Sobolev homeomorphic extensions

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ Sobolev homeomorphic extensions © 2021 European Mathematical Society Published version Koski, Aleksis; Onninen, Jani Koski, A., & Onninen, J. (2021). Sobolev homeomorphic extensions. Journal of the European Mathematical Society, 23(12), 4065-4089. https://doi.org/10.4171/JEMS/1099 2021 © 2021 European Mathematical Society Published by EMS Press. This work is licensed under a CC BY 4.0 license. J. Eur. Math. Soc. 23, 4065–4089 (2021) DOI 10.4171/JEMS/1099 Aleksis Koski Jani Onninen Sobolev homeomorphic extensions Received December 4, 2018; revised December 17, 2019 Abstract. Let Xand Ybe `-connected Jordan domains, `2N, with rectifiable boundaries in the complex plane. We prove that any boundary homeomorphism 'W@Xonto ! @Yadmits a Sobolev homeomorphic extension hWXonto !Yin W1;1.X;C/. If instead Xhas s-hyperbolic growth with s > p1, we show the existence of such an extension in the Sobolev class W1;p.X;C/for p2.1; 2/. Our examples show that the assumptions of rectifiable boundary and hyperbolic growth cannot be relaxed. We also consider the existence of W1;2-homeomorphic extensions with given boundary data. Keywords. Sobolev homeomorphisms, Sobolev extensions, Douglas condition 1. Introduction Throughout this text Xand Yare `-connected Jordan domains, `D1; 2; : : :, in the complex plane C. Their boundaries @Xand @Yare thus disjoint unions of `simple closed curves or points. If `D1, these domains are simply connected and will just be called Jordan domains. In the simply connected case, the Jordan–Schönflies theorem states that every homeomorphism 'W@Xonto ! @Yadmits a continuous extension hWX!Ywhich takes Xhomeomorphically onto Y. In the first part of this paper we focus on a Sobolev variant of the Jordan–Schönflies theorem. The most pressing demand for studying such variants comes from the variational approach to geometric function theory [3,19,33] and nonlinear elasticity [2,5,8]. Both theories share the ideas associated to determining the infimum of a given energy functional EXŒh DZX E.x; h; Dh/ dx(1.1) Aleksis Koski: Department of Mathematics and Statistics, P.O. Box 35 (MaD), FI-40014 University of Jyväskylä, Finland; aleksis.k[email protected] Jani Onninen: Department of Mathematics, Syracuse University, Syracuse, NY 13244, USA, and Department of Mathematics and Statistics, P.O. Box 35 (MaD), FI-40014 University of Jyväskylä, Finland; [email protected] Mathematics Subject Classification (2020): Primary 46E35, 58E20 A. Koski, J. Onninen 4066 among orientation preserving homeomorphisms hWXonto ! Yin the Sobolev space W1;p.X;Y/with given boundary data 'W@Xonto ! @Y. We denote that class of mappings by H1;p '.X;Y/. Naturally, a fundamental question is whether the class H1;p '.X;Y/is non-empty. Question 1.1. Under what conditions does a boundary homeomorphism 'W@Xonto ! @Y admit a homeomorphic extension hWXonto ! Yof Sobolev class W1;p.X;C/? A necessary condition is that 'is the Sobolev trace of some (possibly nonhomeomorphic) mapping in W1;p.X;C/. Hence to solve Question 1.1 one could first study the following natural subquestion: Question 1.2. Suppose that a homeomorphism 'W@X!@Yadmits a W1;p-extension to X. Does it then follow that 'also admits a homeomorphic W1;p-extension? Our main results, Theorem 1.8 and its multiply connected variant (Theorem 1.11), give an answer to these questions when p2Œ1; 2/. The construction of such extensions is important not only to ensure the well-posedness of the related variational questions, but also for example due to the fact that various types of extensions were used to provide approximation results for Sobolev homeomorphisms [16,18]. We touch upon variational topics in Section 7, where we provide an application of one of our results. Apart from Theorem 1.11 and its proof (§6), the rest of the paper deals with the simply connected case. Let us start by considering the above questions in the well-studied setting of the Dirichlet energy, corresponding to pD2above. The Radó [32], Kneser [26] and Choquet [7] theorem asserts that if YR2is a convex domain then the harmonic extension of a homeomorphism 'W@X!