proposition:

Proof of Theorem 1.1
By repeating the proofs of corresponding result in [27] we can formulate the following result.

Proposition 4.1 Assume that is a Jordan curve in R 3 and assume that X
Assume that X is Hölder continuous in an arc T p ⊂ T containing p in its interior. If the arc T p of T is mapped onto the arc p ⊂ so that p ∈ C 1,α , 0 < α < 1, then X is 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 statement in [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 stated as 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 conformal mapping, in view of the original Kellogg's theorem [7], we can assume that the domain is On the other hand, the minimising harmonic mapping has the local representation (2.4). Here 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). Because 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 p ⊂ ∂ . Therefore by Theorem 4.1, having in mind the notation from Sect. 2.1, the mapping is C 1,α in a neighborhood of p, provided the boundary arc is of the same class. But we do not know that X (T p ) ∈ C 1,α . We only know that p is a priori in C ∞ (D) and p = f p (T p ) ∈ C 1,α . This will be enough for the proof.

Proof of Lipschitz continuity
We will prove the following lemma needed in the sequel.
Proof We use the notation from Sect. 2.1. The constant C that appear in the proof is not the same and its value can vary from one to the another appearance. Assume also q ∈ = ∂ , and, by using a rotation and a translation (if it is necessary) we can assume that q = 0, and the tangent line of at q is the real axis. Post-composing by a such Euclidean isometry, the Euclidean harmonicity is preserved. Then in a small neighborhood of q, has the following parameterization γ ( Assume also that, p = 1 and f (1) = q = 0. And assume that for a small angle = p = {e iθ : |θ | } we have f ( ) ⊂ γ (−x 0 , x 0 ). We can assume also that x 0 is a small enough positive constant global for all points q ∈ ∂ . We want to localize the problem. We only need to prove that f is C 1,α in a small neighborhood of 1. We also work with f p = f • p : D → f (A p ) instead of f , where p ( p/| p|) = p, and assume that γ (−x 0 , x 0 ) ⊂ ∂ A p for every p ∈ ∂A ρ . We will from time to time use notation f instead of f p , since they behave in the same way in a small neighborhood of p, because p is a priori in C ∞ Thus, there exists a function x : → R so that ).
We will also from time to time use notation Because p ∈ C 1,α we have as in [27, eq. 3], the following relation The constant C and t 0 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 Now, the following sequence of the inequalities follow from (4.2), (4.4) and Lemma 3.2. and and so Here where L is defined in Lemma 3.2.
In order to continue we collect some results from [27] and [7]. 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 the unit disk, so that |F| M in D.
Further assume that for a constants 0 δ, 0 η, μ π/2 so that for almost every −δ t, s δ we have Then for ζ = τ e is , with |s| δ/2, 1/2 τ 1 we have the estimates and (4.10) and

Assume that f is a holomorphic mapping defined in the unit disk so that
where 0 < μ < 1 and z ∈ D δ . Then the radial limit exists for every θ ∈ (−δ + s 0 , δ + s 0 ) and we have there the inequality where N depends on M and μ. The converse is also true. μ ∈ (0, 1). Assume that f is continuous harmonic mapping on the closed unit disk and satisfies on a small arc = {e iθ : |θ − s 0 | < δ} the condition:

for almost every points s and t. Then f satisfies the Hölder condition
We now reformulate a result of Privalov [7,p. 414,Theorem 5] in its local form (w.r.t. the boundary). μ ∈ (0, 1). Assume that f = u+iv is a holomorphic bounded function defined on the unit disk D and assume that u satisfies the condition |u(e it ) − u(e is )| M|e it − e is | μ , for almost every s and t so that |s − s 0 | < δ and |t − s 0 | < δ. Then there is a constant N depending on M and μ so that
Repeating the proof of the preceding lemma, we also obtain the following pointwise statement.
Then from (4.14) and Lemma 4.4 we get that F p is C 0,(1+α)β in D p, . Let Then we also have (4.17) Since the right hand side of (4.17) is bounded, it follows that G p is (1 + α)β Hölder continuous. Namely
Since f ∈ C ∞ (A ρ ) we get f ∈ C 0,1 (A ρ ) as claimed, and thus the proof of Lemma 4.2 is finished.

