---
abstract: 'We prove that planar homeomorphisms can be approximated by diffeomorphisms in the Sobolev space ${\mathscr{W}}^{1,2}$ and in the Royden algebra. As an application, we show that every discrete and open planar mapping with a holomorphic Hopf differential is harmonic.'
address:
- 'Department of Mathematics, Syracuse University, Syracuse, NY 13244, USA and Department of Mathematics and Statistics, University of Helsinki, Finland'
- 'Department of Mathematics, Syracuse University, Syracuse, NY 13244, USA'
- 'Department of Mathematics, Syracuse University, Syracuse, NY 13244, USA'
author:
- Tadeusz Iwaniec
- 'Leonid V. Kovalev'
- Jani Onninen
date: 'June 26, 2010'
title: |
    Hopf differentials and\
    smoothing Sobolev homeomorphisms
---

[^1]

[^2]

[^3]

Introduction
============

It is a fundamental property of Sobolev spaces ${\mathscr{W}}^{1,p}$, $1\le p<\infty$, that any element can be approximated strongly (i.e., in the norm) by $\mathscr C^{\infty}$ smooth functions, or by piecewise affine ones. In the context of vector-valued Sobolev functions, that is, mappings in ${\mathscr{W}}^{1,p}(\Omega,{\mathbb{R}}^n)$, invertibility comes into play. Indeed, the studies of invertible Sobolev mappings are of great importance in nonlinear elasticity [@Ba0; @FG; @MST; @Sv]. The following natural question was put forward by John M. Ball.

[@Ba] If $u \in {\mathscr{W}}^{1,p}(\Omega, {\mathbb{R}}^n)$ is invertible, can $u$ be approximated in $ {\mathscr{W}}^{1,p}$ by piecewise affine invertible mappings?

J. Ball attributes this question to L.C. Evans, who was led to it through his investigation of the partial regularity of minimizers [@Ev] of neohookean energy functionals [@Ba1; @BPO1; @CL; @SiSp]. We provide an affirmative solution of the Ball-Evans problem in the case $p=n=2$. The most general formulation of our result administers Royden algebras ${\mathscr{A}}(\Omega)$ and ${\mathscr{A}}_\circ (\Omega)$, see Section \[secroy\]. We write $${\mathcal{E}}[h]={\mathcal{E}}_\Omega[h]:= {\lVertDh\rVert}^2_{\mathscr L^2(\Omega)}= \int_\Omega {\lvertDh(z)\rvert}^2\, {\textnormal d}z$$ where ${\lvertDh\rvert}$ is the Hilbert-Schmidt norm of the differential.

\[thmapprox\] Let $h \colon \Omega {\stackrel{{\rm \tiny{onto}}}{\longrightarrow}}\Omega^\ast$ be a homeomorphism of Sobolev class $\mathscr W^{1,2}_{\operatorname{loc}} (\Omega, \Omega^\ast)$. Then for every $\epsilon >0$ there exist a diffeomorphism $H \colon \Omega{\stackrel{{\rm \tiny{onto}}}{\longrightarrow}}\Omega^\ast$ such that

(i) $H-h \in {\mathscr{A}}_\circ (\Omega)$

(ii) $ {\lVertH-h\rVert}_{{\mathscr{A}}(\Omega)}\le \epsilon$

(iii) ${\mathcal{E}}[H] \le {\mathcal{E}}[h]$.

Part (iii) is nontirivial only in the finite energy case, ${\mathcal{E}}_\Omega [h]< \infty$. Let us note that the existence of smooth approximation implies the existence of piecewise-affine approximation, since a diffeomorphism can be triangulated. (In the converse direction, a piecewise-affine mapping can be smoothed in dimensions less than four [@Mu], but not in general.) Partial results toward the Ball-Evans problem were obtained in [@Mo] (for planar bi-Sobolev mappings that are smooth outside of a finite set) and in [@BM] (for planar bi-Hölder mappings, with approximation in the Hölder norm). The articles [@Ba; @SS] illustrate the difficulty of preserving invertibility in the process of smoothing a Sobolev homeomorphism.

We also give an application of Theorem \[thmapprox\] to a problem that originated in a series of papers by Eells, Lemaire and Sealey [@EL2; @EL3; @Se]. It concerns the nonlinear differential equation $$\label{heq2}
\frac{\partial}{ \partial \bar z} \left( h_z \overline{h_{\bar z}} \right)=0$$ for mappings defined in a domain in the complex plane ${\mathbb{C}}$. Naturally, the Sobolev space ${\mathscr{W}}^{1,2}_{\operatorname{loc}}(\Omega, {\mathbb{C}})$ should be considered as the domain of definition of equation . This places $ h_z \overline{h_{\bar z}}$ in $\mathscr L^1_{\operatorname{loc}} (\Omega)$, so the complex Cauchy-Riemann derivative $\frac{\partial}{\partial \bar z}$ applies in the sense of distribution. By Weyl’s lemma $h_z \overline{h_{\bar z}}$ is a holomorphic function.

The expression $Q_h:=h_z \overline{h_{\bar z}}\, {\textnormal d}z \otimes {\textnormal d}z$ is known as the Hopf differential of $h$ (named after H. Hopf, who employed a similar device, e.g., in [@Ho Chapter VI]). It is clear that $Q_h$ is a holomorphic quadratic differential whenever $h$ is harmonic, which is a general fact about energy-stationary mappings between Riemannian manifolds [@EL1 (10.5)], [@Job] and [@Stb]. Eells and Lemaire inquired about the possibility of a converse result, e.g., for mappings with finite energy and almost-everywhere positive Jacobian [@EL2 (2.6)]. In this setting a counterexample was provided by Jost [@Jo], who also proved the existence of ${\mathscr{W}}^{1,2}$-solutions of  in every homotopy class of mappings between compact Riemann surfaces. A more restricted form of the Eells-Lemaire problem, [@EL3 (5.11)] and [@Se], imposed the additional assumption that $h$ is a quasiconformal homeomorphism, and was settled by Hélein [@He] in the affirmative. Here we dispose with the quasiconformality condition and treat general planar homeomorphisms of finite energy. Since the inverse of such a homeomorphism need not be in any Sobolev class [@HK], some difficulties are to be expected. They shall be overcome with the aid of our approximation theorem \[thmapprox\].

\[thmhopf\] Every continuous, discrete and open mapping $h$ of Sobolev class ${\mathscr{W}}^{1,2}_{\operatorname{loc}}(\Omega, {\mathbb{C}})$ that satisfies equation  is harmonic.

The failure of Theorem \[thmhopf\] for uniform limits of homeomorphisms should be mentioned. This is illustrated by Example \[ex\].

Background {#secroy}
==========

Let $\Omega$ be a bounded domain in ${\mathbb{R}}^2 \simeq {\mathbb{C}}$, nonempty open connected set. We consider a class ${\mathscr{A}}(\Omega)$ of uniformly continuous functions $h \colon {\Omega} \to {\mathbb{C}}$ having finite Dirichlet energy, and furnish it with the norm $$\| h\|_{{\mathscr{A}}(\Omega)} = \| h \|_{\mathscr C ({\Omega})} + \| Dh \|_{\mathscr L^2 (\Omega)} < \infty$$ ${\mathscr{A}}(\Omega)$ is a commutative Banach algebra with the usual multiplication of functions in which ${\lVerth_1h_2\rVert}_{{\mathscr{A}}(\Omega)} \le {\lVerth_1\rVert}_{{\mathscr{A}}(\Omega)} {\lVerth_2\rVert}_{{\mathscr{A}}(\Omega)}$. The closure of $\mathscr C^\infty_\circ (\Omega)$ in ${\mathscr{A}}(\Omega)$ will be denoted by ${\mathscr{A}}_\circ (\Omega)$. Suppose, to look at more specific situation, that $\Omega = {\mathbb{U}}$ is a Jordan domain; that is, a simply connected open set whose boundary $\Gamma= \partial {\mathbb{U}}$ is a closed Jordan curve. By goodness of the Carathéodory extension theorem [@Pob p. 18], there is a homeomorphism $\varphi \colon \overline{\mathbb D} \stackrel{\textrm{\tiny{onto}}}{\longrightarrow} \overline{{\mathbb{U}}}$ of the closed unit disk $\overline{{\mathbb{D}}}= \{ \xi \colon {\lvert\xi\rvert} \le 1 \}$ that is conformal in ${\mathbb{D}}$. After the change of variable, $z= \varphi (\xi)$, we obtain a function $H(\xi)= h\big(\varphi (\xi)\big)$ in ${\mathscr{A}}({\mathbb{D}})$. The operation $$\mathbf{T}_\varphi \colon {\mathscr{A}}({\mathbb{U}}) \to {\mathscr{A}}({\mathbb{D}})$$ so defined is an isometry; ${\lVert\mathbf{T}_\varphi h\rVert}_{{\mathscr{A}}({\mathbb{D}})} = {\lVerth\rVert}_{{\mathscr{A}}({\mathbb{U}})}$. Furthermore, $$\mathbf{T}_\varphi \colon {\mathscr{A}}_\circ ({\mathbb{U}}) \to {\mathscr{A}}_\circ ({\mathbb{D}})$$

\[prpoisson\] Let ${\mathbb{U}}$ be a Jordan domain. There is (unique) bounded linear operator $$\mathbf{P}_{{\mathbb{U}}} \colon {\mathscr{A}}({\mathbb{U}}) \to {\mathscr{A}}({\mathbb{U}})$$ such that $$\begin{cases} \mathbf{P}_{{\mathbb{U}}} - \mathbf Id\colon {\mathscr{A}}({\mathbb{U}}) \to {\mathscr{A}}_\circ ({\mathbb{U}})\\
\Delta \circ \mathbf P_{{\mathbb{U}}}=0
\end{cases}$$

We name $\mathbf P_{{\mathbb{U}}}$ the [*Poisson operator*]{}. The energy of $\mathbf P_{{\mathbb{U}}} h$ does not exceed that of $h$. This fact is known as [*Dirichlet’s principle*]{} $$\int_{{\mathbb{U}}} {\lvertD \mathbf P_{{\mathbb{U}}} h\rvert}^2 \le \int_{{\mathbb{U}}} {\lvertD h\rvert}^2$$ The proof of this proposition reduces to the case when ${\mathbb{U}}={\mathbb{D}}$, by conformal change of variables. A routine verification of this case is left to the reader. We only indicate that the less familiar property $\mathbf{P}_{{\mathbb{D}}} h -h \in {\mathscr{A}}_\circ ({\mathbb{D}})$, for $h\in {\mathscr{A}}({\mathbb{D}})$, needs to be justified.