@Yis a univalent map from Xonto Y. Moreover, by a theorem of Lewy [29], this univalent harmonic map has a nonvanishing Jacobian and is therefore a real analytic diffeomorphism in X. However, such an extension is not guaranteed to have finite Dirichlet energy in X. The class of boundary functions which admit a harmonic extension with finite Dirichlet energy was characterized by Douglas [9]. The Douglas condition for a function 'W@Donto ! @Yreads Z@DZ@Dˇˇˇˇ './ './ ˇˇˇˇ 2 jdjjdj<1:(1.2) The mappings satisfying this condition are exactly the ones that admit an extension with finite W1;2-norm. Among these extensions is the harmonic extension of ', known to have the smallest Dirichlet energy. Note that the Dirichlet energy is also invariant with respect to a conformal change of variables in the domain X. Therefore thanks to the Riemann Mapping Theorem, when considering Question 1.1 in the case pD2, we may assume that XDDwithout loss of generality. Now, there is no problem to answer Question 1.1 when pD2and Yis Lipschitz. Indeed, for any Lipschitz domain there exists a global bi-Lipschitz change of variables ˆWC!Cfor which ˆ.Y/is the unit disk. Since the finiteness of the Dirichlet energy is preserved under a bi-Lipschitz change of variables in the target, we may reduce Sobolev homeomorphic extensions 4067 Question 1.1 to the case when XDYDD, for which the Radó–Kneser–Choquet theorem and the Douglas condition provide an answer. In other words, if Yis Lipschitz then the following are equivalent for a boundary homeomorphism 'W@D!@Y: (1) 'admits a W1;2-Sobolev homeomorphic extension hWDonto ! Y. (2) 'admits W1;2-Sobolev extension to D. (3) 'satisfies the Douglas condition (1.2). In the case when 1p < 2, the problem is not invariant under a conformal change of variables in X. However, when Xis the unit disk and Yis a convex domain, a complete answer to Question 1.1 was provided by the following result of Verchota [38]. Proposition 1.3. Let Ybe a convex domain, and let 'W@Donto ! @Ybe any homeomorphism. Then the harmonic extension of 'lies in the Sobolev class W1;p.D;C/for all 1p < 2. This result was further generalized in [15,20,24]. The case p > 2 will be discussed in Subsection 2.3. Our main purpose is to provide a general study of Question 1.1 in the case when 1p < 2. Considering now the endpoint case pD 1, we find that Question 1.1 is equivalent to the question of finding a homeomorphic Lipschitz map extending the given boundary data '. In this case the Kirszbraun extension theorem [25] shows that a boundary map 'W@Donto ! @Yadmits a Lipschitz extension if and only if 'is a Lipschitz map itself. When Xis the unit disk, a positive answer to Question 1.2 is given by the following recent result by Kovalev [28]. Theorem 1.4 (pD 1). Let 'W@D!Cbe a Lipschitz embedding. Then 'admits a homeomorphic Lipschitz extension to the whole plane C. Let us return to the case of the Dirichlet energy (see (1)–(3) above). The equivalence of a W1;2-Sobolev extension and a W1;2-Sobolev homeomorphic extension for non-Lipschitz targets is a more subtle question. In this perspective, a slightly more general class of domains is the class of inner chordarc domains studied in geometric function theory [17,31,35–37]. By definition [36], a Jordan domain Ywith rectifiable boundary is inner chordarc if there exists a constant Csuch that for every pair of points y1; y22@Y one has jy1y2jCY.y1;y2/, where Y.y1;y2/denotes the infimal length of curves contained in Ywith endpoints y1and y2. For example, an inner chordarc domain may have inward cusps on the boundary, as opposed to Lipschitz domains. According to a result of Väisälä [36], the inner chordarc condition is equivalent to the requirement that there exists a homeomorphism ‰WYonto ! D,C1-diffeomorphic in Y, such that the norms of both the gradient matrices D‰ and .D‰/1are bounded from above. Surprisingly, the following example shows that, unlike for Lipschitz targets, the answer to Question 1.2 for pD2is in general negative when the target is only inner chordarc. A. Koski, J. Onninen 4068 Example 1.5. There is an inner chordarc domain Yand a homeomorphism 'W@Donto !@Y satisfying the Douglas condition (1.2) which does not admit a homeomorphic extension hWDonto ! Yin W1;2.D;Y/. In [4], as a part of studies of mappings with smallest mean distortion, it was proved that for C1-smooth Ythe Douglas condition (1.2) can be equivalently formulated in terms of the inverse mapping '1W@Yonto ! @D: Z@YZ@Yˇˇlog j'1./ '1./jˇˇjdjjdj<1:(1.3) It was recently shown that for inner chordarc targets this condition is necessary and sufficient for 'to admit a W1;2-homeomorphic extension [27]. We extend this result both to cover rectifiable targets and to give a global homeomorphic extension as follows. Theorem 1.6 (pD2). Let Ybe a Jordan domain with @Yrectifiable. Every 'W@Donto ! @Ysatisfying (1.3)admits a homeomorphic extension hWC!Cof class W1;2 loc .C;C/. Without the rectifiability of @Y, Question 1.2 will in general have a negative answer for all p2. This follows from the following example of Zhang [40]. Example 1.7. There exists a Jordan domain Yand a homeomorphism 'W@Donto ! @Y which has a W1;2-Sobolev extension to Dbut has no homeomorphic