The minimizer is C 1,˛ up to the boundary
We continue to use the notation from Sects. 2.1 and 4.1. The constant C that appear in the proof is not the same and its value can vary from one to the another appearance, but it is global and the same for all points of ∂A ρ . Assume that f = u + iv : A ρ → is a diffeomorphic minimizer, where A ρ = {z : ρ < |z| < 1}, we need to show that is C 1,α (A ρ ), provided that ∂ ∈ C 1,α . We only need to prove that f is C 1,α in small neighborhood of p ∈ ∂A ρ . We also work with f p = f • p : D → f (A p ) instead of f , where p (1) = 1 as in the previous part of the paper. We will show that f p ∈ C 1,α (D p ), where D p = {z = re is+is 0 : 1/2 r < 1, s ∈ (− , )}, where p/| p| = e is 0 , and > 0 is a small enough positive constant valid for all points p ∈ ∂A ρ .
Then as in [27] we get Recall now that Then from (4.13) and (4.16) we get and for z ∈ A p ∩ A 1 . Therefore from (4.29), (4.30) and (4.39), in a small −neighborhood of p = 1, we get the inequalities for almost every t ∈ (− , ). Further as in [27, p. 325-326] we obtain that and almost every t, s ∈ (− , ). The function k 3 has the same behavior a priori. From this it follows that This concludes the case c 0. Now we continue to prove the case c 0. This case we use of the parameterization By using this we get that ϕ(A ρ ) = is a doubly connected minimal surface bounded by two Jordan curves where 1 and 2 are the inner and outer boundaries of . Moreover ∂ ∈ C 1,α if and only if ∂ ∈ C 1,α . This time Proposition 4.1 will imply the result. From Proposition 4.1 we obtain that f ∈ C 1,α ( −1 p (A p )), where A p is a small neighborhood of a fixed point p. Notice that A p = p/| p|A 1 or A p = p/| p|A ρ , where A 1 and A ρ are fixed domains, whose boundary contains a Jordan arc , whose interior contains 1 respectively ρ. Since A ρ = −1 p (A p ), we get that f ∈ C 1,α (A ρ ) as claimed.
Thus we have finished the proof of Theorem 1.1.

Proof of Theorem 1.2
Having proved Theorem 1.1 this proof is a simple matter. We can reformulate a similar statement to Proposition 4.1, which is valid for higher-differentiability of the mapping, and then use the harmonic mapping ϕ defined (4.45), in order to get that ϕ ∈ C n,α (A ρ ), provided that ∂ ∈ C n,α (A ρ ), which is equivalent with the condition ∂ ∈ C n,α (A ρ ).

Concluding remark
We expect that the following statement is true: 1 Assume that f : D → is a energy minimal diffeomorphism of the energy between two domains with C 1,α boundaries. If Mod(D) Mod( ) then the diffeomorphic minimizer of Dirichlet energy, which is shown to have a C 1,α extension up to the boundary is diffeomorphic on the boundary also and the extension is C 1,α .
This conjecture is motivated by the existing result described in Proposition 1.3 and the example presented in (2.13) of the unique minimizer (up to the rotation) of Dirichlet energy between annuli A r and A R , that maps the outer boundary onto the outer boundary (see [1] for details). The mapping is a diffeomorphism between A r and A R , provided that R < 2r 1 + r 2 . (6.1) If R = 2r 1+r 2 , and 0 < r < 1, then the mapping w(z) = r 2 + |z| 2 z(1 + r 2 ) is a harmonic minimizer (see [1]) of the Euclidean energy of mappings between A(r , 1) and A( 2r 1+r 2 , 1), however |w z | = |wz| = 1 1 + r 2 for |z| = r , and so w is not bi-Lipschitz.
Note that (6.1) is satisfied provided that Mod A r Mod A R . The inequality (6.1) (with instead of <) is necessary and sufficient for the existence of a harmonic diffeomorphism between A r and A R a conjecture raised by Nitsche in [28] and proved by Iwaniec, Kovalev and Onninen in [11], after some partial results given by Lyzzaik [26], Weitsman [36] and Kalaj [19]. If R > 2r 1 + r 2 , then the minimizer of Dirichlet energy throughout the deformations D(A r , A R ) is not a diffeomorphism (see [1] and [3, Example 1.2]).
We want to refer to one more interesting behavior that minimizers of energy share with conformal mappings. Namely, if f is a diffeomorphic minimizer of Dirichlet energy between the domains A ρ and ( , 1 ) so that and 1 are convex, then f (tT) is convex for t ∈ (ρ, 1) [22]. Further if and 1 are circles, then f (tT) is a circle [23].

2)
where l τ = | f (k τ )| denote the length of f (k τ ) and A(r ) is the area of f ( r ).
In order to deal with the inner boundary, we take the composition which maps the annulus A ρ into = {1/(z − a) : z ∈ }. Here a is a point inside the inner Jordan curve. Then = ( , 1 ) is a doubly connected domain with C 1,α boundary. Now we construct a conformal mapping 1 between the domain ( ) and the unit disk and repeat the previous case in order to get that the inequality (3.3) does hold in both boundary components.