Let $\Omega$ be a domain in ${\mathbb{C}}$ and ${\mathbb{U}}\subset \overline{{\mathbb{U}}} \subset \Omega$ a Jordan domain. There exists (unique) bounded linear operator $$\mathbf R_{{\mathbb{U}}} \colon {\mathscr{A}}(\Omega) \to {\mathscr{A}}(\Omega)$$ such that, for every $h\in {\mathscr{A}}(\Omega)$ $$\begin{cases} \mathbf R_{{\mathbb{U}}}h=h \qquad \mbox{ on }\quad \Omega \setminus {\mathbb{U}}\\
\Delta  \mathbf R_{{\mathbb{U}}}h= 0  \qquad \mbox{ in } \quad {\mathbb{U}}\end{cases}$$ The Laplace equation yields ${\mathcal{E}}_{\Omega} [\mathbf R_{{\mathbb{U}}} h] \le {\mathcal{E}}_{\Omega} [h]$. Equality occurs if and only if $h$ is harmonic in ${\mathbb{U}}$.

A short proof of this corollary runs somewhat as follows. The unique harmonic extension of $h \colon \partial {\mathbb{U}}\to {\mathbb{C}}$ inside ${\mathbb{U}}$ given by $\mathbf P_{{\mathbb{U}}} h$ has the property that $\mathbf P_{{\mathbb{U}}} h-h \in {\mathscr{A}}_\circ ({\mathbb{U}})$. Therefore, the zero extension of $\mathbf P_{{\mathbb{U}}} h -h$ outside ${\mathbb{U}}$, denoted by $[\mathbf P_{{\mathbb{U}}}h-h]_\circ$, belongs to ${\mathscr{A}}(\Omega)$. We define $$\mathbf{R}_{{\mathbb{U}}} h:= [\mathbf P_{{\mathbb{U}}} h -h]_\circ +h  \in {\mathscr{A}}(\Omega)$$ The desired properties of the operator $\mathbf R_{{\mathbb{U}}}$ so defined are automatically fulfilled in view of Proposition \[prpoisson\].

\[RKC\] Let $\Omega$ be a domain in ${\mathbb{C}}$ and ${\mathbb{U}}\subset \overline{{\mathbb{U}}} \subset \Omega$ a Jordan domain. Suppose that $h\in \mathscr {\mathscr{A}}(\Omega)$ is a homeomorphism of $\Omega$ onto $h(\Omega)$ and $h({\mathbb{U}})$ is convex. Then $\mathbf R_{\mathbb{U}}h$ is homeomorphism in $\Omega$ and is a harmonic diffeomorphism in ${\mathbb{U}}$.

The injectivity of $\mathbf R_{\mathbb{U}}h$ is the content of the Radó-Kneser-Choquet Theorem [@Dub p. 29]. Furthermore, planar harmonic homeomorphisms are $\mathscr C^\infty$-smooth diffeomorphisms according to Lewy’s theorem [@Dub p. 20].

Smoothing Sobolev homeomorphisms, Theorem \[thmapprox\]
=======================================================

We may and do assume that $h$ is not harmonic, since otherwise $H =h$ satisfies the desired properties by Lewy’s theorem (mentioned above). Let $z_\circ \in \Omega$ be a point such that $h$ fails to be harmonic in any neighborhood of $z_\circ$. By choosing the origin of the coordinate system we ensure that $h(z_\circ)$ does not lie on the boundary of any dyadic squares associated with the coordinate system.

Let us choose and fix any $\epsilon >0$. The construction of $H$ proceeds in 5 steps. We construct homeomorphisms $h_k \colon \Omega {\stackrel{{\rm \tiny{onto}}}{\longrightarrow}}\Omega^\ast$, $k=0, \dots, 5$ such that $h_0=h$, $h_k\in h_{k-1}+{\mathscr{A}}_\circ(\Omega)$, $h_5$ is a diffeomorphism, and ${\lVerth_k-h_{k-1}\rVert}$ is bounded by a multiple of $\epsilon$ for each $k$. In each step we modify the previous construction to gain better regularity. In steps 1, 2 and 4 we use harmonic replacement according to Proposition \[RKC\]. In steps 3 and 5 we smoothen the mapping near the boundaries of the domains in which harmonic replacement was performed. The result of each step is denoted by $h_1,\dots,h_5$. The $h\in {\mathscr{A}}(\Omega)$ requires a few additional details, which are provided at the end of each step.

We begin with a decomposition of the target domain $$\Omega^\ast = \bigcup_{\nu=1}^\infty \overline{{\mathbb{Q}}_{\nu}}$$ into closed nonoverlapping dyadic squares $\overline{{\mathbb{Q}}_{\nu}} \subset \Omega^\ast$. This decomposition is made by selecting the maximal dyadic squares that lie in $\Omega^\ast$. Thus the cover of $\Omega^\ast$ by such squares is locally finite. The preimage of ${\mathbb{Q}}_\nu$ under $h$, denoted by ${\mathbb{U}}_\nu$, is a Jordan domain in $\Omega$. Hereafter ${\mathbb{U}}_\nu$ will be referred to as the curved-square. In fact to every partion of $\Omega^\ast$ into closed squares there will correspond a partition of $\Omega$ into closed curved-squares via the mapping $h \colon \Omega \stackrel{\textrm{\tiny{onto}}}{\longrightarrow} \Omega^\ast$, for example: $$\Omega = \bigcup_{\nu=1}^\infty \overline{{\mathbb{U}}_{\nu}}$$ [**Step 1.**]{} For each ${\mathbb{U}}_\nu$ we replace $h \colon \overline{{\mathbb{U}}_\nu} \stackrel{\textrm{\tiny{onto}}}{\longrightarrow} \overline{{\mathbb{Q}}_{\nu}}$ with a piecewise harmonic homeomorphism $h_1 \colon \overline{{\mathbb{U}}_\nu} \stackrel{\textrm{\tiny{onto}}}{\longrightarrow} \overline{{\mathbb{Q}}_{\nu}} $ that coincides with $h$ on $\partial  \overline{{\mathbb{U}}_{\nu}}$. To this effect we partition the square $ \overline{{\mathbb{Q}}_{\nu}} $, $$\overline{{\mathbb{Q}}_{\nu}} =  \overline{{\mathbb{Q}}^1_{\nu}} \cup  \overline{{\mathbb{Q}}^2_{\nu}} \cup \dots \cup  \overline{{\mathbb{Q}}^n_{\nu}}, \qquad (n=n_\nu=4^{k_\nu})$$ into congruent dyadic squares $ \overline{{\mathbb{Q}}^i_{\nu}}$, $i=1, \dots, n$. The number $n$, depending on $\nu$, will be determined later. For the moment fix $\nu$ and look at the homeomorphisms $$h \colon  \overline{{\mathbb{U}}^i_\nu} = h^{-1}(\overline{{\mathbb{Q}}^i_{\nu}}) \stackrel{\textrm{\tiny{onto}}}{\longrightarrow} \overline{{\mathbb{Q}}^i_{\nu}}$$ These mappings belong to the Royden algebra ${\mathscr{A}}( \overline{{\mathbb{U}}^i_\nu})$. With the aid of Propositions \[prpoisson\] and \[RKC\] we replace each $h \colon \overline{{\mathbb{U}}^i_\nu} \to  \overline{{\mathbb{Q}}^i_\nu}$ with a harmonic homeomorphism $h^i_\nu \colon \overline{{\mathbb{U}}^i_\nu} \stackrel{\textrm{\tiny{onto}}}{\longrightarrow}   \overline{{\mathbb{Q}}^i_\nu}$ that coincides with $h$ on $\partial  \overline{{\mathbb{U}}^i_\nu}$, $i=1,2, \dots , n$. Such mappings are $\mathscr C^\infty$-smooth diffeomorphisms $h^i_\nu \colon {{\mathbb{U}}^i_\nu} \stackrel{\textrm{\tiny{onto}}}{\longrightarrow}  {{\mathbb{Q}}^i_\nu}$. Moreover, $h^i_\nu -h \in {\mathscr{A}}_\circ ({\mathbb{U}}_\nu^i)$ and $$\begin{cases}
{\mathcal{E}}_{{\mathbb{U}}^i_\nu}[h_\nu^i] \le {\mathcal{E}}_{{\mathbb{U}}^i_\nu}[h] \qquad \mbox{for } 1,2, \dots , n \\
 {\mathcal{E}}_{\partial {\mathbb{U}}^i_\nu}[h_\nu^i] = {\mathcal{E}}_{\partial {\mathbb{U}}^i_\nu}[h], \quad \mbox{ because } h_\nu^i=h \mbox{ on } \partial {\mathbb{U}}_\nu^i
\end{cases}$$ We obtain a piecewise harmonic homeomorphism by gluing $h^i_\nu$ together along the common boundaries of ${\mathbb{U}}^i_\nu$. Denote it by $$h^n_{\nu} \colon  \overline{{\mathbb{U}}_\nu}  \stackrel{\textrm{\tiny{onto}}}{\longrightarrow} \overline{{\mathbb{Q}}_{\nu}}$$ $$h^n_{\nu} \in h + {\mathscr{A}}_\circ ({\mathbb{U}}_\nu)$$ Precisely we define $$h_\nu^n=h+ \sum_{i=1}^n [h_\nu^i-h]_\circ$$ Here and in the sequel the notation $[\varphi]_\circ$ for $\varphi\in {\mathscr{A}}_\circ({\mathbb{U}})$ stands for zero extension of $\varphi$ to the entire domain $\Omega$. Obviously $[\varphi]_\circ \in {\mathscr{A}}_\circ(\Omega)$. The above construction depends on the number $n$. For $\nu$ fixed we actually have a sequence $\{h^n_{\nu}\}_{n=1,2, \dots}$ that is bounded in ${\mathscr{A}}({\mathbb{U}}_\nu)$. However, we have uniform bounds independent of $n$, $${\lVerth^n_{\nu}\rVert}_{\mathscr C(\overline{{\mathbb{U}}_\nu})} \le \operatorname{diam}{\mathbb{Q}}_\nu$$ and $${\mathcal{E}}_{{\mathbb{U}}_\nu} [h^n_{\nu}] \le {\mathcal{E}}_{{\mathbb{U}}_\nu} [h]$$ The key observation is that $$\begin{cases}
h^n_{\nu}-h \in {\mathscr{A}}_\circ ({\mathbb{U}}_\nu)\\
\lim\limits_{n \to \infty} {\lVerth^n_{\nu}-h\rVert}_{{\mathscr{A}}({\mathbb{U}}_\nu)}=0
\end{cases}$$ Indeed, for $z\in \overline{{\mathbb{U}}^i_\nu}$ we have $${\lverth^n_{\nu}(z)-h(z)\rvert} \le \operatorname{diam}{\mathbb{Q}}_\nu^i = \frac{1}{\sqrt{n}} \operatorname{diam}{\mathbb{Q}}_\nu$$ Thus $h^n_{\nu } \rightrightarrows h$ uniformly on $\overline{{\mathbb{U}}_\nu}$ as $n \to \infty$. On the other hand the differential matrices $Dh^n_{\nu}$ are bounded in $\mathscr L^2 ({\mathbb{U}}_\nu, {\mathbb{R}}^{2\times 2})$. Their weak limit exits and is exactly equal to $Dh$, because the mappings converge uniformly to $h$. Therefore, $$\begin{split}
\int_{{\mathbb{U}}_\nu} {\lvertDh^n_{\nu}-Dh\rvert}^2 & = \int_{{\mathbb{U}}_\nu} \Big({\lvertDh^n_{\nu}\rvert}^2 + {\lvertDh\rvert}^2-2 \langle Dh^n_{\nu}, Dh \rangle  \Big)\\
& \le  2 \int_{{\mathbb{U}}_\nu} \Big({\lvertDh\rvert}^2 -   \langle Dh^n_{\nu}, Dh \rangle  \Big) \\
& = 2 \int_{{\mathbb{U}}_\nu} \langle Dh, Dh - Dh^n_{\nu} \rangle \longrightarrow 0
\end{split}$$ We can now determine the number $n=n_\nu$ of congruent dyadic squares in ${\mathbb{Q}}_\nu$, simply requiring that $$\begin{cases}
 \operatorname{diam}{\mathbb{Q}}^i_\nu \le {\epsilon} \qquad \mbox{ for every } i=1,2, \dots, n_\nu\\
{\lVertDh^n_{\nu}-Dh\rVert}_{\mathscr L^2 (\overline{{\mathbb{U}}_\nu})} \le \epsilon \cdot 2^{-\nu}
\end{cases}$$ Fix such $n=n_\nu$ and abbreviate the notation for $h^{n_\nu}_{\nu}$ to $h^\nu$. We obtain a homeomorphism $$h_1:=h+ \sum_{\nu =1}^\infty [h^\nu-h]_\circ \in h+ {\mathscr{A}}_\circ (\Omega)$$ where we recall that $ [h^\nu-h]_\circ$ stands for the zero extension of $h^\nu-h$ to the entire domain $\Omega$. Clearly, $h_1$ is harmonic in each ${\mathbb{U}}_\nu^i$, $\nu=1,2, \dots$, $i=1,2, \dots , n_\nu$ and we have $${\lVerth_1-h\rVert}_{\mathscr C (\Omega)} \le \sup \{\operatorname{diam}{\mathbb{Q}}_\nu^i \colon \nu=1,2, \dots, \; \; i=1, \dots, n_\nu \} < {\epsilon}$$ $${\lVerth_1-h\rVert}_{{\mathscr{A}}(\Omega)} \le {\epsilon}+\sum_{\nu =1}^\infty  {\lVertDh^\nu-Dh\rVert}_{\mathscr L^2({\mathbb{U}}_\nu)} \le {\epsilon}+ \sum_{\nu=1}^\infty \epsilon \cdot 2^{-\nu} = 2 \epsilon$$ For further considerations it will be convenient to number the squares ${\mathbb{Q}}_\nu^i$ and their preimages ${\mathbb{U}}_\nu^i$ using only one index. These sets will be respectively denoted by ${\mathbb{Q}}^\alpha$ and ${\mathbb{U}}^\alpha$, $\alpha=1,2, \dots$. For the record, $$\label{doeq}
\operatorname{diam}{\mathbb{Q}}^\alpha \le {\epsilon}, \qquad \alpha =1,2, \dots$$ Summing up the energy inequalities for the mappings $h^i_\nu \colon {\mathbb{U}}_\nu^i \to {\mathbb{Q}}_\nu^i$ we see that the total energy of $h_1$ does not exceed the energy of $h$. Even more, since $h$ was assumed to be not harmonic, there is at least one region ${\mathbb{U}}_\nu^i$ for which $h \colon {\mathbb{U}}_\nu^i \to {\mathbb{Q}}_\nu^i$ was not harmonic. Consequently, its harmonic replacement results in strictly smaller energy. Hence $$\label{36p}{\mathcal{E}}_{\Omega} [h_1] < {\mathcal{E}}_{\Omega} [h], \qquad \mbox{so let } \delta= {\lVertDh\rVert}_{\mathscr L^2(\Omega)} - {\lVertDh_1\rVert}_{\mathscr L^2(\Omega)} >0$$