extension in the class W1;1.D;C/. We now return to the case when 1p < 2. In this case it is natural to ask under which conditions on the domains Xand Y, any homeomorphism 'W@Xonto ! @Yadmits aW1;p-Sobolev homeomorphic extension. Proposition 1.3 already implies that this is the case for XDDand Yconvex. Example 1.7, however, implies that this result does not hold in general for nonrectifiable targets Y. A general characterization is provided by the following theorem. Theorem 1.8 (1p < 2). Let Xand Ybe Jordan domains in the plane with @Yrectifiable. Let 'W@Xonto ! @Ybe a homeomorphism. Then there is a homeomorphic extension hWXonto ! Ysuch that (1) h2W1;1.X;C/, provided @Xis rectifiable, and (2) h2W1;p.X;C/for 1<p<2, provided Xhas s-hyperbolic growth with s > p 1. Definition 1.9. Let Xbe a domain in the plane. Choose a point x02X. We say that X has s-hyperbolic growth,s2.0; 1/, if hX.x0; x/ Cdist.x0; @X/ dist.x; @X/1s for all x2X:(1.4) Here hXstands for the quasihyperbolic metric on Xand dist.x; @X/is the Euclidean distance from xto the boundary. The constant Cis allowed to depend on s; x0;and the domain Xbut not on the point x. Sobolev homeomorphic extensions 4069 It is easily verified that this definition does not depend on the choice of x0. Recall that if is a domain, the quasihyperbolic metric his defined by [13] h.x1; x2/Dinf 2Z 1 dist.x; @X/jdxj; x1; x22; (1.5) where is the family of all rectifiable curves in joining x1and x2. Definition 1.9 is motivated by the following example. For s2.0; 1/ we consider the Jordan domain Xswhose boundary is given by the curve sD ¹.x; y/ 2CW 1x1; y D jxjsº[¹z2CW jzij D 1; Im.z/ 1º: Fig. 1. The Jordan domain Xs. In particular, the boundary of Xsis locally Lipschitz except at the origin. Near the origin the boundary of Xsbehaves like the graph of the function jxjs. Then one can verify that the boundary of Xshas t-hyperbolic growth for every ts. Note that the smaller the number s, the sharper the cusp is. The results of Theorem 1.8 are sharp, as described by the following result. Theorem 1.10. (1) There exists a Jordan domain Xwith nonrectifiable boundary and a homeomorphism 'W@X!@Dsuch that 'does not admit a continuous extension to Xin the Sobolev class W1;1.X;C/. (2) For every p2.1; 2/ there exists a Jordan domain Xwhich has s-hyperbolic growth, with p1Ds, and a homeomorphism 'W@X!@Dsuch that 'does not admit a continuous extension to Xin the Sobolev class W1;p.X;C/. A. Koski, J. Onninen 4070 To conclude, as promised earlier, we extend our main result to the case where the domains are not simply connected. The following generalization of Theorem 1.8 holds. Theorem 1.11. Let Xand Ybe multiply connected Jordan domains with @Yrectifiable. Let 'W@Xonto ! @Ybe a homeomorphism which maps the outer boundary component of Xto the outer boundary component of Y. Then there is a homeomorphic extension hWXonto ! Ysuch that (1) h2W1;1.X;C/, provided @Xis rectifiable, and (2) h2W1;p.X;C/for 1<p<2, provided Xhas s-hyperbolic growth with s > p 1. 2. Preliminaries 2.1. The Dirichlet problem Let be a bounded domain in the complex plane. A function uW!Rin the Sobolev class W1;p loc ./,1 < p < 1, is called p-harmonic if div jrujp2ruD0: (2.1) We call 2-harmonic functions simply harmonic. There are two formulations of the Dirichlet boundary value problem for the p-harmonic equation (2.1). We first consider the variational formulation. Lemma 2.1. Let uı2W1;p./ be a given Dirichlet data. There exists precisely one function u2uıCW1;p ı./ which minimizes the p-harmonic energy: ZjrujpDinf ²ZjrwjpWw2uıCW1;p ı./³: Here W1;p ı./ denotes the completion of compactly supported smooth functions in  with respect to the W1;p./ Sobolev norm. The variational formulation coincides with the classical formulation of the Dirichlet problem. Lemma 2.2. Let Cbe a bounded Jordan domain and uı2W1;p./ \C./. Then there exists a unique p-harmonic function u2W1;p./ \C./ such that uj@ Duıj@. For the proofs of these facts we refer to [18]. 2.2. The Radó–Kneser–Choquet Theorem Lemma 2.3. Consider a Jordan domain XCand a bounded convex domain YC. Let hW@Xonto !@Ybe a homeomorphism and HWX!Cits harmonic extension. Then H is a C1-diffeomorphism of Xonto Y. For the proof of this lemma we refer to [11,21]. The following p-harmonic analogue of the Radó–Kneser–Choquet Theorem is due to Alessandrini and Sigalotti [1] (see also [22]). Sobolev homeomorphic extensions 4071 Proposition 2.4. Let Xbe a Jordan domain in C,1 < p < 1, and hDuCiv WX!C a continuous mapping whose coordinate functions are p-harmonic. Suppose that Yis convex and hW@Xonto ! @Yis a homeomorphism. Then his a diffeomorphism from X onto Y. 2.3. Sobolev homeomorphic extensions onto a Lipschitz target Combining the results in this section allows us to easily solve Question 1.2 for convex targets. Proposition 2.5. Let Xand Ybe Jordan domains in the plane with Yconvex, and let 1 < p < 1. Suppose that 'W@Xonto !