n-Harmonic mappings between annuli

Proposition 1.3. (Transition to the inverse Map) Let $f \in \mathcal{W}_{loc}^{1,n-1}(\mathbb{Y},\mathbb{X})$ be a homeomorphism of finite outer distortion between bounded domains, with $\mathbb{K}_I(y,f) \in \mathcal{L}^1(\mathbb{Y})$ . Then the inverse map $h = f^{-1}$ : $\mathbb{X} \xrightarrow{onto} \mathbb{Y}$ belongs to the Sobolev class $\mathcal{W}^{1,n}(\mathbb{X},\mathbb{Y})$ and we have (1.11) $$n^{\frac{n}{2}} \int_{\mathbb{Y}} \mathbb{K}_{I}(y, f) dy = \int_{\mathbb{X}} \|Dh(x)\|^{n} dx$$ This identity gains in significance if we realize that the polyconvex variational integrand in the left hand side turns into a convex one, a rarity that one can exploit when studying quasiconformal mappings of smallest $\mathcal{L}^1$ -mean distortion. From yet another perspective, it is worth mentioning the classical Teichmüller theory which is concerned, broadly speaking, with extremal mappings between Riemann surfaces. The extremal Teichmüller mappings are exactly the ones whose distortion function has the smallest possible $\mathcal{L}^{\infty}$ -norm. The existence and uniqueness of such an extremal quasiconformal map within a given homotopy class of quasiconformal mappings is the heart of Teichmüller's theory. Now, in view of the identity (1.11), minimizing the $\mathcal{L}^1$ -norm of the inner distortion function offers a study of n-harmonic mappings. Is there any better motivation?

Theorem 1.4. Let $\mathbb{X} \subset \mathbb{R}^n$ be a ball with a k-dimensional closed disk removed, and let $\mathbb{Y} \subset \mathbb{R}^n$ be a ball with a (k+1)-dimensional closed disk removed, $1 \leq k < n-1$ . Then every homeomorphism $h: \mathbb{X} \xrightarrow{onto} \mathbb{Y}$ has infinite n-harmonic energy. Note that both $\mathbb{X}$ and $\mathbb{Y}$ are topological annuli; that is, homeomorphic images of a spherical annulus $\mathbb{A} = \{x \colon r < |x| < R\}$ . Let us view the disks removed from the balls as cracks. It can be easily shown, by means of an example, that mappings of finite conformal energy may outstretch an (n-1)-dimensional crack into an n-dimensional hole. However, Theorem 1.4 ensures us that, in principle, mappings of finite energy cannot increase the dimension of lower dimensional cracks, a fact highly nontrivial to observe and prove [31]. From now on we assume without explicit mention that the domains $\mathbb{X}$ and $\mathbb{Y}$ admit at least one homeomorphism $h : \mathbb{X} \xrightarrow{\text{onto}} \mathbb{Y}$ in the Sobolev space $\mathscr{W}^{1,n}(\mathbb{X},\mathbb{Y})$

Theorem 1.5. [30, Theorem 1.1] For the above-mentioned pair of domains $\mathbb{X}$ and $\mathbb{Y}$ , there exists a nonnegative continuous function $\eta = \eta(x)$ defined on $\overline{\mathbb{X}}$ such that (1.16) $$\operatorname{dist}(h_{j}(x), \partial \mathbb{Y}) \leqslant \eta(x) \| Dh_{j} \|_{\mathscr{L}^{n}(\mathbb{X})}, \qquad \eta \equiv 0 \text{ on } \partial \mathbb{X}.$$ ^&lt;sup>1</sup>Homeomorphisms converging weakly in $\underline{\mathscr{W}}^{1,n}(\mathbb{X},\mathbb{Y})$ also converge c-uniformly, so their limits are still continuous, taking $\mathbb{X}$ into $\overline{\mathbb{Y}}$ . 1.6. ANNULI 7 In particular, (1.17) $$\operatorname{dist}(h(x), \partial \mathbb{Y}) \leqslant \eta(x) \sup_{j \geqslant 1} \|Dh_j\|_{\mathscr{L}^n(\mathbb{X})}.$$ The weak limit h actually covers the target domain, but this may fail if $\mathbb{X}$ and $\mathbb{Y}$ have only one boundary component.

Theorem 1.6. [30, Theorem 1.4] The mapping h is continuous and $\mathbb{Y} \subset h(\mathbb{X}) \subset \overline{\mathbb{Y}}$ . Furthermore, there exists a measurable mapping $f : \mathbb{Y} \to \mathbb{X}$ , such that $$h \circ f = \mathrm{id} : \mathbb{Y} \to \mathbb{Y},$$ everywhere on Y. This right inverse mapping has bounded variation, $$||f||_{BV(Y)} \le \int_{\mathbb{X}} ||Dh(x)||^{n-1} dx.$$ As noted in [30, Remark 9.1] the weak limit h is monotone in the sense of C.B. Morrey [37].

Theorem 1.7. The mapping h is monotone, meaning that for every continuum $\mathbb{K} \subset \mathbb{Y}$ its preimage $h^{-1}(\mathbb{K}) \subset \mathbb{X}$ is also a continuum; that is compact and connected. The proof of this theorem is presented in Section 3.3.