[**Step 2.**]{} Denote by $\mathcal F=\{{\mathbb{Q}}^\alpha \colon \alpha=1,2, \dots\}$ the family of all open squares ${\mathbb{Q}}^\alpha \subset \overline{{\mathbb{Q}}^\alpha} \subset \Omega^\ast$ that are build in Step 1 for the construction of the mapping $h_1 \colon \Omega \to \Omega^\ast$. Let $\mathcal V$ be the set of vertices of these squares. Whenever two squares ${\mathbb{Q}}^\alpha, {\mathbb{Q}}^\beta \in \mathcal F$, $\alpha \ne \beta$, meet along their boundaries the intersection $I^{\alpha , \beta}= \partial {\mathbb{Q}}^\alpha \cap \partial {\mathbb{Q}}^\beta$ is either a point in $\mathcal V$ or a closed interval with endpoints in $\mathcal V$. Denote by $\mathcal J \subset \{I^{\alpha , \beta} \colon \alpha\ne \beta,\;  \alpha, \beta=1,2, \dots \}$ the subfamily of all such intersections, excluding empty set and vertices. For each interval $I^{\alpha , \beta} \in \mathcal J$ we shall construct a doubly convex lens-shaped region $\mathbb L^{\alpha, \beta}$ with $I^{\alpha, \beta}$ as its axis of symmetry in the following way. Let $R$ be a number greater than the length of $I^{\alpha, \beta}$ to be chosen later. There exist exactly two open disks of radius $R$ for which $I^{\alpha, \beta}$ is a chord. Let $\mathbb L_R^{\alpha, \beta}$ be their intersection. This is a symmetric doubly convex lens of curvature $\frac{1}{R}$. Thus $\mathbb L_R^{\alpha, \beta}$ is bounded by two circular arcs $\gamma^{\alpha, \beta}= {\mathbb{Q}}^\alpha \cap \partial \mathbb L_R^{\alpha, \beta}$ and $\gamma^{\beta, \alpha}={\mathbb{Q}}^\beta \cap \partial \mathbb L_R^{\alpha, \beta} $. As the curvature of the lens approaches zero the area of $\mathbb L_R^{\alpha, \beta}$ tends to $0$. This allows us to choose $R$ depending on $\alpha$ and $\beta$ so that the lenses $\mathbb L^{\alpha, \beta}=\mathbb L_R^{\alpha, \beta}$ have the following property. $$\label{shouldnumber}
\int_{\mathbb K^{\alpha, \beta}} {\lvertDh_1\rvert}^2 < \frac{\epsilon^2}{2^{\alpha+\beta}}, \quad \mbox{ where } \mathbb K^{\alpha, \beta}=h^{-1}_1(\mathbb L_R^{\alpha, \beta})$$ The lenses $\mathbb L^{\alpha, \beta}$ are disjoint because the opening angle of each lens is at most $\pi/3$ and their axes are either parallel or orthogonal. However, the closures of the lenses considered here may have a common point that lies in $\mathcal V$. On each $\mathbb K^{\alpha, \beta}$ we replace $h_1$ by the harmonic extension of its restriction to $\partial \mathbb K^{\alpha, \beta}$. Thus we obtain a homeomorphism $h_2^{\alpha, \beta} \colon \overline{\mathbb K^{\alpha, \beta}} {\stackrel{{\rm \tiny{onto}}}{\longrightarrow}}\mathbb \overline{L^{\alpha, \beta}}$ of class $h_1+ {\mathscr{A}}_\circ (\mathbb K^{\alpha, \beta})$. By Proposition \[RKC\] the mappings $h_2^{\alpha, \beta} \colon \mathbb K^{\alpha, \beta} {\stackrel{{\rm \tiny{onto}}}{\longrightarrow}}\mathbb L^{\alpha, \beta}$ are diffeomorphisms. Finally, we define $$h_2=h_1+\sum_{\alpha, \beta} [h_2^{\alpha, \beta}-h_1]_\circ \in h_1+ {\mathscr{A}}_\circ (\Omega) = h+ {\mathscr{A}}_\circ (\Omega)$$ and observe that, from , $${\lVerth_2-h_1\rVert}_{\mathscr C (\Omega)}\le \sup_{\alpha, \beta} \operatorname{diam}\left( \mathbb L^{\alpha, \beta} \right) \le \epsilon.$$ Also,   and Dirichlet’s principle imply $$\begin{split}
 \int_{\Omega} {\lvertDh_2-Dh_1\rvert}^2 &\le \sum_{\alpha, \beta} \int_{\mathbb K^{\alpha, \beta}}  2\left({\lvertDh_2\rvert}^2+{\lvertDh_1\rvert}^2\right) \le  4 \sum_{\alpha, \beta =1}^\infty    \int_{\mathbb K^{\alpha, \beta}}  {\lvertDh_1\rvert}^2 \\&\le  4 \sum_{\alpha, \beta =1}^\infty \frac{\epsilon^2}{2^{\alpha+\beta}} =4{\epsilon^2}
 \end{split}$$ Thus $${\lVerth_2-h_1\rVert}_{{\mathscr{A}}(\Omega)} \le \epsilon + 2 \epsilon = 3 \epsilon$$

The boundary of $\mathbb K^{\alpha, \beta}$ consists of two $\mathscr C^\infty$-smooth arcs $\Gamma^{\alpha, \beta}$ and $\Gamma^{\beta, \alpha}$ which share common endpoints, called the apices of $\mathbb K^{\alpha, \beta}$. These are preimages of $\gamma^{\alpha, \beta}$ and $\gamma^{\beta, \alpha}$ under the mapping $h_1$, respectively. Outside of the apices, the homeomorphism $h_2\colon \overline{\mathbb K^{\alpha,\beta}}{\stackrel{{\rm \tiny{onto}}}{\longrightarrow}}\overline{\mathbb L^{\alpha,\beta}}$ is $C^{\infty}$ smooth with positive Jacobian. The smoothness is a classical result of Kellogg; [*a harmonic function with $\mathscr C^\infty$-smooth values on a smooth part of the boundary is $\mathscr C^\infty$-smooth up to this part of the boundary*]{} [@GTb Theorem 6.19]. The positivity of the Jacobian on such part of the boundary follows from the convexity of its image, see [@Dub p. 116].