@Yis a homeomorphism. Then there exists a continuous gWX!Cin W1;p.X;C/such that g.x/ D'.x/ on @Xif and only if there exists a homeomorphism hWX!Yin W1;p.X;C/such that h.x/ D'.x/ on @X. Proof. The “if” part is immediate. For the “only if” part we write gDuıCivı2 W1;p.X;C/\C.X;C/and consider the unique p-harmonic functions uand vwhich coincide with uıDRe 'and vıDIm 'respectively on @X. First, these classical solutions agree with the variational ones (see Lemmas 2.1 and 2.2). In particular, we have ZXjrujpZXjruıjpand ZXjrvjpZXjrvıjp: Second, according to Proposition 2.4 the mapping h2W1;p.X;C/is a homeomorphism. Now, replacing the convex Yby a Lipschitz domain offers no challenge. Indeed, this follows from a global bi-Lipschitz change of variables ˆWC!Cfor which ˆ.Y/is the unit disk. If the domain in Proposition 2.5 is the unit disk D, then the existence of a finite p-harmonic extension can be characterized in terms of a Douglas type condition. If 1 < p < 2, then such an extension exists for an arbitrary boundary homeomorphism (Proposition 1.3) and if 2p < 1the extension exists if and only the boundary homeomorphism 'W@Donto ! @Ysatisfies Z@DZ@Dˇˇˇˇ './ './ ˇˇˇˇ p jdjjdj<1:(2.2) For the proof of this last fact we refer to [34, pp. 151–152]. 2.4. Carleson measures and the Hardy space Hp Roughly speaking, a Carleson measure on a domain Gis a measure that is bounded from above by the Hausdorff 1-measure on @Gnear the boundary of G. We will need the notion of a Carleson measure only on the unit disk D. A. Koski, J. Onninen 4072 Definition 2.6. Let be a Borel measure on D. Then is a Carleson measure if there is a constant C > 0 such that .S.// C for every  > 0. Here S./ D ¹rei˛ W1 < r < 1;   < ˛ <  Cº: Carleson measures have many applications in harmonic analysis. A celebrated result by L. Carleson [6] (see also [10, Theorem 9.3]) tells us that a Borel measure on Dis a bounded Carleson measure if and only if the injective mapping from the Hardy space Hp.D/into the measurable space Lp .D/is bounded. Proposition 2.7. Let be a Borel measure on the unit disk D. Let 0 < p < 1. Then there exists a constant C > 0 such that ZDjf .z/jpd.z/1=p CkfkHp.D/for all f2Hp.D/ if and only if is a Carleson measure. Recall that the Hardy space Hp.D/,0 < p < 1, is the class of holomorphic functions fon the unit disk satisfying kfkHp.D/WD sup 0r<11 2 Z2 0jf .rei /jpd1=p <1: Note that kkHp.D/is a norm when p1, but not when 0<p<1. 3. Sobolev integrability of the harmonic extension At the end of this section we prove our main result in the simply connected case, Theorem 1.8. The proof will be based on a suitable reduction of the target domain to the unit disk, and the following auxiliary result which concerns the regularity of harmonic extensions. Theorem 3.1. Let Xbe a Jordan domain and 'W@X!@Dan arbitrary homeomorphism. Let hdenote the harmonic extension of 'to X, which is a homeomorphism from Xto D. Then the following hold. (1) If the boundary of Xis rectifiable, then h2W1;1.X;C/. (2) If Xhas s-hyperbolic growth, then h2W1;p.X;C/for p < s C1. This theorem will be a direct corollary of the following theorem and the two propositions after it. Sobolev homeomorphic extensions 4079 Fig. 4. The portions of height kget mapped onto slices with side length dk. and  k, this gives the estimate ZSkjDHjpdz .RSkjDHjdz/p jSkjp1c.Rk 0dkdy/p p1 k.P1 jDkj/.p1/=s Dcdp kk P1 jDkj : Now by our choice of kDk2=10, we see that P1 jDkjis comparable to 1=k, so by adding up we obtain the estimate ZSkSkjDHjpdz c 1 X kD1 dp k k:(4.1) However, our choice of dkD.log.100 Ck//1=p ensures that the right hand side of (4.1) diverges. It follows that Hcannot lie in W1;p.Xs;C/, which completes the proof. 5. The case pD2 In this section we address Theorem 1.6 as well as Examples 1.5 and 1.7. Example 1.5.For this example, let first ˆfor any 2.0; 1 denote the conformal map ˆ.z/ Dlog1z 3 defined on the unit disk and having target YWD ˆ.D/. In fact, Yis a domain with smooth boundary apart from one point at which it has an outer cusp of degree =.1 C/ (i.e. it is bi-Lipschitz equivalent to the domain X=.1C/ pictured in Figure 1). A. Koski, J. Onninen 4080 Since ˆis conformal and maps the unit disk into a set of finite measure, it lies in the Sobolev space W1;2.D;C/. However, it does not admit a homeomorphic extension to the whole plane in the Sobolev class W1;2 loc .C;C/. The reason is a modulus of continuity estimate for any homeomorphism in W1;2 loc .C;C/. Indeed, let !z.t/ denote