Theorem 1.8. Let $\mathbb{A} = \mathbb{A}(r,R)$ and $\mathbb{A}^ = \mathbb{A}(r_,R_)$ be planar annuli, $0 < r < R < \infty$ and $0 < r_ < R_* < \infty$ . We have: 1.7. HAMMERING A PART OF AN ANNULUS INTO A CIRCLE, n=2  FIGURE 1. Hammering the inner ring into the unit circle. Case 1. (Within the Nitsche bound) If $$\frac{R}{r} \geqslant \frac{1}{2} \left( \frac{R}{r} + \frac{r}{R} \right)$$ then the harmonic homeomorphism $$h^{\circ}(z) = \frac{r}{2} \left( \frac{z}{r} + \frac{r}{\bar{z}} \right), \quad h^{\circ} \colon \mathbb{A} \xrightarrow{\text{onto}} \mathbb{A}$$ attains the smallest energy among all homeomorphisms $h: \mathbb{A} \xrightarrow{\text{onto}} \mathbb{A}^*$ , and as such is unique up to a conformal automorphism of $\mathbb{A}$ . <u>Case 2.</u> (Below the Nitsche bound) If $$\frac{R}{r} < \frac{1}{2} \left( \frac{R}{r} + \frac{r}{R} \right)$$ then the infimum energy among all homeomorphisms $h: \mathbb{A} \xrightarrow{\text{onto}} \mathbb{A}^*$ is not attained. Let a radius $r < \sigma < R$ be determined by the equation $$\frac{R}{r} = \frac{1}{2} \left( \frac{R}{\sigma} + \frac{\sigma}{R} \right)$$ -critial Nitsche configuration. Then the following mapping $$h^{\circ}(z) = \begin{cases} r_ \frac{z}{|z|} & r < |z| \leqslant \sigma \\ \frac{r}{2} \left(\frac{z}{\sigma} + \frac{\sigma}{\bar{z}}\right) & \sigma \leqslant |z| < R \end{cases}$$ is a $\mathcal{W}^{1,2}$ -limit of homeomorphisms $h_j: \mathbb{A} \xrightarrow{\text{ontq}} \mathbb{A}$ , and its energy is smaller than that of any homeomorphism $h: \mathbb{A} \xrightarrow{\text{ontq}} \mathbb{A}$ . The proof of this theorem was first given in [2]; here in Section 10 we present another one, based on free Lagrangians. Let us emphasize that this result does not rule out the existence of univalent harmonic mappings from $\mathbb{A}$ onto $\mathbb{A}$ , simply because harmonic homeomorphisms need not be the ones that minimize the energy. Nonexistence of harmonic homeomorphisms between such annuli $\mathbb{A}$ and $\mathbb{A}$ was conjectured by J. C. C. Nitsche [41]. After several partial results were obtained in [17, 34, 35, 50], the conjecture was proved in [25, 26]. The connection between the Nitsche conjecture and minimal surfaces is further explored in [28]. Hammering also occurs in free boundary problems for minimal graphs, where it is called edge-creeping [9, 21, 46]. Similar hammering phenomena will be observed in higher dimensions as well.

Theorem 1.9. Let $Mod \mathbb{A}^* > Mod \mathbb{A}$ . Then for n = 2, 3, the n-harmonic radial map $h^{\circ} = \lambda \aleph_{-}(kx)$ assumes the minimum conformal energy within all homeomorphisms. Such a minimizer is unique up to a conformal automorphism of $\mathbb{A}$ . Surprisingly, for $n \ge 4$ the answer will depend on how wide is the target annulus $\mathbb{A}^*$ , relatively to $\mathbb{A}$ .

Theorem 1.10. For dimensions $n \geqslant 4$ , there exists a function $\mathcal{N}^{\dagger} = \mathcal{N}^{\dagger}(t)$ , $t < \mathcal{N}^{\dagger}(t) < \infty$ for t > 0, see Figure 2, such that: if (1.32) $\operatorname{Mod} \mathbb{A} \leqslant \operatorname{Mod} \mathbb{A}^ \leqslant \mathcal{N}^{\dagger}(\operatorname{Mod} \mathbb{A})$ -upper Nitsche bound for $n \geqslant 4$ , then the map $h^{\circ} : \mathbb{A} \to \mathbb{A}$ is a unique (up to an automorphism of $\mathbb{A}$ ) minimizer of the conformal energy among all homeomorphisms.

Theorem 1.11. In dimensions $n \ge 4$ , there are annuli $\mathbb{A}$ and $\mathbb{A}$ such that no radial stretching from $\mathbb{A}$ onto $\mathbb{A}$ minimizes the conformal energy.

Theorem 1.12. Under the condition at (1.34) there exist unique k > 0 and $\lambda > 0$ such that $h^{\circ}(x) = \lambda H_{+}(kx)$ takes $\mathbb{A}$ homeomorphically onto $\mathbb{A}^{}$ . This map is a unique (up to a conformal automorphism of $\mathbb{A}$ ) minimizer of the conformal energy among all homeomorphisms of $\mathbb{A}$ onto $\mathbb{A}^{}$ .  FIGURE 2. The Nitsche functions. Rather unexpectedly, the injectivity of the weak limit of a minimizing sequence of homeomorphisms $h_j \colon \mathbb{A} \xrightarrow{\text{onto}} \mathbb{A}^*$ will fail once the lower bound in (1.34) is violated. Let us look more closely at the critical configuration of annuli; that is, $$(1.36) \quad \mathcal{N}_{\dagger}(\operatorname{Mod} \mathbb{A}') = \operatorname{Mod} \mathbb{A}^, \quad \left\{ \begin{array}{c} \text{where } \mathbb{A}' = \mathbb{A}(1, R) \text{ and} \\ \mathbb{A}^ = \mathbb{A}(1, R_), \ R_ = H_+(R) \end{array} \right.$$ Thus the critical Nitsche map $\aleph_+(x) = H_+(|x|) \frac{x}{|x|}$ , defined for $1 \leq |x| \leq R$ takes $\mathbb{A}'$ homeomorphically onto $\mathbb{A}^*$ . The Jacobian determinant of $\mathbb{N}$ vanishes for |x| = 1. Now let us built a pair of annuli below the lower bound at (1.34), simply by pasting an additional spherical ring to $\mathbb{A}'$ along the unit sphere. Thus, we consider a slightly fatter annulus $$\mathbb{A}=\mathbb{A}\left(r,R\right)=\mathbb{A}(r,1]\cup\mathbb{A}[1,R)$$ As in dimension n=2, we have

Theorem 1.13. The following deformation (1.37) $$h^{\circ}(x) \stackrel{\text{def}}{=} \left\{ \begin{array}{ll} \frac{x}{|x|} & r < |x| \leqslant 1, & hammering part \\ \aleph_{+}(x) & 1 \leqslant |x| \leqslant R, & n\text{-harmonic part} \end{array} \right.$$ is a $\mathcal{W}^{1,n}$ -limit of homeomorphisms $h: \mathbb{A} \xrightarrow{\operatorname{onto}} \mathbb{A}$ and its energy is smaller than that of any homeomorphism from $\mathbb{A}$ onto $\mathbb{A}$ . Among all such mappings $h^{\circ}$ is unique up to conformal automorphisms of $\mathbb{A}$ .

Theorem 1.14. Let $\mathbb{A}$ and $\mathbb{A}$ be spherical annuli in $\mathbb{R}^n$ , n = 2, 3. Then for every homeomorphism $h: \mathbb{A} \xrightarrow{\text{onto}} \mathbb{A}$ we have $$(1.47) \ \mathscr{F}_h \stackrel{\mathrm{def}}{=\!\!\!=} \int_{\mathbb{A}} \frac{\|\,Dh\,\|^{\,n}}{|h|^n} \geqslant \left(n-1+\alpha^2\right)^{\frac{n}{2}} \operatorname{Mod}\mathbb{A}\,, \quad \text{ where } \alpha = \frac{\operatorname{Mod}\mathbb{A}^*}{\operatorname{Mod}\mathbb{A}}$$ Equality holds for the power stretching $h(x) = r_* r^{-\alpha} |x|^{\alpha-1} x$ , uniquely up to conformal automorphisms of $\mathbb{A}$ . It is somewhat surprising that in dimensions $n \ge 4$ , this feature of $h^{\alpha}$ no longer holds when the target annulus is conformally too fat. We have the following result.