In conclusion, $h_2$ is locally bi-Lipschitz in $\Omega \setminus h^{-1}(\mathcal V)$. The exceptional set $h^{-1}(\mathcal V)$ is discrete because $\mathcal V$ is.\
By  we have $$\label{39}
{\lVertDh_2\rVert}_{\mathscr L^2(\Omega)} \le {\lVertDh_1\rVert}_{\mathscr L^2(\Omega)} \le {\lVertDh\rVert}_{\mathscr L^2(\Omega)}  - \delta$$

[**Step 3.**]{} First we cover the set of vertices $\mathcal V$ by disks $\{\mathbb D_v \colon v \in \mathcal V\}$ centered at $v$ with radii small enough so that $$\label{ddiam}
\operatorname{diam}\mathbb D_v\le \epsilon,$$ and $\{ 3\mathbb D_v \colon v \in \mathcal V \}$ is a disjoint collection of disks in $\Omega^\ast$. Moreover, their preimages under $h_2$ must satisfy $$\label{star}
\sum_{v \in \mathcal V} \int_{h_2^{-1}(3 {\mathbb{D}}_\nu)} {\lvertDh_2\rvert}^2 < {\epsilon}^2$$ Denote by $\tilde{\Omega}^\ast = \Omega^\ast \setminus \bigcup_{v\in \mathcal V} \overline{{\mathbb{D}}}_v$ and $\tilde{\Omega}= \Omega \setminus \bigcup_{v \in \mathcal V} h_2^{-1} (\overline{{\mathbb{D}}}_v)$. Our focus for a while will be on one of the circular sides of a lens $\mathbb L^{\alpha, \beta}$, say $$\gamma^{\alpha, \beta}= {\mathbb{Q}}^\alpha \cap \partial \mathbb L^{\alpha, \beta} \subset {\mathbb{Q}}^\alpha$$ We truncate it near the endpoints by setting $\tilde{\gamma}^{\alpha, \beta}= \tilde{\Omega} \cap \gamma^{\alpha, \beta}$. Such truncated open arcs are mutually disjoint; even more, their closures are isolated continua in $\Omega^\ast$. This means that there are disjoint neighborhoods of them. We are actually interested in a neighborhood of $\tilde{\gamma}^{\alpha, \beta}$ of the shape of a thin [*concavo-convex lens*]{} that we shall denote by $\tilde{\mathbb L}^{\alpha, \beta}$. By definition, $\tilde{\gamma}^{\alpha, \beta} \subset \tilde{\mathbb L}^{\alpha, \beta} \subset {\mathbb{Q}}^\alpha$. The construction of such lens goes as follows. Let $a$ and $b$ denote the endpoints of $\tilde{\gamma}^{\alpha, \beta}$, we assemble two circular arcs $\tilde{\gamma}_+^{\alpha, \beta}$ and $\tilde{\gamma}_-^{\alpha, \beta}$ with endpoints at $a$ and $b$ to form together with their endpoints a concavo-convex Jordan curve. This Jordan curve constitutes the boundary of a circular lens $\tilde{\mathbb L}^{\alpha, \beta}$. The term concavo-convex lens refers to the configuration in which $\tilde{\mathbb L}^{\alpha, \beta}$ lies in the concave side of the arc $\tilde{\gamma}_-^{\alpha, \beta}$ and convex side of $\tilde{\gamma}_+^{\alpha, \beta}$. It is clear that such lenses can be made arbitrarily thin so that $\tilde{\mathbb L}^{\alpha, \beta} \subset \tilde{\Omega}^\ast$ and the closures of $\tilde{\mathbb L}^{\alpha, \beta}$ will still be isolated continua in $\Omega^\ast$. From now on we fix the family $\{\tilde{\mathbb L}^{\alpha, \beta} \colon \alpha \ne \beta \}$ of such concavo-convex lenses associated with the arcs $\tilde{\gamma}^{\alpha, \beta}$. We then look at their preimages ${\mathbb{U}}^{\alpha, \beta} = h_2^{-1} ( \tilde{\mathbb L}^{\alpha, \beta})$ and the $\mathscr C^\infty$-smooth arcs $\Upsilon^{\alpha, \beta} =  h_2^{-1} ( \tilde{\gamma}^{\alpha, \beta})$. The endpoints of $\Upsilon^{\alpha, \beta}$ lie in $\partial {\mathbb{U}}^{\alpha, \beta}$. Moreover, $\Upsilon^{\alpha, \beta}$ splits ${\mathbb{U}}^{\alpha, \beta}$ into two disjoint subdomains ${\mathbb{U}}_+^{\alpha, \beta}$ and ${\mathbb{U}}_-^{\alpha, \beta}$ such that ${\mathbb{U}}^{\alpha, \beta} \setminus \Upsilon^{\alpha, \beta} = {\mathbb{U}}_+^{\alpha, \beta} \cup {\mathbb{U}}_-^{\alpha, \beta}$. Here we have a homeomorphism $h_2\colon {\mathbb{U}}^{\alpha, \beta} {\stackrel{{\rm \tiny{onto}}}{\longrightarrow}}\tilde{\mathbb L}^{\alpha, \beta} $ which is $\mathscr C^\infty$-diffeomorphism on $\overline{{\mathbb{U}}}_+^{\alpha, \beta}$ and $\mathscr C^\infty$-diffeomorphism on $\overline{{\mathbb{U}}}_-^{\alpha, \beta}$. Therefore, for some positive number $M_{\alpha, \beta}$, we have pointwise inequlities ${\lvertDh_2\rvert} \le {M_{\alpha, \beta}}$ and $\det Dh_2 \ge \frac{1}{M_{\alpha, \beta}}$ in both $\overline{{\mathbb{U}}}_+^{\alpha, \beta}$ and $\overline{{\mathbb{U}}}_-^{\alpha, \beta}$. Having established such a deformation of lenses and their preimages under $h_2$, we apply Corollary \[muncor\]. We infer that there is also a constant $M'_{\alpha, \beta}>0$ with the following property: to every neighborhood of $\Upsilon^{\alpha, \beta}$, say an open connected set ${\mathbb{U}}_\circ^{\alpha, \beta}\subset {\mathbb{U}}^{\alpha, \beta}$ that contains $\Upsilon^{\alpha, \beta}$, there corresponds a $\mathscr C^\infty$-diffeomorphism, denoted by $h_3 \colon {\mathbb{U}}^{\alpha, \beta} {\stackrel{{\rm \tiny{onto}}}{\longrightarrow}}\tilde{\mathbb L}^{\alpha, \beta}$, such that $$\begin{cases}
 h_3(z)=h_2(z) \quad \mbox{ for } z\in {\mathbb{U}}^{\alpha, \beta}\setminus {{\mathbb{U}}}_\circ^{\alpha, \beta}\\
 {\lvertDh_3\rvert} \le {M'_{\alpha, \beta}}\quad \mbox{ and } \quad \det Dh_3 \ge \frac{1}{M'_{\alpha, \beta}} \quad \mbox{in } {\mathbb{U}}^{\alpha, \beta}
 \end{cases}$$ We emphasize that $M'_{\alpha, \beta}$ is independent of the neighborhood ${{\mathbb{U}}}_\circ^{\alpha, \beta}$. We choose and fix ${{\mathbb{U}}}_\circ^{\alpha, \beta}$ thin enough to satisfy

-   $\overline{{\mathbb{U}}}_\circ^{\alpha, \beta} \subset {{\mathbb{U}}}^{\alpha, \beta} \cup \overline{\Upsilon}^{\alpha, \beta}$

-   ${\lvert{{\mathbb{U}}}_\circ^{\alpha, \beta}\rvert} \le [M_{\alpha, \beta}+M'_{\alpha, \beta}]^{-2} \epsilon^2\,  2^{-\alpha - \beta}$

-   $\sup_{{\mathbb{U}}^{\alpha, \beta}_\circ} {\lvert{Dh_2}\rvert} \le M_{\alpha, \beta}$

-   , we also assume that ${\lvert{{\mathbb{U}}}_\circ^{\alpha, \beta}\rvert} \le [M'_{\alpha, \beta}]^{-2}\delta^2\,  4^{-\alpha - \beta-1}$

Recall that $\delta$ was defined by  and later appeared in . This is certainly possible; for instance, take ${{\mathbb{U}}}_\circ^{\alpha, \beta}$ to be the preimage under $h_2$ of a sufficiently thin concavo-convex lens containing $\tilde{\gamma}^{\alpha, \beta}$. We call $h_3 \colon {\mathbb{U}}^{\alpha, \beta} {\stackrel{{\rm \tiny{onto}}}{\longrightarrow}}\tilde{\mathbb L}^{\alpha, \beta}$ a smoothing of $h_2 \colon {\mathbb{U}}^{\alpha, \beta} {\stackrel{{\rm \tiny{onto}}}{\longrightarrow}}\tilde{\mathbb L}^{\alpha, \beta}$ associated with a given arc $\Upsilon^{\alpha, \beta}= h_2^{-1}(\tilde{\gamma}^{\alpha, \beta})$. We now define a homeomorphism $h_3\colon \Omega{\stackrel{{\rm \tiny{onto}}}{\longrightarrow}}\Omega^*$ by the rule $$h_3=\begin{cases} \text{smoothing of $h_2$} \quad &\text{in ${\mathbb{U}}^{\alpha,\beta}$} \\
h_2 & \text{in $\Omega\setminus \bigcup_{\alpha,\beta}{\mathbb{U}}^{\alpha,\beta}$}
\end{cases}$$ It belongs to $h_2+{\mathscr{A}}_\circ (\Omega)$. Obviously $h_3$ is a $\mathscr C^{\infty}$-diffeomorphism in $\widetilde{\Omega}$. We have for every $z\in\Omega$ $$\begin{split}
{\lverth_3(z)-h_2(z)\rvert}&\le \begin{cases} \operatorname{diam}\widetilde{\mathbb L}^{\alpha,\beta} \quad &\text{for } z\in{\mathbb{U}}^{\alpha,\beta} \\
0 & \text{otherwise}
\end{cases} 
\\ &\le \operatorname{diam}{\mathbb{Q}}^\alpha \le {\epsilon}
\end{split}$$ see . Hence ${\lVerth_3-h_2\rVert}_{\mathscr C({\Omega})}\le{\epsilon}$. As regards the energy of $h_3-h_2$ we find that $$\label{step3chain}
\begin{split}
{\mathcal{E}}_\Omega[h_3-h_2] &= \sum_{\alpha,\beta} \int_{{{\mathbb{U}}}_\circ^{\alpha,\beta}}{\lvertDh_3-Dh_2\rvert}^2
\\
&\le \sum_{\alpha,\beta} {\lvert{{\mathbb{U}}}_\circ^{\alpha,\beta}\rvert}\sup_{{{\mathbb{U}}}_\circ^{\alpha,\beta}}
\left({\lvertDh_3\rvert}+{\lvertDh_2\rvert}\right)^2
\\
& \le \sum_{\alpha,\beta} {\lvert{{\mathbb{U}}}_\circ^{\alpha,\beta}\rvert} [M_{\alpha,\beta}'+M_{\alpha, \beta}]^2\le \sum_{\alpha, \beta =1}^\infty \frac{\epsilon^2}{2^{\alpha+ \beta}} \le {\epsilon^2} 
\end{split}$$ These estimates sum up to $${\lVerth_3-h_2\rVert}_{{\mathscr{A}}(\Omega)} \le {\epsilon}+ {\epsilon}= 2 \epsilon$$ Let us record for subsequent use the following estimate, obtained from  and . $$\begin{aligned}
\label{mes}
\sum_{v \in \mathcal V} \int_{h_3^{-1}(3 {\mathbb{D}}_\nu)} {\lvertDh_3\rvert}^2  &\le \sum_{v \in \mathcal V}\left( \int_{h_3^{-1}(3 {\mathbb{D}}_\nu) \setminus h_2^{-1}(3 {\mathbb{D}}_\nu)  } {\lvertDh_3\rvert}^2 +\int_{h_2^{-1}(3 {\mathbb{D}}_\nu)} {\lvertDh_3\rvert}^2  \right) \nonumber \\
\le \int_{\{h_3\ne h_2\}} {\lvertDh_3\rvert}^2 &+ 2\sum_{v \in \mathcal V} \left(\int_{h_2^{-1}(3{\mathbb{D}}_v)} {\lvertDh_3-Dh_2\rvert}^2 +   \int_{h_2^{-1}(3 {\mathbb{D}}_\nu)} {\lvertDh_2\rvert}^2    \right) \nonumber\\
&\le   \sum_{\alpha, \beta} \left(M'_{\alpha, \beta} \right)^2 {\lvert{\mathbb{U}}^{\alpha, \beta}_\circ\rvert} + 2\epsilon^2 + 2 \epsilon^2 \le 5 \epsilon^2\end{aligned}$$ For the energy of $h_3$, we observe that $$\begin{split}
{\lVertDh_3\rVert}_{\mathscr L^2(\Omega)} & \le {\lVertDh_3\rVert}_{\mathscr L^2(\Omega\setminus \cup {\mathbb{U}}_\circ^{\alpha, \beta})} + \sum_{\alpha, \beta} {\lVertDh_3\rVert}_{\mathscr L^2({\mathbb{U}}_\circ^{\alpha, \beta})} \\
&=   {\lVertDh_2\rVert}_{\mathscr L^2(\Omega\setminus \cup {\mathbb{U}}_\circ^{\alpha, \beta})} +  \sum_{\alpha, \beta} {\lVertDh_3\rVert}_{\mathscr L^2({\mathbb{U}}_\circ^{\alpha, \beta})} \\
&\le {\lVertDh_2\rVert}_{\mathscr L^2(\Omega)} + \sum_{\alpha, \beta} {\lvert{\mathbb{U}}_\circ^{\alpha, \beta}\rvert}^{1/2}  \sup_{{\mathbb{U}}^{\alpha, \beta}_\circ} {\lvertDh_3\rvert}\\
&\le   {\lVertDh\rVert}_{\mathscr L^2(\Omega)} - \delta + \frac{\delta}{2}  
\end{split}$$ Thus $$\label{lastadd}
{\lVertDh_3\rVert}_{\mathscr L^2(\Omega)} \le  {\lVertDh\rVert}_{\mathscr L^2(\Omega)} -  \frac{\delta}{2}$$