the modulus of continuity of gWC!Cat a point z, !z.t/ Dosc B.z;t/ gDsup¹jg.x1/g.x2/jW x1; x22B.z; t/º: If gis a homeomorphism in W1;2 loc .C;C/, then Zr 0 !z.t/2 tdt < 1:(5.1) Proof of (5.1).Since gis a homeomorphism, we have osc B.z;t/ gosc @B.z;t/ g: According to Sobolev’s inequality on spheres, for almost every t > 0 we obtain osc @B.z;t/ g[email protected];t/jDgj: These together with Hölder’s inequality imply !z.t/ Dosc B.z;t/ gosc @B.z;t/ gC[email protected];t/jDgj21=2 ; and therefore for almost every t > 0 we have !z.t/2 t[email protected];t/jDgj2; where Cis independent of z. Integrating this from 0to r > 0 yields the claim (5.1). Now, since the map ˆfor 1does not satisfy the modulus of continuity estimate (5.1) at the boundary point zD1, it is not possible to extend ˆeven locally as a W1;2-homeomorphism around the point zD1. To address the exact claim of Example 1.5, we now define an embedding 'W@D!C as follows. Fixing 2.0;1, in the set ¹[email protected]/ 0ºwe let '.z/ Dˆ.z/. We also map the complementary set ¹[email protected]/ < 0ºsmoothly into the complement of Y, and in such a way that '.@D/becomes the boundary of a Jordan domain Q Y(see Figure 5). It is now easy to see that 'satisfies the Douglas condition (1.2). Indeed, since ˆis in the Sobolev space W1;2.D;C/, its restriction to the boundary must necessarily satisfy the Douglas condition. Since 'agrees with this boundary map in a neighborhood of zD1, verifying the finiteness of the integral in (1.2) poses no difficulty in this neighborhood. On the rest of @Dwe may choose 'to be locally Lipschitz, which shows that (1.2) is Sobolev homeomorphic extensions 4081 Fig. 5. The Jordan domains Yand Q Y. necessarily satisfied for '. Hence we have found a map from @Dinto the boundary of the chordarc domain Q Ywhich admits a W1;2-extension to Dbut not a homeomorphic one. Example 1.7.In [40], Zhang constructed an example of a Jordan domain, which we shall denote by Y, such that the conformal map gWD!Ydoes not admit a W1;1homeomorphic extension to the whole plane. We shall not repeat this construction here, but will instead briefly show how it relates to our questions. The domain Yis constructed in such a way that there is a boundary arc @Yover which one cannot extend the conformal map geven locally as a W1;1-homeomorphism. The complementary part of the boundary, @Yn, is piecewise linear. Hence we may employ the same argument as in the previous example. We choose a Jordan domain Q Yin the complement of Ywhose boundary consists of the arc and, say, a piecewise linear curve. We then define a boundary map 'W@D!@Q Yso that it agrees with gin a neighborhood of the set g1./ and is locally Lipschitz everywhere else. With the same argument as before, this boundary map must satisfy the Douglas condition (1.2). Hence this boundary map admits a W1;2-extension to Dbut not even a W1;1-homeomorphic extension. Naturally the boundary of the domain Q Yis quite ill-behaved, in particular nonrectifiable (though its Hausdorff dimension is still 1). Proof of Theorem 1.6.Let W@D!@Ydenote a constant speed parametrization of the rectifiable curve @Y. Let GWC!Cbe the homeomorphic Lipschitz extension of given by Theorem 1.4. Denoting fWD '1ı, we find by a change of variables that Z@YZ@Yˇˇlog j'1./ '1./jˇˇjdjjdj D Z@DZ@Dˇˇlog jf .z/ f .!/jˇˇjdzjjd!j: Now the result of Astala, Iwaniec, Martin and Onninen [4, Theorems 11.4 and 9.1] shows that the inverse map f1W@D!@Dsatisfies the Douglas condition (1.2). Thus f1 extends to a harmonic W1;2-homeomorphism H1from Dto Dby the RKC Theorem (Lemma 2.3). Letting hWD GıH1, we find that hlies in W1;2.D;C/since Gis Lipschitz. Moreover, the boundary values of hare equal to ı.'1ı/1D', giving us a homeomorphic extension of 'in W1;2.D;C/. To further extend 'into the complement of D, assume first without loss of generality that 02Y. We now let .z/ D1=z denote the inversion with respect to the unit circle, which is a diffeomorphism in Cn¹0º. The map WD ı'ıis then a homeomorphism A. Koski, J. Onninen 4082 from @Dto @.Y/. Note that since is the identity on @D, we also have Dı'. Since is locally bi-Lipschitz in Cn¹0º, there is L > 1 such that is L-bi-Lipschitz in a neighborhood of @.Y/. Hence we may estimate that Z@.Y/Z@.Y/ˇˇlog j 1.˛/  1.ˇ/jˇˇjd˛jjdˇj DZ@.Y/Z@.Y/ˇˇlog j'1..˛// '1..ˇ//jˇˇjd˛jjdˇj DZ@YZ@Yˇˇlog j'1./ '1./jˇˇj0./jj0./jjdjjdj L2Z@YZ@Yˇˇlog j'1./ '1./jˇˇjdjjdj<1: This shows that satisfies condition (1.3), and hence the earlier part of the proof shows that we may extend as a W1;2-homeomorphism Q hfrom Dto the Jordan domain bounded by @.Y/. Hence ıQ hıis a homeomorphism from CnDto CnY, equal to 'on the boundary, and in W1;2.U; C/for any bounded subset UCnDdue to the bi-Lipschitz bounds on in Cn¹0º. This concludes the proof. 