Theorem 1.15. For each $n \ge 4$ , there exists $\alpha_n > 1$ such that (1.47) holds whenever (1.48) $$\alpha \stackrel{\text{def}}{=} \frac{\operatorname{Mod} \mathbb{A}^*}{\operatorname{Mod} \mathbb{A}} < \alpha_n$$ The power stretching $h(x) = r_* r^{-\alpha} |x|^{\alpha-1} x$ is the only minimizer of $\mathscr{F}_h$ modulo conformal automorphism of $\mathbb{A}$ . Examples will be given to show that the extremals are no longer power stretchings if $$\frac{\operatorname{Mod} \mathbb{A}^*}{\operatorname{Mod} \mathbb{A}} \geqslant \sqrt{\frac{n-1}{n-3}}$$ In other words, the upper bound for $\alpha$ in (1.48) lies in the interval $1 < \alpha_n < \sqrt{\frac{n-1}{n-3}}$ . See Chapter 12 for more precise estimates of $\alpha_n$ . Moreover, if Mod $\mathbb{A}^*$ is too large relative to Mod $\mathbb{A}$ , then the extremals cannot be found even within general radial mappings, see Chapter 14.

Theorem 1.16. Under the Nitsche bounds $$(1.63) \aleph_{\dagger}(\operatorname{Mod} \mathbb{A}) \leqslant \operatorname{Mod} \mathbb{A}^{*} \leqslant \aleph^{\dagger}(\operatorname{Mod} \mathbb{A})$$ the $\mathcal{L}^1(\mathbb{Y})$ -norm of the inner distortion $\mathbb{K}_I(y,f)$ assumes its minimum value on a mapping $f \colon \mathbb{A}^ \xrightarrow{\operatorname{onto}} \mathbb{A}$ whose is inverse is a radial n-harmonic mapping $h^{\circ} \colon \mathbb{A} \xrightarrow{\operatorname{onto}} \mathbb{A}$ . Such an extremal mapping f is unique up to a conformal automorphism of $\mathbb{A}$ .

Theorem 1.17. If the domain annulus $\mathbb{A}^*$ is too thin relative to the target annulus $\mathbb{A}$ ; precisely, under the condition (1.64) $$\operatorname{Mod} \mathbb{A}^* < \aleph_{\dagger}(\operatorname{Mod} \mathbb{A})$$ -below the Nitsche bound. then the infimum of the $\mathscr{L}^1(\mathbb{A}^)$ -norm of $\mathbb{K}_I(y,f)$ is not attained among homeomorphisms $f \colon \mathbb{A}^ \xrightarrow{\text{onto}} \mathbb{A}$ . Nevertheless, we were able to find the infimum of the $\mathcal{L}^1$ -norms and identify the minimizing sequences. The weak BV-limits of such sequences and the underlying concept of their distortion (to be defined) are worth carrying out.

Proposition 2.1. If $\epsilon$ is small enough then there is no n-harmonic homeomorphism $h \colon \mathcal{A} \xrightarrow{\operatorname{into}} \mathbb{V}_{\epsilon}$ such that $\mathbb{S} \subset h(\mathcal{A})$ . PROOF. We shall not give any explicit bound for $\epsilon$ , but instead we argue by contradiction. Suppose that for every positive integer $\ell$ , thus $\epsilon=1/\ell$ , there exists an n-harmonic homeomorphism $$h_{\ell} = (h_{\ell}^1, \dots, h_{\ell}^n) \colon \mathcal{A} \xrightarrow{\text{onto}} h_{\ell}(\mathcal{A}), \quad \text{such that} \quad \mathbb{S} \subset h_{\ell}(\mathcal{A}) \subset \mathbb{V}_{\epsilon}, \quad \epsilon = \frac{1}{\ell}.$$ We appeal to the $\mathscr{C}^{1,\alpha}$ -regularity theory of n-harmonic mappings [13, 36, 47, 48, 49]. Accordingly, $h_{\ell}$ are uniformly bounded in $\mathscr{C}^{1,\alpha}$ -norm on every compact subset of $\mathcal{A}$ . Moreover, there is a subsequence, again denoted by $\{h_{\ell}\}$ , which converges together with the first order derivatives to a mapping $h: \mathcal{A} \to \mathbb{S}$ uniformly on compact subsets. The limit mapping $h = (h^1, \ldots, h^n)$ still satisfies the n-harmonic equation. We aim to show that $Dh \equiv 0$ in $\mathcal{A}$ , meaning that h is constant. Recall that $Dh: \mathcal{A} \to \mathbb{R}^{n \times n}$ is (Hölder) continuous. The computation below is certainly valid in the open region where $Dh \neq 0$ , because h is $\mathscr{C}^{\infty}$ -smooth in such region. We have $$F(h(x)) = 0$$ for all $x \in \mathcal{A}$ . Applying partial differentiation $\frac{\partial}{\partial x_{\nu}}$ and chain rule yields (2.1) $$\sum_{i=1}^{n} \frac{\partial F}{\partial y_i} \frac{\partial h^i}{\partial x_{\nu}} = 0, \quad \text{for } \nu = 1, 2, \dots, n.$$ On the other hand, the n-harmonic equation takes the form (2.2) $$\sum_{\nu=1}^{n} \frac{\partial}{\partial x_{\nu}} \left[ \lambda^{n-2} \frac{\partial h^{i}}{\partial x_{\nu}} \right] = 0 \quad \text{for } i = 1, 2, \dots, n, \quad \text{where } \lambda = \| Dh(x) \|$$ or, equivalently (2.3) $$\sum_{\nu=1}^{n} \left[ (n-2)\lambda^{n-3} \frac{\partial \lambda}{\partial x_{\nu}} \frac{\partial h^{i}}{\partial x_{\nu}} + \lambda^{n-2} \frac{\partial^{2} h^{i}}{\partial x_{\nu} \partial x_{\nu}} \right] = 0 \quad i = 1, \dots, n.$$ We multiply these equations by $\frac{\partial F}{\partial y_i}$ , sum them up with respect to i, and use the identity (2.1) to obtain (2.4) $$\lambda^{n-2} \sum_{i=1}^{n} \sum_{\nu=1}^{n} \frac{\partial F}{\partial y_i} \frac{\partial^2 h^i}{\partial x_{\nu} \partial x_{\nu}} = 0.$$ Next we differentiate (2.1) with respect to $\nu$ and sum up the equations, (2.5) $$\sum_{i=1}^{n} \sum_{\nu=1}^{n} \left( \frac{\partial F}{\partial y_i} \frac{\partial^2 h^i}{\partial x_\nu \partial x_\nu} + \sum_{j=1}^{n} \frac{\partial^2 F}{\partial y_i \partial y_j} \frac{\partial h^j}{\partial x_\nu} \frac{\partial h^i}{\partial x_\nu} \right) = 0.$$ Here the double sum for the first term in (2.5) vanishes due to (2.4), so we have $$\sum_{\nu=1}^{n} \left( \sum_{i,j} \frac{\partial^{2} F}{\partial y_{i} \partial y_{j}} \frac{\partial h^{i}}{\partial x_{\nu}} \frac{\partial h^{j}}{\partial x_{\nu}} \right) = 0.$$ Since the Hessian matrix of F is positive definite, this equation yields $\frac{\partial h^i}{\partial x_{\nu}} = 0$ for all $i, \nu = 1, 2, \dots, n$ , as desired. To reach a contradiction we look more closely at the homeomorphisms $h_{\ell} \colon \mathcal{A} \xrightarrow{\text{onto}} h_{\ell}(\mathcal{A})$ . Choose and fix an (n-1)-dimensional hypersurface $\Sigma \subset \mathcal{A}$ (topological sphere) which separates the boundary components of $\mathcal{A}$ . Its images $h_{\ell}(\Sigma) \subset h_{\ell}(\mathcal{A})$ separate the boundary components of $h_{\ell}(\mathcal{A})$ . Thus, in particular, $$0 < \liminf_{\ell \to \infty} [\operatorname{diam} h_{\ell}(\Sigma)] = \operatorname{diam} h(\Sigma) = 0$$ a clear contradiction. It is not so clear, however, how thick $\mathcal{A}$ should be to ensure existence of n-harmonic homeomorphisms $h \colon \mathcal{A} \xrightarrow{\text{onto}} \mathcal{A}$ . In dimension n = 2 the condition $\text{Mod } \mathcal{A}^* \geqslant \text{Mod } \mathcal{A}$ is sufficient [24] and is not far from being necessary [27]. Questions of existence of harmonic diffeomorphisms between surfaces are treated in [33].