[**Step 4.**]{} We have already upgraded the mapping $h$ to a homeomorphism $h_3 \colon \Omega \to \Omega^\ast$ such that $h_3\in h+{\mathscr{A}}_\circ(\Omega)$ and $$\label{sstep4}
\begin{split}
{\lVerth_3-h\rVert}_{{\mathscr{A}}(\Omega)} & \le {\lVerth_3-h_2\rVert}_{{\mathscr{A}}(\Omega)} + {\lVerth_2-h_1\rVert}_{{\mathscr{A}}(\Omega)} + {\lVerth_1-h\rVert}_{{\mathscr{A}}(\Omega)}\\ & \le 2 \epsilon + 3 \epsilon + 2\epsilon = 7 \epsilon
\end{split}$$ Moreover, $h_3$ is a $\mathscr C^\infty$-diffeomorphism on $\Omega \setminus \bigcup_{v\in \mathcal V} h_2^{-1} (\overline{{\mathbb{D}}}_v)$. We now define a homeomorphism $h_4 \colon \Omega {\stackrel{{\rm \tiny{onto}}}{\longrightarrow}}\Omega^\ast$ by performing harmonic replacement of $h_3$ on each set $h_3^{-1}(2 {\mathbb{D}}_v)$. This gives us a $\mathscr C^\infty$-diffeomorphism $h_4 \colon h_3^{-1} (2 \overline{{\mathbb{D}}}_v)\to 2 \overline{{\mathbb{D}}}_v$, see Step 2 for details. For each $z\in \Omega$ $${\lverth_4(z)-h_3(z)\rvert} \le \begin{cases} 2 \operatorname{diam}{\mathbb{D}}_v \quad & \mbox{ if } z\in h_3^{-1} (2{\mathbb{D}}_v) \\
0 & \mbox{ otherwise}\end{cases} \quad \le 2{\epsilon}$$ Hence ${\lVerth_4-h_3\rVert}_{\mathscr C (\Omega)} \le 2{\epsilon}$. Using  we estimate the energy as follows. $$\begin{split}
{\mathcal{E}}_\Omega [h_4-h_3] &= \sum_{v\in \mathcal V} {\mathcal{E}}_{h_3^{-1}(2{\mathbb{D}}_v)}[h_4-h_3]\\
& \le 2 \sum_{v\in \mathcal V} \left( {\mathcal{E}}_{h_3^{-1}(2{\mathbb{D}}_v)}[h_4] + {\mathcal{E}}_{h_3^{-1}(2{\mathbb{D}}_v)}[h_3]  \right)\\
& \le 4  \sum_{v\in \mathcal V} {\mathcal{E}}_{h_3^{-1}(2{\mathbb{D}}_v)}[h_3] \le 20\epsilon^2
\end{split}$$ Thus, by  $$\label{sstep5} {\lVerth_4-h_3\rVert}_{{\mathscr{A}}(\Omega)} \le \epsilon + \sqrt{20} \epsilon \le 6 \epsilon$$ By virtue of Dirichlet’s principle and  we have $$\label{wearehungry}
{\lVertDh_4\rVert}_{\mathscr L^2(\Omega)} \le {\lVertDh_3\rVert}_{\mathscr L^2(\Omega)} \le {\lVertDh\rVert}_{\mathscr L^2(\Omega)}  - 
\frac{\delta}{2}$$

[**Step 5.**]{} The final step consists of smoothing $h_4$ in a neighborhood of each smooth Jordan curves $C_v= \partial h_3^{-1}(2{\mathbb{D}}_v)$. We proceed in much the same way as in Step 3, but we appeal to Corollary \[muncircor\] instead of Corollary \[muncor\]. By smoothing $h_4$ in a sufficiently thin neighborhood of each $C_v$ we obtain a $\mathscr C^{\infty}$-diffeomorphism $h_5 \colon \Omega {\stackrel{{\rm \tiny{onto}}}{\longrightarrow}}\Omega^\ast$, $h_5\in h_4 + {\mathscr{A}}_\circ (\Omega)$ such that $$\label{needsharpen}
{\lVerth_5-h_4\rVert}_{{\mathscr{A}}(\Omega)} \le {\epsilon}$$ We now recapitulate the estimates ,  and  to obtain a $\mathscr C^\infty$-diffeomorphism in $\Omega$ $$H:=h_5\in h+{\mathscr{A}}(\Omega)$$ such that $$\begin{split}
{\lVertH-h\rVert}_{{\mathscr{A}}(\Omega)} & \le {\lVerth_5-h_4\rVert}_{{\mathscr{A}}(\Omega)} +  {\lVerth_4-h_3\rVert}_{{\mathscr{A}}(\Omega)} +  {\lVerth_3-h\rVert}_{{\mathscr{A}}(\Omega)}\\
& \le \epsilon + 6 \epsilon + 7 \epsilon  =14 \epsilon     
\end{split}$$ which is as strong as (ii) in Theorem \[thmapprox\].\
To obtain the desired energy estimate ${\mathcal{E}}_\Omega [h_5] \le {\mathcal{E}}_\Omega [h]$, we need to sharpen the energy part in . By narrowing further the neighborhoods of $C_v$ we can be make the energy ${\mathcal{E}}_\Omega[h_5-h_4]$ as small as we wish; for example to obtain $${\lVertDh_5-Dh_4\rVert}_{\mathscr L^2 (\Omega)}< \frac{\delta}{2}$$ This is enough to conclude that $${\lVertDH\rVert}_{\mathscr L^2(\Omega)} \le {\lVertDh\rVert}_{\mathscr L^2(\Omega)}$$ because of .

Hopf differentials, Theorem \[thmhopf\]
=======================================

A quadratic differential on a domain $\Omega$ in the complex plane ${\mathbb{C}}$ takes the form $Q=F(z)\, {\textnormal d}z \otimes {\textnormal d}z$, where $F$ is a complex function on $\Omega$. Given a conformal change of the variable $z$, $z= \varphi(\xi)$, where $\varphi \colon \Omega' \to \Omega$, the pull back $$\varphi^\sharp (Q)= F\big(\varphi(\xi)\big)\, {\textnormal d}\varphi \otimes {\textnormal d}\varphi = F \big( \varphi (\xi) \big) \dot{\varphi}^2 (\xi) \, {\textnormal d}\xi \otimes {\textnormal d}\xi$$ defines a quadratic differential on $\Omega'$. It is plain that for a complex harmonic function $h \colon \Omega \to {\mathbb{C}}$ the associated Hopf differential $$Q_h= h_z \overline{h_{\bar z}}\,  {\textnormal d}z \otimes {\textnormal d}z$$ is holomorphic, meaning that $$\label{heq0}
\frac{\partial}{\partial \bar z} \left(h_z \overline{h_{\bar z}} \right)=0$$ Conversely, if a Hopf differential $Q_h=h_z \overline{h_{\bar z}}\,  {\textnormal d}z \otimes {\textnormal d}z$ is holomorphic for some $\mathscr C^1$-mapping $h$, then $h$ is harmonic at the points where the Jacobian determinant $J(z,h):=\det Dh= {\lverth_z\rvert}^2- {\lverth_{\bar z}\rvert}^2 \ne 0$, see [@EL1 10.5] and our Remark \[notredable\]. Here the assumption that $J(z,h) \ne 0$ is critical. Let us illustrate it by the following.

\[ex\] Consider a mapping $h\in \mathscr C^{1,1} ({\mathbb{C}}_\circ)$ defined on the punctured plane ${\mathbb{C}}_\circ = {\mathbb{C}}\setminus \{0\}$ by the rule $$\label{heq1}
h(z)= \begin{cases} \frac{z}{|z|}& \mbox{ for } 0< {\lvertz\rvert} \le 1 \\
\frac{1}{2} \left(z+ \frac{1}{\bar z}\right)\qquad  &  \mbox{ for } 1 \le{\lvertz\rvert} < \infty \end{cases}$$ Direct computation shows that $$h_z (z)= \begin{cases} \frac{1}{2}|z|^{-1}& \mbox{ for } 0< {\lvertz\rvert} \le 1 \\
\frac{1}{2} \qquad  &  \mbox{ for } 1 \le{\lvertz\rvert} < \infty \end{cases}$$ and $$h_{\bar z} (z)= \begin{cases} -\frac{1}{2}|z| {\bar z}^{-2}& \mbox{ for } 0< {\lvertz\rvert} \le 1 \\
-\frac{1}{2}\bar z^{-2} \qquad  &  \mbox{ for } 1 \le{\lvertz\rvert} < \infty \end{cases}$$ Thus $$\label{tristars}
Q_h = - \frac{{\textnormal d}z \otimes {\textnormal d}z}{4z^2} \qquad \mbox{ in } {\mathbb{C}}_\circ$$ It may be worth mentioning that the mapping $h$ in  is the unique (up to rotation of $z$) minimizer of the Dirichlet energy $$\mathcal E [H]= \int_{{\mathbb{A}}} {\lvertDH\rvert}^2$$ over the annulus ${\mathbb{A}}= A(r,R)=\{z \colon r< {\lvertz\rvert} < R  \}$, $0<r<1<R$, subject to all weak limits of homeomorphisms $H \colon {\mathbb{A}}{\stackrel{{\rm \tiny{onto}}}{\longrightarrow}}A(1, R_\ast)$, where $R_\ast = \frac{1}{2} \left(R + \frac{1}{R}\right)$, see [@AIM]. Note that the Hopf differential of  is real along the boundary circles of ${\mathbb{A}}$. The concentric circles are horizontal trajectories of $Q_h$. In fact this is a general property of minimizers [@Job Lemma 1.2.5]. The general pattern is that with the loss of injectivity comes the loss of the Lagrange-Euler equation for the extremal mapping.