6. The multiply connected case: Proof of Theorem 1.11 In this section we consider multiply connected Jordan domains Xand Yof the same topological type. Any such domain can be equivalently obtained by removing from a simply connected Jordan domain the same number, say 0` < 1, of closed disjoint topological disks or single points. Throughout what follows, we will assume that none of the boundary components of Xand Yare single points – this case will only be addressed at the very end of the proof. If `D1, the resulting doubly connected domain is conformally equivalent to a circular annulus AD ¹z2CWr < jzj< 1ºwith some 0<r<1. In fact, if `1then every .` C1/-connected Jordan domain can be mapped by a conformal mapping onto a circular domain [14]. An .` C1/-connected circular domain consists of the domain bounded by the boundary of the unit disk Dand kother circles in the interior of D. This conformal equivalence to circular domains will be used in certain parts of the proof. The conformal mappings between multiply connected Jordan domains extend continuously up to the boundaries. The idea of the proof of Theorem 1.11 is simply to split the multiply connected domains Xand Yinto simply connected parts and apply Theorem 1.8 in each of these parts. Let us consider first the case where Xand Yare doubly connected. 6.1. Doubly connected Xand Y Case 1: pD1. Suppose that the boundary of Xis rectifiable. We split Xinto two rectifiable simply connected domains as follows. Take a line Lpassing through any point in the Sobolev homeomorphic extensions 4083 bounded component of CnX. Then there exist two open line segments I1and I2on L that are contained in Xand have endpoints on different components of the boundary of X. These segments split the domain Xinto two Jordan domains X1and X2with rectifiable boundaries. For kD1;2, let pkdenote the endpoint of Iklying on the inner boundary of Xand Pk the endpoint on the outer boundary. We let qkD'.pk/and QkD'.Pk/, where 'denotes the given boundary map from the statement of Theorem 1.11. We would now simply like to connect qkto Qkby a rectifiable curve kinside Yin such a way that 1and 2do not intersect. It is quite obvious this can be done but we provide a proof regardless. Let YCdenote the Jordan domain bounded by the outer boundary of Y. Take a conformal map gCWD!YC. Then g0 Cis in the Hardy space H1since @Y1is rectifiable, and by [10, Theorem 3.13] we find that gCmaps the segment Œ0; g1 C.Qk/ into a rectifiable curve in YC. Let C kdenote the image of the segment Œ.1 /g1 C.Qk/; g1 C.Qk/ under gCfor a sufficiently small . Then C kis a rectifiable curve connecting Qkto an interior point QC kof Yif is small enough. With a similar argument, possibly adding a Möbius transformation to the argument to invert the order of the boundaries, one finds a rectifiable curve  kconnecting qkto an interior point q k. For small enough the four curves constructed here do not intersect. If denotes the union of these four curves, we may now use the path-connectedness of the domain Ynto join the points QC 1and q 1with a smooth simple curve inside Y that does not intersect . By combining the curves C 1and  1one obtains a rectifiable simple curve 1connecting Q1and q1. Using the fact that Ynis doubly connected, we may now join QC 2and q 2with a smooth curve that does not intersect 1or . This yields a rectifiable simple curve 2connecting Q2and q2. This proves the existence of the curves kwith the desired properties. These curves split Yinto two simply connected Jordan domains Y1and Y2. We may now extend the homeomorphism 'to map the boundary of Xkto the boundary of Ykhomeomorphically. The exact parametrization which maps the segments Ikto the curves kdoes not matter. The rest of the claim follows directly from the first part of Theorem 1.8, giving us a homeomorphic extension of 'in the Sobolev class W1;1.X;C/, as claimed. Case 2: 1 < p < 2. Suppose that Xhas s-hyperbolic growth. Then we take an annulus A centered at the origin such that there exists a conformal map gWA!X. By a result of Gehring and Osgood [12], the quasihyperbolic metrics hXand hAare comparable via the conformal map g. This shows that for any fixed x02Aand all x2Awe have hA.x0; x/ C hX.g.x0/; g.x// C dist.g.x/; @X/1s:(6.1) Let now ACdenote the simply connected domain obtained by intersecting Aand the upper half-plane. We claim that the domain XCWD g.AC/has s-hyperbolic growth as well. A. Koski, J. Onninen 4084 To prove this claim, fix x02ACand take an arbitrary x2A. Let dDdist.x; @AC/. We aim to establish the inequality hAC.x0; x/ C dist.g.x/; @XC/1s:(6.2) Note that ACis bi-Lipschitz equivalent to the