Lemma 3.1. Every n-harmonic mapping $h \in \mathcal{W}^{1,n}_{loc}(\mathbb{X},\mathbb{R}^n)$ solves the generalized n-harmonic equation (3.6). PROOF. We consider again the perturbed mappings $h_{\epsilon}(x) = h(x+\epsilon \eta(x))$ which coincide with h outside a subdomain $\mathbb{U}$ compactly contained in $\mathbb{X}$ . Applying the integral form (3.1) of the n-harmonic equation, but with the test function $h - h_{\epsilon} \in \mathscr{W}^{1,n}_{\circ}(\mathbb{U},\mathbb{R}^n)$ in place of $\eta$ , 1 we estimate the energy of h over $\mathbb{U}$ as follows $$\mathcal{E}_{h} = \int_{\mathbb{U}} \|Dh\|^{n} = \int_{\mathbb{U}} \|Dh\|^{n-2} \langle Dh, Dh \rangle$$ $$= \int_{\mathbb{U}} \|Dh\|^{n-2} \langle Dh, Dh_{\epsilon} \rangle \leqslant \left( \int_{\mathbb{U}} \|Dh\|^{n} \right)^{\frac{n-1}{n}} \left( \int_{\mathbb{U}} \|Dh_{\epsilon}\|^{n} \right)^{\frac{1}{n}}.$$ Hence $$\mathcal{E}_h \leqslant \mathcal{E}_{h_{\epsilon}}$$ , with equality at $\epsilon = 0$ . This means $\frac{d}{d\epsilon}\mathcal{E}_{h_{\epsilon}}=0$ at $\epsilon=0$ . Equivalently, h satisfies the generalized n-harmonic equation. ^&lt;sup>1</sup>This is justified because $\mathscr{C}^{\infty}_{\circ}(\mathbb{U},\mathbb{R}^n)$ is dense in $\mathscr{W}^{1,n}_{\circ}(\mathbb{U},\mathbb{R}^n)$ . In dimension n=2, the generalized harmonic equation reduces to (3.7) $$\operatorname{Div}\left(D^{*}h\,Dh - \frac{1}{2}\|Dh\|^{2}I\right) = 0.$$ This equation is known as Hopf-Laplace equation [11]. In complex notation it takes the form (3.8) $$\frac{\partial}{\partial \bar{z}} \left( h_z \overline{h_{\bar{z}}} \right) = 0, \qquad z = x_1 + i x_2.$$ If, by some reason, $h \in \mathcal{C}^2(\mathbb{X}, \mathbb{C})$ then (3.8) reads as (3.9) $$J(z,h)\Delta h = 0, \quad \text{where } \Delta = 4\frac{\partial^2}{\partial z \partial \bar{z}}.$$ There exist diverse non-harmonic solutions to (3.9), many of them of great interest in the theory of minimal surfaces and some with potential applications to nonlinear elasticity (elastic plates), see [11]. Equally in higher dimensions, the n-harmonic mappings of Sobolev class $\mathcal{W}_{loc}^{1,n}(\mathbb{X},\mathbb{R}^n)$ are only particular solutions to the generalized n-Laplacian.

Proposition 3.2. The equilibrium solution for mappings that are slipping along the boundaries satisfies, in addition to (3.6), the following condition $$(3.13) D^*h \cdot Dh \colon \mathbf{T}_r \partial \mathbb{X} \to \mathbf{T}_r \partial \mathbb{X}$$ equivalently, $$(3.14) D^*h \cdot Dh \colon \mathbf{N}_x \partial \mathbb{X} \to \mathbf{N}_x \partial \mathbb{X}.$$ where $\mathbf{T}_x \partial \mathbb{X}$ and $\mathbf{N}_x \partial \mathbb{X}$ designate the tangent and normal spaces at $x \in \partial \mathbb{X}$ . In dimension n=2 this amounts to saying that the Hopf quadratic differential $h_z\overline{h_{\bar{z}}}\,\mathrm{d}z^2$ is real along the boundary components of $\mathbb{X}$ . In an annulus $\mathbb{A}=\{z\colon r<|z|< R\}$ every Hopf differential which is real on $\partial\mathbb{A}$ takes the form $$h_z \overline{h_{\bar{z}}} \equiv \frac{c}{z^2}$$ , c is a real constant see [11]. One of the solutions is the mapping we are already encountered in (1.20) $$h(z) = \begin{cases} \frac{z}{|z|} & \frac{1}{R} < |z| \leqslant 1, & \text{hammering part} \\ \frac{1}{2} \left( z + \frac{1}{\overline{z}} \right) & 1 \leqslant |z| \leqslant R, & \text{harmonic part} \end{cases}$$ In higher dimensions the radial mapping $h(x) = H(|x|) \frac{x}{|x|}$ in the annulus $A = \{x : r < |x| < R\}$ also complies with the boundary condition (3.13). Indeed, the Cauchy-Green tensor takes the form $$D^*h \cdot Dh = \frac{H^2}{|x|^2}I + \left(\dot{H}^2 - \frac{H^2}{|x|}\right)\frac{x \otimes x}{|x|^2}.$$ The normal vector field at $\partial \mathbb{A}$ is $\mathfrak{n} = \mathfrak{n}(x) = x$ , and we see that $$[D^*h(x) \cdot Dh(x)] \mathfrak{n} = \lambda \mathfrak{n}, \qquad \lambda = [\dot{H}(|x|)]^2.$$ For any tangent vector $\mathfrak{t} = \mathfrak{t}(x)$ , we have $(x \otimes x)\mathfrak{t} = 0$ so $$[D^*h(x) \cdot Dh(x)] \mathfrak{t} = \frac{H^2}{|x|^2} \mathfrak{t}.$$ The corresponding singular values of the right Cauchy-Green tensor are $\dot{H}(x)$ and $\frac{H(|x|)}{|x|}$ ; principal stretches. Before moving to a study of n-harmonic mappings between spherical annuli, let us fulfill the promise of proving that weak $\mathcal{W}^{1,n}$ -limits of homeomorphisms are indeed monotone.