Properties of the function $h$ with holomorphic Hopf differential $Q= h_z \overline{h_{\bar z}} \, {\textnormal d}z \otimes {\textnormal d}z$ are of interest in the studies of harmonic mappings [@EL2; @EL3; @Job; @Sc; @Se], minimal surfaces [@DHKWb; @Stb] and Teichmüller theory [@GLb]. In this section we prove Theorem \[thmhopf\] which imposes fairly minimal assumptions that imply harmonicity of $\mathscr W^{1,2}$-solution to the equation . Some elements of the proof go back to [@RS; @RW].

As a consequence of the Stoilow factorization theorem [@AIM p. 56] the branch set of $h$ is discrete, hence removable for continuous harmonic functions. Thus we assume that $h \colon \Omega {\stackrel{{\rm \tiny{onto}}}{\longrightarrow}}\Omega^\ast$ is a homeomorphism of Sobolev class ${\mathscr{W}}^{1,2}_{\operatorname{loc}} (\Omega, \Omega^\ast)$ such that $$h_z \overline{h_{\bar z}} =F(z) \quad \mbox{ is holomorphic in } \; \Omega$$ By virtue of Theorem \[thmapprox\], there exists a sequence of diffeomorphisms $h^j \colon \Omega {\stackrel{{\rm \tiny{onto}}}{\longrightarrow}}\Omega^\ast$ converging $c$-uniformly and strongly in ${\mathscr{W}}^{1,2}_{\operatorname{loc}} (\Omega, \Omega^\ast)$ to $h$. Denote by $$h_z^j h^j_{\bar z}=:F^j \in \mathscr L_{\operatorname{loc}}^1(\Omega)$$ Thus $F^j \to F$ strongly in $ \mathscr L_{\operatorname{loc}}^1(\Omega)$. Let us first dispose of an easy case.\
[**Case 0.**]{} [*The homogeneous equation $F \equiv 0$*]{}. Since $h^j$ are diffeomorphisms the Jacobian determinant $J(z,h^j)= {\lverth_z^j\rvert}^2-{\lverth_{\bar z}^j\rvert}^2$ is either positive everywhere in $\Omega$ or negative everywhere in $\Omega$. Let us settle the case when $J(z,h^j)>0$ for infinitely many indices $j=1,2,\dots$. For such $j$ we have ${\lverth^j_z\rvert}>{\lverth^j_{\bar z}\rvert}$, which yields ${\lverth^j_{\bar z}\rvert}^2\le {\lverth^j_z h^j_{\bar z}\rvert}$. Passing to the $\mathscr L^1$-limit we obtain $${\lverth_{\bar z}\rvert}^2\le {\lverth_z h_{\bar z}\rvert} = {\lvertF(z)\rvert}\equiv 0.$$ Thus $h$ is holomorphic, by Weyl’s lemma. Similarly, in case $J(z,h^j)<0$ for infinitely many indices $j=1,2,\dots$, we find that $h$ is antiholomorphic.

\[rem\] We observe, based on the above arguments, that for this homogeneous equation $h_z\overline{h_{\bar z}}\equiv 0$ every solution $h\in {\mathscr{W}}_{\operatorname{loc}}^{1,2}(\Omega)$ obtained as the weak ${\mathscr{W}}^{1,2}$-limit of homeomorphisms is either holomorphic or antiholomorphic. The situation is dramatically different if $h_z\overline{h_{\bar z}}\not\equiv 0$; some topological assumption on $h$ are necessary, as illustrated in Example \[ex\].

[**Case 1.**]{} [*Nonhomogeneous equation $F\not\equiv 0$*]{}. The function $F$, being holomorphic, may vanish only at isolated points. Since isolated points are removable for bounded harmonic functions, it suffices to consider the set where $F\ne 0$. Proceeding further in this direction, we may and do assume that $F(z)\equiv 1$ (by a conformal change of the $z$-variable) and $h$ is a ${\mathscr{W}}^{1,2}$-homeomorphism in the closure of the unit square ${\mathbb{Q}}=\{x+iy\colon 0<x<1, 0<y<1\}$. The problem now reduces to establishing that the equation $$\label{eeq3}
h_{z}\overline{h_{\bar z}}\equiv 1$$ implies $\Delta h=0$. This will be proved indirectly by means of the energy-minimizing property $$\label{ineq4}
\mathcal E_{{\mathbb{Q}}}[h]\le \mathcal E_{{\mathbb{Q}}}[H]$$ where $H\colon {\mathbb{Q}}\to h({\mathbb{Q}})$ is any homeomorphism in $h+{\mathscr{A}}_\circ({\mathbb{Q}})$; in particular, $H=h$ on $\partial{\mathbb{Q}}$. Indeed, if $h$ were not harmonic, we would be able to decrease its energy by harmonic replacement (Propositions \[prpoisson\] and \[RKC\]), contradicting .

Proof of the inequality 
------------------------

With the aid of the approximation theorem we need only prove  for mappings $H\in h+ {\mathscr{A}}_\circ({\mathbb{Q}})$ that are diffeomorphisms on ${\mathbb{Q}}$. From now on we assume that this is the case. Denote ${\mathbb{Q}}^*=h({\mathbb{Q}})=H({\mathbb{Q}})$. We consider a sequence $h^j\in h+{\mathscr{A}}_\circ({\mathbb{Q}})$ of diffeomorphisms $h^j\colon {\mathbb{Q}}{\stackrel{{\rm \tiny{onto}}}{\longrightarrow}}{\mathbb{Q}}^*$ converging in ${\mathscr{A}}({\mathbb{Q}})$ to $h$. Moreover we may also assume that $Dh^j\to Dh$ almost everywhere in ${\mathbb{Q}}$ by passing to a subsequence if necessary. Now the sequence $\chi^j\colon\overline{{\mathbb{Q}}}\to\overline{{\mathbb{Q}}}$ of self-homeomorphisms of the closed unit disk given by $\chi^j=H^{-1}\circ h^j$, where $\chi^j=\operatorname{id}$ on $\partial{\mathbb{Q}}$, is converging uniformly on $\overline{{\mathbb{Q}}}$ to $\chi=H^{-1}\circ h$. It is important to observe that $\chi\in {\mathscr{W}}_{\operatorname{loc}}^{1,2}({\mathbb{Q}})$ and $\chi^j$ converges to $\chi$ in ${\mathscr{W}}^{1,2}({\mathbb{Q}}')$ on any compactly contained subdomain ${\mathbb{Q}}'\Subset {\mathbb{Q}}$. Since $h^j$ and $(\chi^j)^{-1}$ are diffeomorphisms on ${\mathbb{Q}}'$ and $\chi^j({\mathbb{Q}}')$, respectively, the chain rule can be applied to the composition $H=h^j\circ (\chi^j)^{-1}$. For $w\in \chi^j({\mathbb{Q}}')$ we have $$\begin{split}
\frac{\partial H(w)}{\partial w} &= h_z^j(z)\frac{\partial (\chi^j)^{-1}}{\partial w}+h_{\bar z}^j(z)\frac{\partial (\chi^j)^{-1}}{\partial \bar w} \\
\frac{\partial H(w)}{\partial \bar w} &= h_z^j(z)\frac{\partial (\chi^j)^{-1}}{\partial \bar w}+h_{\bar z}^j(z)\overline{\frac{\partial (\chi^j)^{-1}}{\partial w}}
\end{split}$$ where $z=(\chi^j)^{-1}(w)$.

The partial derivatives of $(\chi^j)^{-1}$ at $w$ can be expressed in terms of $\chi_{_z}^j(z)$ and $\chi_{\bar z}^j(z)$ by the rules $$\begin{split}
\frac{\partial (\chi^j)^{-1}}{\partial w} &= \frac{\chi_z^j(z)}{J(z,\chi^j)} \\
\frac{\partial (\chi^j)^{-1}}{\partial \bar w} &= - \frac{\chi_{\bar z}^j(z)}{J(z,\chi^j)}
\end{split}$$ where the Jacobian determinant $J(z,\chi^j)$ is strictly positive. This yields $$\begin{split}
\frac{\partial H}{\partial w} & =\frac{h_z^j\overline{\chi_z^j}-h_{\bar z}^j\overline{\chi_{\bar z}^j}}{J(z,\chi^j)} \\
\frac{\partial H}{\partial \bar w} & =\frac{h_{\bar z}^j{\chi_z^j}-h_{\bar z}^j{\chi_{\bar z}^j}}{J(z,\chi^j)}
\end{split}$$ We compute the energy integral of $H$ over the set $\chi^j({\mathbb{Q}}')$ by substitution $w=\chi^j(z)$, $$\begin{split}
\mathcal E_{\chi^j({\mathbb{Q}}')}[H]&=2\int_{\chi^j({\mathbb{Q}}')} \left({\lvertH_w\rvert}^2+{\lvertH_{\bar w}\rvert}^2\right)\,{\textnormal d}w \\
&=2\int_{{\mathbb{Q}}'}\frac{{\lverth_z^j \overline{\chi_z^j} - h_{\bar z}^j \overline{\chi_{\bar z}^j}\rvert}^2+
{\lverth_{\bar z}^j \chi_z^j - h_{z}^j \chi_{z}^j\rvert}^2}{{\lvert\chi_z^j\rvert}^2-{\lvert\chi_{\bar z}^j\rvert}^2}\,{\textnormal d}z
\end{split}$$ On the other hand, the energy of $h^j$ over the set ${\mathbb{Q}}'$ is $$\mathcal E_{{\mathbb{Q}}'}[h^j]=2\int_{{\mathbb{Q}}'}\left({\lverth_z^j\rvert}^2+{\lverth_{\bar z}^j\rvert}^2\right)\,{\textnormal d}z$$ Subtract these two integrals to obtain $$\label{cchain}
\begin{split}
\mathcal E_{{\mathbb{Q}}}[H]-\mathcal E_{{\mathbb{Q}}'}[h^j] &\ge \mathcal E_{\chi^j({\mathbb{Q}}')}[H]-\mathcal E_{{\mathbb{Q}}'}[h^j] \\
&=  4\int_{{\mathbb{Q}}'}  \frac{\left({\lverth_z^j\rvert}^2+{\lverth_{\bar z}^j\rvert}^2\right)\cdot {\lvert\chi_{\bar z}^j\rvert}^2
-2\operatorname{Re}\left[h_z^j\overline{h_{\bar z}^j} \overline{\chi_z^j}\chi_{\bar z}^j\right]}{{\lvert\chi_z^j\rvert}^2-{\lvert\chi_{\bar z}^j\rvert}^2}\,{\textnormal d}z \\
& \ge 4 \int_{{\mathbb{Q}}'} \frac{ 2{\lverth_z^jh_{\bar z}^j\rvert} {\lvert\chi_{\bar z}^j\rvert}^2
-2\operatorname{Re}\left[h_z^j\overline{h_{\bar z}^j} \overline{\chi_z^j}\chi_{\bar z}^j\right]}{{\lvert\chi_z^j\rvert}^2-{\lvert\chi_{\bar z}^j\rvert}^2}\,{\textnormal d}z \\
& = 4 \int_{{\mathbb{Q}}'} \left[\frac{{\lvert\chi_z^j-\sigma^j(z) \chi_{\bar z}^j\rvert}^2}{{\lvert\chi_z^j\rvert}^2-{\lvert\chi_{\bar z}^j\rvert}^2} -1 \right]
\, {\lverth_z^jh_{\bar z}^j\rvert} \,{\textnormal d}z
\end{split}$$ where we have introduced the notation $$\sigma^j = \sigma^j(z) = \begin{cases}
{h_z^j\overline{h_{\bar z}^j}}{\, {\lverth_z^jh_{\bar z}^j\rvert}^{-1}} \qquad &\text{if } h_z^jh_{\bar z}^j\ne 0 \\
1 & \text{otherwise.}
 \end{cases}$$ Note that ${\lvert\sigma^j\rvert}=1$ and $\sigma^j\to 1$ almost everywhere.