unit disk, implying that hAC.x0; x/ is comparable to log.1=d/. The boundary of ACcontains two line segments on the real line; let us denote them by I1and I2. Note that we have the estimate dist.g.x/; @XC/dist.g.x/; @X/: (6.3) If it happened that dDdist.x; @A/, meaning that the closest point to xon @ACis not on I1or I2, then the hyperbolic distances hAC.x0; x/ and hA.x0; x/ are comparable and by the inequalities (6.1) and (6.3) the inequality (6.2) holds. Hence it is enough to prove (6.2) when dDdist.x;I1[I2/. We may also assume that dis small. Due to the geometry of the half-annulus AC, the projection of xto the real line lies on either I1or I2, and the vertical line segment Lxbetween xand its projection lies in ACand has length d. Letting Ddenote the distance from xto @ACn.I1[I2/, we see that Dd. We may now reiterate the proof of (3.4) to find that jg0.z/j  C dist.z; @A/log 1 1s.dist.z; @A/1/ for z2A. We should mention that the simply connectedness assumption used in the proof of (3.4) may be circumvented by using the equivalence of the quasihyperbolic metrics under ginstead of passing to the hyperbolic metric. Hence dist.g.x/; @XC/ZLxjg0.z/jjdzj  Cd Dlog 1 1s.1=D/ : From this we find that (6.2) is equivalent to log.1=d/ CD1slog.1=D/ d1s; which is true since Dd. Hence (6.2) holds, and this implies that XChas s-hyperbolic growth by reversing the argument that gives (6.1). We define Xsimilarly. Hence we have split Xinto two simply connected domains with s-hyperbolic growth. On the image side, we may split Yinto two simply connected domains with rectifiable boundary as in Case 1. Extending 'in an arbitrary homeomorphic way between the boundaries of these domains and applying part 2 of Theorem 1.8(2) gives a homeomorphic extension of 'in W1;p.X;C/whenever s > p 1. Sobolev homeomorphic extensions 4085 6.2. The general case Case 3: pD1. Assume that Xand Yare `-connected Jordan domains with rectifiable boundaries. By induction, we may assume that the result of Theorem 1.11 holds for .` 1/-connected Jordan domains. Hence we are only required to split Xand Yinto two domains with rectifiable boundary, one which is doubly connected and the other is .` 1/-connected. We hence describe how to ‘isolate’ a given inner boundary component X0from an `- connected Jordan domain X. Let Xouter ¤X0denote the outer boundary component of X. Take a small neighborhood of X0inside X. Let 0be a piecewise linear Jordan curve contained in this neighborhood and separating X0from the other boundary components of X. Let also 1be a piecewise linear Jordan curve inside Xand in a small enough neighborhood of Xouter so that all the inner boundary components of Xare inside 1. Take y0and y1on 0and 1respectively, and connect them with a piecewise linear curve ˛y not intersecting any boundary components of X. Choose z0on 0close to y0and z1on 1close to y1so that we may connect z0and z1by a piecewise linear curve ˛zarbitrarily close to ˛ybut intersecting neither ˛ynor any boundary components of X. Since the region bounded by Xouter and 1is doubly connected, by the construction in Case 1 we may connect y1and z1to any two given points y2and z2on the boundary Xouter via nonintersecting rectifiable curves ˇyand ˇzlying inside this region. Let now denote the union of the curves ˇy,ˇz,˛y,˛z, and the curve 0 0obtained by taking 0and removing the part between y0and z0. By construction contains two arbitrary points on Xouter and separates the domain Xinto a doubly connected domain with inner boundary component X0and an .n 1/-connected Jordan domain. Since is rectifiable, both of these domains are also rectifiable. Applying the same construction for Y, we may separate the boundary component '.X0/of Yby a rectifiable curve 0. Since the boundary points y2and z2above were arbitrary, we may assume that 0intersects the outer boundary of Yat the points '.y2/ and '.z2/. Extending 'to a homeomorphism from onto 0and applying the induction assumptions now gives a homeomorphic extension in the class W1;1.X;C/. Case 4: 1<p<2. We still have to deal with the case where Xhas s-hyperbolic growth and is `-connected. By the same arguments as in the previous case, it will be enough to split Xinto a doubly connected and an .` 1/-connected domain with s-hyperbolic growth. Since Xis `-connected, there exists a domain such that every boundary component of is a circle and there is a conformal map gW!X. Let be a piecewise linear simple curve with both endpoints on the outer boundary of such that separates one of the inner boundary components of @ from the others, which implies that the curve  splits into a doubly connected set 1and an .