Lemma 5.1. Let $h(x) = H(|x|) \frac{x}{|x|}$ be a radial stretching of class $\mathscr{C}^2(\mathbb{A}, \mathbb{R}^n)$ , where $\mathbb{A} = \{x : a < |x| < b\}$ . Then, with the notation |x| = t, we have $$\frac{n\dot{H}}{n-1}\operatorname{div} \|Dh\|^{n-2}Dh =$$ $$-\frac{x}{t^{n+1}}\frac{d}{dt}\left\{\left[(n-1)H^2 + t^2\dot{H}^2\right]^{\frac{n-2}{2}}\left(H^2 - t^2\dot{H}^2\right)\right\}$$

Proposition 5.3. Every radial n-harmonic mapping in the annulus $\mathbb{A} = \{x \; ; \; a < |x| < b\}$ takes the form $$(5.24) q(x) = \lambda \cdot h(kx), \lambda \in \mathbb{R}, k > 0$$ where $h \in \mathscr{C}^{\infty}(\mathbb{R}^n)$ , is one of the four principal n-harmonics. It should be observed, as a corollary, that radial n-harmonics are $\mathscr{C}^{\infty}$ smooth in the entire space $\mathbb{R}^n_{\circ}$ .

Proposition 6.1. The following point-wise estimates hold $$(6.33) |h_N| \geqslant |d|h| \wedge \star dt |$$ $$(6.34) |h_T|^{n-1} \geqslant |h|^{n-1} \left| dt \wedge h^{\sharp} \omega \right|$$ PROOF. Formula (6.30) holds for scalar functions in place of h we apply it to |h| $$(6.35) |h_N| \geqslant |h|_N| = |d|h| \wedge \star dt$$ establishing the inequality (6.33). The proof of the estimate (6.34) is more involved. Using (6.32), we are reduced to showing that (6.36) $$\left| \frac{dh \wedge dt}{n-1} \right|^{n-1} \geqslant \left| \sum_{i=1}^{n} (-1)^{i} \frac{h^{i}}{|h|} dh^{1} \wedge \dots \wedge dh^{i-1} \wedge dh^{i+1} \wedge \dots \wedge dh^{n} \wedge dt \right|$$ By Schwarz inequality it suffices to prove that $$(6.37) \left| \frac{dh \wedge dt}{n-1} \right|^{n-1} \geqslant \left( \sum_{i=1}^{n} \left| dh^{1} \wedge \dots \wedge dh^{i-1} \wedge dh^{i+1} \wedge \dots \wedge dh^{n} \wedge dt \right|^{2} \right)^{\frac{1}{2}}$$ Note that here in both sides only spherical derivatives of h are significant; the terms containing $h_N dt$ vanish after wedging them with dt. This observation permits us to replace $dh^i$ by the covectors $\mathfrak{a}_i = dh^i - h_N^i dt$ . We view $\mathfrak{a}_i$ as elements of the space $\bigwedge^1(\mathbb{R}^{n-1})$ . Once this interpretation is accepted, the proof continues via an algebraic inequality.

Lemma 6.2. For every n-tuple of covectors in $\mathbb{R}^{n-1}$ , $\mathfrak{a}_1,...,\mathfrak{a}_n \in \bigwedge^1(\mathbb{R}^{n-1})$ , we have $$(6.38) |\mathfrak{a}_{1}|^{2} + \dots + |\mathfrak{a}_{n}|^{2} \geqslant (n-1) \left( \sum_{i=1}^{n} |\mathfrak{a}_{1} \wedge \dots \wedge \mathfrak{a}_{i-1} \wedge \mathfrak{a}_{i+1} \wedge \dots \wedge \mathfrak{a}_{n}|^{2} \right)^{\frac{1}{n-1}}$$ PROOF. We look at the matrix M of size $n \times (n-1)$ whose rows are made of the covectors $\mathfrak{a}_1,...,\mathfrak{a}_n$ . Let $\mathfrak{b}_2,...,\mathfrak{b}_n \in \mathbb{R}^n$ denote the column vectors of M (6.39) $$\begin{bmatrix} -- & \mathfrak{a}_1 & -- & \\ -- & \mathfrak{a}_2 & -- & \\ & \ddots & \\ -- & \mathfrak{a}_n & -- \end{bmatrix} = M = \begin{bmatrix} | & | & | & | \\ | & | & | & | \\ \mathfrak{b}_2 & \mathfrak{b}_3 \cdots \mathfrak{b}_n & \\ | & | & | & | \end{bmatrix}$$ In the left hand side of (6.38) we are actually dealing with the square of the Hilbert-Schmidt norm of M (6.40) $$|\mathfrak{a}_1|^2 + \dots + |\mathfrak{a}_n|^2 = \text{Tr } (M^*M) = |\mathfrak{b}_2|^2 + \dots + |\mathfrak{b}_n|^2$$ On the other hand, the sum in the right hand side of (6.38) is the Hilbert-Schmidt square of the norm of the cross product of the column vectors $$\sum_{i=1}^{n} |\mathfrak{a}_{1} \wedge ... \wedge \mathfrak{a}_{i-1} \wedge \mathfrak{a}_{i+1} \wedge ... \wedge \mathfrak{a}_{n}|^{2} = |\mathfrak{b}_{2} \times ... \times \mathfrak{b}_{n}|^{2} \leqslant |\mathfrak{b}_{2}|^{2} \cdots |\mathfrak{b}_{n}|^{2}$$ by Hadamard's inequality. The inequality between arithmetic and geometric means yields (6.41) $$|\mathfrak{b}_2|^2 \cdots |\mathfrak{b}_n|^2 \leqslant \left(\frac{|\mathfrak{b}_2|^2 + \dots + |\mathfrak{b}_n|^2}{n-1}\right)^{n-1}$$ From here Lemma 6.2 is immediate. This also completes the proof of Proposition 6.1.