Upon using Hölder’s inequality we continue the chain  as follows. $$\label{cchain2}
\ge \; 4\frac{\left[\int_{{\mathbb{Q}}'}\left| \chi_z^j-\sigma^j\chi_{\bar z}^j\right| \,\sqrt{{\lverth_z^jh_{\bar z}^j\rvert}}\,{\textnormal d}z\right]^2}{\int_{{\mathbb{Q}}'} J(z,h^j)\,{\textnormal d}z}
-4\int_{{\mathbb{Q}}'} {\lverth_z^jh_{\bar z}^j\rvert}.$$ The denominator in  is at most $1$ because $$\int_{{\mathbb{Q}}'} J(z,h^j)\,{\textnormal d}z = {\lvert\chi^j({\mathbb{Q}}')\rvert}\le{\lvert{\mathbb{Q}}\rvert}=1.$$ Therefore, $$\mathcal E_{{\mathbb{Q}}}[H]-\mathcal E_{{\mathbb{Q}}'}[h^j]\ge 4\left[\int_{{\mathbb{Q}}'}\left| \chi_z^j-\sigma^j\chi_{\bar z}^j\right|\,\sqrt{{\lverth_z^jh_{\bar z}^j\rvert}}\,{\textnormal d}z  \right]^2 -4\int_{{\mathbb{Q}}'}{\lverth_z^jh_{\bar z}^j\rvert}\, {\textnormal d}z.$$ It is at this point that we can pass to the limit as $j\to\infty$, to obtain $$\label{cchain3}
\mathcal E_{{\mathbb{Q}}}[H]-\mathcal E_{{\mathbb{Q}}'}[h]\ge 4\left[\int_{{\mathbb{Q}}'}{\lvert\chi_z-\chi_{\bar z}\rvert}\, {\textnormal d}z \right]^2-4{\lvert{\mathbb{Q}}'\rvert}.$$ Since ${\mathbb{Q}}'$ was an arbitrary compactly contained subdomain of ${\mathbb{Q}}$, the estimate  remains valid with ${\mathbb{Q}}'$ replaced by ${\mathbb{Q}}$. $$\label{cchain4}
\begin{split}
\mathcal E_{{\mathbb{Q}}}[H]-\mathcal E_{{\mathbb{Q}}}[h] &\ge 4\left[\int_{{\mathbb{Q}}}\left|\frac{\partial\chi}{\partial y}\right|\,{\textnormal d}x\,{\textnormal d}y\right]^2-4  \\
&\ge 4\int_0^1 \left|\int_0^1 \frac{\partial\chi (x,y)}{\partial y}\,{\textnormal d}y\right|\,{\textnormal d}x - 4 \\
&= 4 \int_0^1 {\lvert\chi(x,1)-\chi(x,0)\rvert}\,{\textnormal d}x-4 =4-4 =0
\end{split}$$ as desired.

\[notredable\] When specialized to the case $h \in \mathscr C^1$, Theorem \[thmhopf\] shows that $h$ is harmonic outside of the zero set of its Jacobian.

Auxiliary smoothing results
===========================

Here we present some results concerning smoothing of piecewise differentiable planar homeomorphisms. They can be found in [@Mu] in greater generality, but since we require quantitative control of derivatives, a self-contained proof is in order. Here it is more convenient to use the operator norm of a matrix, denoted by ${\lVert\cdot\rVert}$. Note that ${\lVertA\rVert} \le {\lvertA\rvert} \le 2 {\lVertA\rVert}$ for $2 \times 2$-matrices.

\[mun\] Let ${\mathbb{U}}\subset{\mathbb{R}}^2$ be a domain containing an open segment $I$ with endpoints on the boundary $\partial {\mathbb{U}}$ which splits ${\mathbb{U}}$ into two subdomains ${\mathbb{U}}_1$ and ${\mathbb{U}}_2$ such that ${\mathbb{U}}\setminus I= {\mathbb{U}}_1\cup {\mathbb{U}}_2$. Suppose that $f\colon \overline{{\mathbb{U}}}{\stackrel{{\rm \tiny{onto}}}{\longrightarrow}}\overline{{\mathbb{U}}^*}\subset{\mathbb{R}}^2$ is a homeomorphism with the following properties:

(i) \[mun2a\] For $j=1,2, \dots$ the restriction of $f$ to $\overline{{\mathbb{U}}_j} $ is $\mathscr C^{\infty}$-smooth, equals the identity on $I$;

(ii) \[mun2\] There is a constant $M>0$ such that for $j=1,2$ the restriction of $f$ to $\overline{{\mathbb{U}}_j}$ satisfies ${\lVertDf\rVert}\le M$ and $\det Df\ge M^{-1}$.

Then for any open set ${\mathbb{U}}_\circ$ with $I\subset {\mathbb{U}}_\circ\subset {\mathbb{U}}$ there is a $\mathscr C^{\infty}$-diffeomorphism $g\colon {\mathbb{U}}\to {\mathbb{U}}^*$ such that

-   $g$ agrees with $f$ on ${\mathbb{U}}\setminus {\mathbb{U}}_\circ$ (and also on $I$);

-   ${\lVertDg\rVert}\le 20M$ and $\det Dg\ge (20M)^{-1}$ on ${\mathbb{U}}$.

Without loss of generality $I\subset {\mathbb{R}}=\{(x,y) \colon y=0\}$. We write $f$ in components as $(u,v)$ where $u$ and $v$ are functions of $x$ and $y$. Let us introduce a notation; given any $\mathscr C^{\infty}$-smooth function $\beta\colon{\mathbb{R}}\to[0,\infty)$, denote $V(\beta)=\{(x,y)\in {\mathbb{R}}^2\colon {\lverty\rvert}<\beta(x)\}$. We can and do choose $\beta$ so that $I\subset V(\beta)\subset {\mathbb{U}}_\circ$, and further scale it down until the following holds. $$\label{beta}
\begin{split}
{\lvert\beta'(x)\rvert} &\le\frac{1}{40M} \qquad \mbox{ for all } x\in {\mathbb{R}}; \\
{\lvertv_x\rvert}&\le \frac{1}{50M^2} \qquad \text{in }V(\beta)\setminus I, \quad \mbox{ because } v(x,0)=0; \\
{\lvertu_x-1\rvert}&\le \frac{1}{10} \qquad \quad \;  \text{ in }V(\beta)\setminus I, \quad \mbox{ because } u(x,0)=x.
\end{split}$$ As a consequence of   and , $$\label{beta1}
v_y \ge \frac{M^{-1}-{\lvertu_y v_x\rvert}}{u_x}\ge \frac{1}{2M}.$$ Since $v$ is also $M$-Lipschitz by , the following double inequality holds in $V(\beta)\setminus I$. $$\label{vest}
\frac{1}{2M} \le \frac{v}{y}\le M.$$

Let us fix be a nondecreasing $\mathscr C^{\infty}$ function $\alpha\colon{\mathbb{R}}\to{\mathbb{R}}$ such that $\alpha(t)=0$ for $t\le 1/3$. Let $\alpha(t)=1$ for $t\ge 2/3$. Moreover, $\alpha'(t)\le 4$ for all $t\in{\mathbb{R}}$ and $\alpha(\infty)=1$, by convention. Now we introduce a modification of $u$ on ${\mathbb{U}}$ by setting $$\tilde u := \alpha(t)u+(1-\alpha(t))x \qquad \text{where } t=\begin{cases}\frac{{\lverty\rvert}}{\beta(x)} \quad & \mbox{ if } \beta(x) \ne 0 \\
\infty & \mbox{ otherwise} \end{cases}.$$ Note that $\tilde u=u$ outside of $V(\beta)$. In $V(\beta)\setminus I$ we compute the derivatives as follows. $$\label{d1}
\begin{split}
\tilde u_x &= - t^2  \alpha'(t) \beta'(x) \frac{u-x}{{\lverty\rvert}} + \alpha(t)u_x+1-\alpha(t) \\
\tilde u_y &= t\alpha'(t)\frac{u-x}{y} + \alpha(t) u_y
\end{split}$$ Since $u$ is $M$-Lipschitz by , we have ${\lvertu-x\rvert}\le M{\lverty\rvert}$. From this,  and  we obtain $$\label{d2}
\frac{8}{10} \le \tilde u_x\le \frac{12}{10},
\quad \text{ and } \quad {\lvert\tilde u_y\rvert}\le 5M,$$ which combined with  yields $$\label{d3}
\tilde u_x v_y-\tilde u_y v_x \ge \frac{8}{10}\frac{1}{2M}- \frac{5M}{50M^2}= \frac{3}{10 M}.$$

Next we modify $v$ on ${\mathbb{U}}$. Specifically, $$\tilde v := \alpha(s) v+(1-\alpha(s))\frac{y}{2M} \qquad \text{where } s= \begin{cases} \frac{3{\lverty\rvert}}{\beta(x)} \quad & \mbox{ if } \beta(x) \ne 0 \\
\infty& \mbox{ otherwise}  \end{cases}$$ Note that $\tilde{v}=v$ outside of $V(\beta/3)$, and on the set $V(\beta/3)$ we already have $\tilde{u} \equiv x$.