` 1/-connected set 2. We claim that the domains X1Dg.1/and X2Dg.2/have s-hyperbolic growth. A. Koski, J. Onninen 4086 The proof of this claim is nearly identical to the arguments in Case 2, so we will summarize it briefly. For X2, we aim to establish the inequality h2.x0; x/ C dist.g.x/; @X2/1s(6.4) for fixed x022and x22. For this inequality, it is only essential to consider xclose to @2. If xis closer to the boundary of the original set @ than to , then the hyperbolic distance between x0and xin 2is comparable to the distance inside the larger set . Then the s-hyperbolic growth of implies (6.4) as in Case 2. If xis closer to but a fixed distance away from the boundary of , then the smoothness of gin compact subsets of implies the result. If xis closest to a line segment in which has its other endpoint on @, then we may employ a similar estimate to that in Case 2, using the bound for jg0.z/jin terms of dist.z; @/, to conclude that (6.4) also holds here. This implies that X2 satisfies (6.4), and hence it has s-hyperbolic growth. The argument for X1is the same. After splitting Xinto two domains of smaller connectivity and s-hyperbolic growth, we split the target Yaccordingly into rectifiable parts using the argument from Case 3. Applying induction on `now proves the result in this case. 6.3. Punctured domains We now address the case where Xand Yare `-connected and where some of the inner boundary components of Xand Ymay be single points. Let these points be x1; : : : ; xN 2Xand y1;: : : ; yN2Y. Without loss of generality we may assume '.xj/Dyjfor all j. Let Q Xdenote the .` N /-connected domain X[¹x1; : : : ; xNºand define Q Ysimilarly. We now consider the boundary map 'j@Q XW@Q X!@Q Yand let Q hWQ X!Q Ydenote the W1;p-homeomorphic extension of this boundary map. If such a map satisfied Q h.xj/Dyj for all jthen we would be done. If not, let UQ Ybe a smooth simply connected domain large enough to contain all the points Q h.x1/; : : : ; Q h.xN/and y1; : : : ; yN. Then consider a diffeomorphic change of variables WU!Uthat is the identity map on the boundary and sends the point Q h.xj/to yjfor every j. Now the map hWD ıQ hjXWX!Yis the desired Sobolev homeomorphic extension of '. This finishes the proof of Theorem 1.11. 7. Monotone Sobolev minimizers The classical harmonic mapping problem deals with the question of whether there exists a harmonic homeomorphism between two given domains. Of course, when the domains are Jordan such a mapping problem is always solvable. Indeed, according to the Riemann Mapping Theorem there is a conformal mapping hWXonto ! Y. Finding a harmonic homeomorphism which coincides with a given boundary homeomorphism 'W@Xonto ! @Yis a more subtle question. If Yis convex, then there always exists a harmonic homeomorphism hWXonto ! Ywith h.x/ D'.x/ on @Xby Lemma 2.3. For a nonconvex target Y, however, Sobolev homeomorphic extensions 4087 there always exists at least one boundary homeomorphism whose harmonic extension takes points in Xbeyond Y. To find a deformation hWXonto !Ywhich resembles harmonic homeomorphisms Iwaniec and Onninen [23] applied the direct method of the calculus of variations and considered minimizing sequences in H1;2 '.X;Y/. They called such minimizers monotone Hopf-harmonics and proved the existence and uniqueness result in the case when Yis a Lipschitz domain and the boundary data 'satisfies the Douglas condition. Note that by the Riemann Mapping Theorem one may always assume that XDD. Theorem 1.6 allows one to go beyond the Lipschitz targets. Indeed, under the assumptions of Theorem 1.6, the class H1;2 '.D;Y/is non-empty. Furthermore, if hı2H1;2 '.D;Y/, then hısatisfies the uniform modulus of continuity estimate jhı.x1/hı.x2/j2CRDjDhıj2 log1 jx1x2j for x1; x22Dsuch that jx1x2j< 1. This follows by taking the global W1;2 loc -homeomorphic extension given by Theorem 1.6 and applying a standard local modulus of continuity estimate for W1;2-homeomorphisms [19, Corollary 7.5.1, p. 155]. Now, applying the direct method of the calculus of variations allows us to find a minimizing sequence in H1;2 '.D;Y/for the Dirichlet energy converges weakly in W1;2.D;C/and uniformly in D. Being a uniform limit of homeomorphisms the limit mapping HWDonto !Ybecomes monotone. Indeed, the classical Youngs approximation theorem [39] asserts that a continuous map between compact oriented topological 2-manifolds (surfaces) is monotone if and only if it is a uniform limit of homeomorphisms. Monotonicity, the concept of Morrey [30], simply means that for a continuous HWX!Ythe preimage H1.yı/of a point yı2Yis a continuum in X. We have thus proved the following result. Theorem 7.1. Let Xand Ybe Jordan domains and assume that @Yis rectifiable. 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