Lemma 7.1. Let $\Phi: [r_, R_] \to \mathbb{R}$ be any integrable function. Then the n-form (7.3) $$\Phi(|h|) dh^1 \wedge \dots \wedge dh^n$$ is a free Lagrangian. Precisely, we have (7.4) $$\int_{\mathbb{A}} \Phi(|h|) J(x,h) \, dx = \omega_{n-1} \int_{r}^{R} \tau^{n-1} \, \Phi(\tau) \, d\tau$$ for every orientation preserving homeomorphism $h \in \mathcal{W}^{1,n}(\mathbb{A},\mathbb{A}^*)$ . This is none other than a general formula of integration by substitution.

Lemma 7.2. The following differential n-form (7.5) $$\sum_{i=1}^{n} \frac{x_i \, dx_1 \wedge \dots \wedge dx_{i-1} \wedge d|h| \wedge dx_{i+1} \wedge \dots \wedge dx_n}{|h| \, |x|^n} = \frac{(d|h|) \wedge \star dt}{|h| \, t^{n-1}}$$ is a free Lagrangian in the class of all homeomorphisms $h \in \mathcal{W}^{1,1}(\mathbb{A}, \mathbb{A}^)$ that preserve the order of the boundary components of the annuli $\mathbb{A}$ and $\mathbb{A}$ . Precisely, we have (7.6) $$\int_{\mathbb{A}} \frac{d|h| \wedge \star dt}{|h| t^{n-1}} = \operatorname{Mod} \mathbb{A}^*$$ PROOF. First observe that the function $|h|: \mathbb{A} \to (r_, R_)$ extends continuously to the closure of $\mathbb{A}$ so that $|h(x)| = r$ for |x| = r and $|h(x)| = R$ for |x| = R. The point is that our arguments will actually work for arbitrary continuous mappings $h: \overline{\mathbb{A}} \to \mathbb{R}^n_{\circ}$ of Sobolev class $\mathscr{W}^{1,1}(\mathbb{A}, \mathbb{R}^n)$ which satisfy the above boundary conditions, not necessarily homeomorphisms. The advantage in such a generality is that we can apply smooth approximation of h without worrying about injectivity. This extends Lemma 7.2 to all permissible mappings. If h is smooth it is legitimate to apply Stokes' theorem. Recalling the form $\omega$ and (6.26), we write the integral at (7.6) as: $$\int_{\mathbb{A}} (d \log |h|) \wedge \omega = \int_{\mathbb{A}} d (\omega \log |h|) = (\log R_) \int_{|x|=R} \omega - (\log r_) \int_{|x|=r} \omega$$ (7.7) $$= \omega_{n-1} \log \frac{R}{r} = \operatorname{Mod} \mathbb{A}^*$$ The same proof works for more general integrals of the form (7.8) $$\int_{\mathbb{A}} \Phi'(|h|) \frac{d|h| \wedge \star dt}{t^{n-1}} = \omega_{n-1} \left[ \Phi(R_) - \Phi(r_) \right]$$ with $\Phi \in \mathscr{C}^1[r_, R_]$ . This also reads as (7.9) $$\int_{\mathbb{A}} \frac{\Phi'(|h|)|h|_{N}}{|x|^{n-1}} = \omega_{n-1} \left[ \Phi(R_{}) - \Phi(r_{}) \right]$$ Another example of a free Lagrangian rests on the topological degree. Let $\Psi: S^{n-1}_t \to \mathbb{S}^{n-1}$ be any smooth mapping of a sphere $S^{n-1}_t = \{x \, ; \, |x| = t\}$ into the unit sphere $\mathbb{S}^{n-1} \subset \mathbb{R}^n$ . Then there is an integer, denoted by $\deg \Psi$ , and called the degree of $\Psi$ , such that (7.10) $$\int_{S_{\star}^{n-1}} \Psi^{\sharp}(\omega) = \omega_{n-1} \operatorname{deg} \Psi$$ where $\omega$ stands for the standard surface measure of the unit sphere $\mathbb{S}^{n-1} \subset \mathbb{R}^n$ , as given in (6.25). Note a general fact that deg $\Psi$ remains unchanged under small perturbations of $\Psi$ . If $\Psi$ is an orientation-preserving diffeomorphism, then deg $\Psi = 1$ . Now a free Lagrangian somewhat dual to that in Lemma 7.2, is obtained as follows.

Lemma 7.3. The following differential n-form (7.11) $$\sum_{i=1}^{n} \frac{h^{i} dh^{1} \wedge \dots \wedge dh^{i-1} \wedge d|x| \wedge dh^{i+1} \wedge \dots \wedge dh^{n}}{|x| |h|^{n}}$$ is a free Lagrangian in the class of all orientation preserving homeomorphism mappings $h \in \mathcal{W}^{1,n-1}(\mathbb{A},\mathbb{A}^*)$ . Precisely, we have (7.12) $$\int_{\mathbb{A}} \frac{dt}{t} \wedge h^{\sharp} \omega = \operatorname{Mod} \mathbb{A}$$ PROOF. As a first step, we approximate h by smooth mappings $h_{\nu}$ converging to h c-uniformly on $\mathbb{A}$ and in the norm of $\mathcal{W}^{1,n-1}(\mathbb{A},\mathbb{R}^n)$ . For $\nu$ sufficiently large we consider the mappings (7.13) $$\Psi = \frac{h_{\nu}}{|h_{\nu}|} : \mathbb{A}' \to \mathbb{S}^{n-1}, \quad \nu = 1, 2, \dots$$ defined on a slightly smaller annulus $\mathbb{A}'=\{x\,;\, r'\leqslant |x|\leqslant R'\}$ , where r< r'< R'< R. The degree of $\Psi$ restricted to any concentric sphere in $\mathbb{A}'$ is equal to 1. Moreover, we have the point-wise identity $\Psi^\sharp\omega=h^\sharp_\nu(\omega)$ . Now we integrate as follows (7.14) $$\int_{\mathbb{A}'} \frac{dt}{t} \wedge \Psi^{\sharp} \omega = \int_{\mathbb{A}'} d\left(\log|t| \, \Psi^{\sharp} \omega\right)$$ because $d\left(\Psi^{\sharp}\omega\right)=\Psi^{\sharp}\left(d\omega\right)=0$ . By Stokes' theorem, the right hand side is equal to: $$\int_{|x|=R'} \log |x| \, \Psi^{\sharp} \omega - \int_{|x|=r'} \log |x| \, \Psi^{\sharp} \omega = \omega_{n-1} \log R' - \omega_{n-1} \log r' = \operatorname{Mod} \mathbb{A}'$$ Passing to the limit as $\nu \to \infty$ we conclude with the formula (7.15) $$\int_{\mathbb{A}^{d}} \frac{dt}{t} \wedge h^{\sharp} \omega = \operatorname{Mod} \mathbb{A}'$$ Finally letting $r' \to r$ and $R' \to R$ yield (7.12). REMARK 7.4. The reader may wish to generalize this lemma to mappings $h: \mathbb{A} \to \mathbb{R}^n_{\circ}$ which are not necessarily homeomorphisms. The degree of $\frac{h}{|h|}$ will emerge as a factor in front of the right hand side of (7.12). The following corollary from Lemmas 7.1, 7.2, 7.3 provide us with three inequalities, which we call free Lagrangian estimates.