Computations similar to  yield (on the set $V(\beta/3)\setminus I$) $$\label{d1v}
\begin{split}
\tilde v_x &= -\frac{1}{3}\alpha'(s) s^2 \frac{v-y}{{\lverty\rvert}} + \alpha(s)v_x; \\
\tilde v_y &= \frac{s\alpha'(s)}{y}\left(v-\frac{y}{2M}\right) + \alpha(s)v_y+\frac{1-\alpha(s)}{2M}.
\end{split}$$ Straightforward estimates based on ,  and  comply $$\label{d4}
\begin{split}
{\lvert\tilde v_x\rvert}& \le \frac{4 M}{3}+\frac{1}{50M^2}<\frac{3M}{2}, \\
\frac{1}{2M}& \le \tilde v_y \le  5M.
\end{split}$$

It remains to check that the mapping $g:=(\tilde u,\tilde v)$, which agrees with $f$ outside of $V(\beta)$, satisfies all the requirements. As regards $\mathscr C^\infty$-smoothness we need only check it on $V(\beta/9)$. But in this neighborhood of $I$ we have a linear mapping, $g(x,y)= \left(x, \frac{y}{2M}\right)$, so $\mathscr C^\infty$-smooth. By virtue of  and  we have ${\lVertDg\rVert}\le 20M$. The desired lower bound for $\det Dg$ follows from  and . Consequently, $g$ is a local diffeomorphism, and since it agrees with $f$ on $\partial V(\beta)$, it is in fact a diffeomorphism, by a topological result: [*a local homeomorphism which shares boundary values with a homeomorphism is injective*]{} [@Mu Lemma 8.2].

We also need a polar version of Proposition \[mun\].

\[muncir\] Let ${\mathbb{U}}\subset{\mathbb{R}}^2$ be a domain containing a circle ${\mathbb{T}}$. Suppose that $f\colon {\mathbb{U}}{\stackrel{{\rm \tiny{onto}}}{\longrightarrow}}{\mathbb{U}}^*\subset{\mathbb{R}}^2$ is a homeomorphism with the following properties:

(i) \[munc1\] The restriction of $f$ to ${\mathbb{T}}$ is the identity mapping;

(ii) \[munc2\] There is a constant $M>0$ such that the restriction of $f$ to either component of ${\mathbb{U}}\setminus {\mathbb{T}}$ is $\mathscr C^{\infty}$-smooth with ${\lVertDf\rVert}\le M$ and $\det Df\ge M^{-1}$.

Then for any open set $W$ with ${\mathbb{T}}\subset W\subset {\mathbb{U}}$ there is a $\mathscr C^{\infty}$-diffeomorphism $g\colon {\mathbb{U}}\to {\mathbb{U}}^*$ such that

-   $g$ agrees with $f$ on ${\mathbb{U}}\setminus W$ and on ${\mathbb{T}}$;

-   ${\lVertDg\rVert}\le 80M$ and $\det Dg\ge (80M)^{-1}$ on ${\mathbb{U}}$.

It is convenient to identify ${\mathbb{R}}^2$ with ${\mathbb{C}}$. Without loss of generality ${\mathbb{T}}=\{z\in{\mathbb{C}}\colon {\lvertz\rvert}=1\}$. Let $\psi(\zeta)=\exp(i\zeta)$. The mapping $F=\psi^{-1}\circ f\circ \psi$ is well-defined in some open horizontal strip $S_h=\{z\in{\mathbb{C}}\colon {\lvert\operatorname{Im}z\rvert}<\epsilon \}$ which we choose thin enough so that $\psi(S_\epsilon)\subset W$ and ${\lvert\psi'\rvert}^2< e^{2\epsilon }\le 2$. Note that $F$ is $2\pi$-periodic and satisfies $${\lVertDF\rVert}\le 2M \quad \text{ and }\quad \det DF\ge (2M)^{-1}.$$

The proof of Proposition \[mun\] applies to $F$ with no changes other than one simplification: $\beta>0$ is now a small positive constant rather than a function. Thus we obtain a diffeomorphism $G$ which agrees with $F$ on ${\mathbb{R}}\cup (S\setminus V(\beta))$ and satisfies ${\lVertDG\rVert}\le 40$ and $\det DG\ge (40M)^{-1}$. Since $F$ was $2\pi$-periodic, so is $G$. Thus, $g:=\psi\circ G\circ \psi^{-1}$ is the desired diffeomorphism.

Our applications require slightly more general versions of Proposition \[mun\] and Corollary \[muncir\], where the separating curve is allowed to have other shapes and $f$ is not required to agree with the identity on the curve.

\[nice\] A parametric curve $\Gamma \colon (0,1) \to {\mathbb{R}}^2$ is [*regular*]{} if $\Gamma$ extends to a bigger interval $(a,b)\supset [0,1]$ so that the extended mapping is a $\mathscr C^\infty$-diffeomorphism onto its image.

Note that a regular curve $\Gamma$ has well-defined endpoints $\Gamma(0)$ and $\Gamma(1)$. Also, $\Gamma$ extends to an injective $\mathscr C^\infty$-mapping $\Phi\colon (0,1)\times (-1,1)\to{\mathbb{R}}^2$ such that ${\lVertD\Phi\rVert}$ and ${\lVert(D\Phi)^{-1}\rVert}$ are bounded. This follows from the existence of a tubular neighborhood of the image of $\Gamma$ [@MRb Theorem 4.26].

Corollaries \[muncor\] and \[muncircor\], given below, generalize Proposition \[mun\] and Corollary \[muncir\] respectively.

\[muncor\] Let ${\mathbb{U}}\subset {\mathbb{R}}^2$ be a domain containing the image of a regular arc $\Gamma$ with endpoints on the boundary $\partial {\mathbb{U}}$ which divides ${\mathbb{U}}$ into two subdomains ${\mathbb{U}}_1$ and ${\mathbb{U}}_2$ such that ${\mathbb{U}}\setminus \Gamma={\mathbb{U}}_1 \cup {\mathbb{U}}_2$. Suppose $f \colon \overline{{\mathbb{U}}} \to \overline{{\mathbb{U}}^\ast} \subset {\mathbb{R}}^2$ is a homeomorphism such that $f \circ \Gamma$ is also regular and the restriction of $f$ to each $\overline{{\mathbb{U}}_i}$ is $\mathscr C^\infty$-smooth and satisfies $${\lvertDf(z)\rvert} \le M, \quad \det Df(z) \ge \frac{1}{M} \quad \mbox{for } z\in {\mathbb{U}}_i$$ where $M$ is a positive constant. Then there is a constant $M'>0$ such that to every open set ${\mathbb{U}}' \subset {\mathbb{U}}$ with $\Gamma \subset {\mathbb{U}}'$ there corresponds a $\mathscr C^\infty$-diffeomorphism $g\colon {\mathbb{U}}{\stackrel{{\rm \tiny{onto}}}{\longrightarrow}}{\mathbb{U}}^\ast$ with the following properties

-   $g(z)=f(z)$ for $z\in {\mathbb{U}}\setminus {\mathbb{U}}'$ (and also on $\Gamma$)

-   ${\lvertDg(z)\rvert} \le M'$ and $\det Dg(z) \ge \frac{1}{M'}$ on ${\mathbb{U}}$.

Let ${\mathbb{Q}}=(0,1)\times (-1,1)$. Let $\Phi$ and $\Psi$ be the extensions of $\Gamma$ and $f\circ \Gamma$ to ${\mathbb{Q}}$ as in Definition \[nice\]. There is a domain $\widetilde{{\mathbb{U}}}$ such that $(0,1)\times \{0\}\subset \widetilde{{\mathbb{U}}} \subset {\mathbb{Q}}$, $\Phi(\widetilde{{\mathbb{U}}})\Subset {\mathbb{U}}'$, and the composition $F:=\Psi^{-1}\circ f\circ \Phi$ is defined in $\widetilde{{\mathbb{U}}}$. Note that $F=\operatorname{id}$ on $(0,1)\times \{0\}$. We apply Proposition \[mun\] (with $\widetilde{{\mathbb{U}}}$ in place of ${\mathbb{U}}$ and with $F$ in place of $f$) and obtain a $\mathscr C^\infty$-diffeomorphism $G \colon \widetilde{{\mathbb{U}}} \to F(\widetilde{{\mathbb{U}}})$. Finally, replace $F$ within $\widetilde{{\mathbb{U}}}$ with the diffeomorphism $g=\Psi\circ G\circ \Phi^{-1}$.

\[muncircor\] Let ${\mathbb{U}}\subset {\mathbb{R}}^2$ be a domain containing the image of a $\mathscr C^\infty$-smooth Jordan curve $\Gamma$ which divides ${\mathbb{U}}$ into two subdomains ${\mathbb{U}}_1$ and ${\mathbb{U}}_2$ such that ${\mathbb{U}}\setminus \Gamma={\mathbb{U}}_1 \cup {\mathbb{U}}_2$. Suppose $f \colon \overline{{\mathbb{U}}} \to \overline{{\mathbb{U}}^\ast} \subset {\mathbb{R}}^2$ is a homeomorphism such that the restriction of $f$ to each $\overline{{\mathbb{U}}_i}$ is $\mathscr C^\infty$-smooth and satisfies $${\lvertDf(z)\rvert} \le M, \quad \det Df(z) \ge \frac{1}{M} \quad \mbox{for } z\in {\mathbb{U}}_i$$ where $M$ is a positive constant. Then there is a constant $M'>0$ such that to every open set ${\mathbb{U}}' \subset {\mathbb{U}}$ with $\Gamma \subset {\mathbb{U}}'$ there corresponds a $\mathscr C^\infty$-diffeomorphism $g\colon {\mathbb{U}}{\stackrel{{\rm \tiny{onto}}}{\longrightarrow}}{\mathbb{U}}^\ast$ with the following properties

-   $g(z)=f(z)$ for $z\in {\mathbb{U}}\setminus {\mathbb{U}}'$ (and also on $\Gamma$)

-   ${\lvertDg(z)\rvert} \le M'$ and $\det Dg(z) \ge \frac{1}{M'}$ on ${\mathbb{U}}$.

The proof of Corollary \[muncor\] is easily adapted to this case.

Concluding remarks
==================

One may wonder whether the proof of Theorem \[thmapprox\] can be extended to the spaces ${\mathscr{W}}^{1,p}$, $1<p<\infty$, by means of the $p$-harmonic replacement in place of Proposition \[RKC\]. Indeed, $p$-harmonic mappings are $\mathscr C^{1, \alpha}$-smooth [@Ur]. However, the injectivity of $p$-harmonic replacement of a homeomorphism is unclear.

Is there a version of the Radó-Kneser-Choquet theorem for $p$-harmonic mappings? That is, does the $p$-harmonic extension of a homeomorphism onto a convex Jordan curve enjoy the injectivity property?

An attempt to extend Theorem \[thmapprox\] to higher dimensions faces another obstacle: the Radó-Kneser-Choquet theorem fails in dimensions $n\ge 3$ as was proved by Laugesen [@La].

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[^1]: Iwaniec was supported by the NSF grant DMS-0800416.

[^2]: Kovalev was supported by the NSF grant DMS-0968756.

[^3]: Onninen was supported by the NSF grant DMS-1001620.