Corollary 7.5. Let h be a homeomorphism between spherical rings $\mathbb{A}$ and $\mathbb{A}$ in the Sobolev class $\mathcal{W}^{1,n}(\mathbb{A},\mathbb{A}^)$ . Then (7.16) $$\int_{\mathbb{A}} \Phi(|h|) |h_N| |h_T|^{n-1} \ge \omega_{n-1} \int_{r}^{R_*} \tau^{n-1} \Phi(\tau) d\tau$$ whenever $\Phi$ is integrable in $[r_, R_]$ . We have the equality in (7.16) if and only if $|h_N| |h_T|^{n-1} = J(x, h)$ . Furthermore, (7.17) $$\int_{\mathbb{A}} \frac{|h_N|}{|h| |x|^{n-1}} \geqslant \operatorname{Mod} \mathbb{A}^*$$ (7.18) $$\int_{\mathbb{A}} \frac{|h_T|^{n-1}}{|h|^{n-1}|x|} \geqslant \operatorname{Mod} \mathbb{A}$$ Note that we have equalities if h is a radial mapping. Other cases of equalities are also possible.

Theorem 8.1. Let $\mathbb{A}$ and $\mathbb{A}$ be spherical rings in $\mathbb{R}^n$ , $n \geq 2$ . Then for every $h \in \mathscr{P}(\mathbb{A}, \mathbb{A}^)$ we have (8.7) $$\int_{\mathbb{A}} \frac{|Dh|^n}{|h|^n} \geqslant \max\{1, \alpha^n\} \operatorname{Mod} \mathbb{A}$$ As for the sharpness of this estimate we note that equality holds for the power stretching $h(x) = |x|^{\alpha-1}x$ . There are, however, other cases of equality in (8.7) if $\operatorname{Mod} \mathbb{A}^* \neq \operatorname{Mod} \mathbb{A}$ .

Proposition 8.2. For each radial stretching we have (8.9) $$\mathscr{F}_h = \int_{\mathbb{A}} \frac{\|Dh\|^n}{|h|^n} \geqslant \int_{\mathbb{A}} \frac{\|Dh^\alpha\|^n}{|h^\alpha|^n} = (\alpha^2 + n - 1)^{\frac{n}{2}} \operatorname{Mod} \mathbb{A}$$ Equality holds modulo rotation of $\mathbb{A}$ . Proof. We express the energy of h as (8.10) $$\mathscr{F}_{h} = \int_{\mathbb{A}} \frac{\left(|h_{N}|^{2} + (n-1)|h_{T}|^{2}\right)^{\frac{n}{2}}}{|h|^{n}}$$ where the radial and spherical derivatives of h are found in terms of the strain function H by using formulas (6.19); (8.11) $$|h_N| = |\dot{H}(t)| \quad \text{and} \quad |h_T| = \frac{|H(t)|}{t}$$ Integration in polar coordinates leads to a line integral $$\mathscr{F}_{h} = \omega_{n-1} \int_{r}^{R} \left( \frac{t^{2} \dot{H}^{2}}{H^{2}} + n - 1 \right)^{\frac{n}{2}} \frac{dt}{t}$$ $$= \omega_{n-1} \log \frac{R}{r} \int \left( X^{2} + n - 1 \right)^{\frac{n}{2}} d\mu$$ $$(8.12)$$ where the integral average is taken with respect to the measure $d\mu = \frac{dt}{t}$ . The key is that $X(t) = \frac{t\dot{H}}{H}$ is a one dimensional free Lagrangian. Indeed, we see that the integral mean (8.13) $$f_r^R X d\mu = \frac{1}{\log \frac{R}{r}} \int_r^R \frac{\dot{H}(t) dt}{H(t)} = \frac{\log \frac{R}{r}}{\log \frac{R}{r}} = \alpha$$ does not depend on H. Next using Jensen's inequality for the convex function $X \to (X^2 + n - 1)^{\frac{n}{2}}$ yields $$\mathscr{F}_{h} \geqslant \left[ \left( \int_{r}^{R} X \, d\mu \right)^{2} + n - 1 \right]^{\frac{n}{2}} \operatorname{Mod} \mathbb{A}$$ $$= \left[ \alpha^{2} + n - 1 \right]^{\frac{n}{2}} \operatorname{Mod} \mathbb{A}$$ (8.14) as desired. Examining these arguments backwards we obtain the uniqueness statement. As a corollary, letting $\alpha$ go to 0, we observe that among all radial deformations $h: \mathbb{A} \to \mathbb{S}^{n-1}$ , the n-harminic energy (8.15) $$\mathscr{E}_h = \int_{\mathbb{A}} \|Dh\|^n \geqslant (n-1)^{\frac{n}{2}} \operatorname{Mod} \mathbb{A}$$ assumes its minimal value for the mapping $h(x) = \frac{x}{|x|}$ . This will be later shown to be true in a larger class of mappings. In dimensions n=2,3 the assertion of Proposition 8.2 remains true for all permissible mappings in the Sobolev class $\mathcal{W}^{1,n}(\mathbb{A},\mathbb{A}^*)$ .

Lemma 9.1. Let $X, Y \ge 0$ and $1 \le \alpha < \alpha_n$ . Then (9.3) $$a = a(\alpha) \stackrel{\text{def}}{=} \frac{(\alpha^2 + n - 1)^{\frac{n-2}{2}} (\alpha^2 - 1)}{\alpha^n} < 1$$ and, we have (9.4) $$\left[ X^2 + (n-1)Y^2 \right]^{\frac{n}{2}} \geqslant a X^n + b X Y^{n-1}$$ where $$(9.5) b = \frac{n\left(\alpha^2 + n - 1\right)^{\frac{n-2}{2}}}{\alpha}$$ Equality holds if and only if $X = \alpha Y$ . PROOF. Because of homogeneity we may assume that Y=1; the case Y=0 is obvious. We are reduced to proving an inequality with one variable $X\geqslant 0$ , (9.6) $$[X^2 + (n-1)]^{\frac{n}{2}} \geqslant aX^n + bX$$ Consider the function (9.7) $$\varphi(X) = (X^2 + n - 1)^{\frac{n}{2}} - aX^n - bX$$ and its two derivatives $$(9.8) \quad \varphi'(X) = n \left(X^2 + n - 1\right)^{\frac{n-2}{2}} X - naX^{n-1} - b$$ $$(9.9) \varphi''(X) = n(n-1) \left[ \left( X^2 + n - 1 \right)^{\frac{n-4}{2}} (X^2 + 1) - aX^{n-2} \right] > 0$$ This letter estimate is guaranteed by our hypothesis that a < 1. Thus $\varphi$ is strictly convex and, therefore, has at most one critical point. The coefficient b has been defined above exactly in a way to ensure that $\varphi'(\alpha) = 0$ . The coefficient a has been chosen so that $\varphi(\alpha) = 0$ . This completes the proof of Lemma 9.1.

Lemma 9.2. Let $X, Y \ge 0$ and $0 \le \alpha \le 1$ . Then $$(9.10) \left[ X^2 + (n-1)Y^2 \right]^{\frac{n}{2}} \geqslant a Y^n + b X Y^{n-1}$$ where (9.11) $$a = (n-1) \left(\alpha^2 + n - 1\right)^{\frac{n-2}{2}} \left(1 - \alpha^2\right)$$ and (9.12) $$b = n\alpha \left(\alpha^2 + n - 1\right)^{\frac{n-2}{2}}$$ Equality holds if and only if $X = \alpha Y$ . PROOF. Because of homogeneity we may assume that Y = 1, the case Y = 0 is obvious. We are reduced to the inequality with one variable $X \ge 0$ , $$(9.13) [X^2 + (n-1)]^{\frac{n}{2}} \geqslant a + bX$$ Consider the function (9.14) $$\varphi(X) = (X^2 + n - 1)^{\frac{n}{2}} - bX$$ and its two derivatives (9.15) $$\varphi'(X) = n(X^2 + n - 1)^{\frac{n-2}{2}}X - b$$ (9.16) $$\varphi''(X) = n(n-1) \left(X^2 + n - 1\right)^{\frac{n-4}{2}} (X^2 + 1) > 0$$ As before, $\varphi$ is strictly convex with one critical point. The coefficient b has been defined exactly to ensure that $\varphi'(\alpha) = 0$ , whereas $a = \varphi(\alpha)$ . This completes the proof of Lemma 9.2. We now proceed to the proof of Theorem 1.15. Consider two cases.

Proposition 12.1. Let $h : \mathbb{A} \to \mathbb{A}$ be a permissible map in $\mathscr{P}(\mathbb{A}, \mathbb{A}^)$ where $$\frac{R}{r} < \frac{R}{r} < \frac{H_-\left(\gamma_n \frac{R}{r}\right)}{H_-(\gamma_n)}$$ Then (12.15) $$\int_{\mathbb{A}} \|Dh\|^n \geqslant \int_{\mathbb{A}} \|Dh^{\circ}\|^n$$ Equality holds if and only if $h(x) = h^{\circ}(x)$ up to a conformal automorphism of $\mathbb{A}$ . PROOF. The characteristic equation (5.28) for the map $h^{\circ}: \mathbb{A} \to \mathbb{A}^*$ defines a positive constant $q = -(n-1)^{\frac{n-2}{2}} \mathcal{L}H$ ; that is, (12.16) $$q \equiv \left[t^2 \dot{H}^2 + (n-1)H^2\right]^{\frac{n-2}{2}} \left(t^2 \dot{H}^2 - H^2\right) > 0$$ Equivalently, (12.17) $$\left(n - 1 + \eta_H^2\right)^{\frac{n-2}{2}} \left(\eta_H^2 - 1\right) = \frac{q}{Hn}$$ Now, consider an arbitrary permissible map $h: \mathbb{A} \to \mathbb{A}$ . We may assume that h is a homeomorphism preserving orientation and the order of the boundary components of the annuli. We introduce a function $\eta = \eta(t)$ implicitly defined for $r_ \leq t \leq R_*$ by the equation, (12.18) $$(n-1+\eta^2)^{\frac{n-2}{2}} (\eta^2 - 1) = \frac{q}{t^n}$$ Note that the equation (12.17) for $\eta_H$ at t=r coincides with (12.18) for $\eta$ at $t=r$ . Hence $\eta(r_)=\eta_H(r)$ . Since $\eta=\eta(t)$ is decreasing we find that for every $r_\leqslant t\leqslant R$ , $$(12.19) 1 < \eta(t) \leqslant \eta(r_*) = \eta_H(r) \leqslant \alpha_n$$ Next we recall the function $a = a(\alpha)$ , $1 \le \alpha < \infty$ , from the formula (9.3). By the definition of $\alpha_n$ at (8.18) it follows that (12.20) $$a(\eta) = \frac{(n-1+\eta^2)^{\frac{n-2}{2}}(\eta^2-1)}{\eta^n} \le 1$$ for all $r_ < t < R$  FIGURE 2. The coefficient a(|h|). Now Lemma 9.1 applies to $X = |h_N|$ , $Y = |h_T|$ and $\alpha = \eta(|h|)$ , $$||Dh||^{n} = \left[ |h_{N}|^{2} + (n-1)|h_{T}|^{2} \right]^{\frac{n}{2}}$$ $$\geqslant a(|\eta|)|h_{N}|^{n} + b(|\eta|)|h_{N}||h_{T}|^{n-1}$$ $$= q \left[ \frac{|h_{N}|}{|h|\eta(|h|)} \right]^{n} + \Phi(|h|)|h_{N}||h_{T}|^{n-1}$$ where $\Phi(|h|) = b(\eta(|h|))$ and b is given by (9.5). According to Lemma 9.1, equality holds at a given point x if and only if $|h_N(x)| = \eta(|h(x)|) |h_T(x)|$ . In particular, it holds almost everywhere for $h = h^{\circ}(x)$ , because $|h_N^{\circ}| = \eta_H |h_T^{\circ}|$ . We now integrate over the annulus $\mathbb{A}$ . The last term at (12.21) is easily handled by the lower bound at (7.16), $$\int_{\mathbb{A}} \Phi(|h|) |h_N| |h_T|^{n-1} \ge \omega_{n-1} \int_{r}^{R} \tau^{n-1} \Phi(\tau) d\tau$$ $$= \int_{\mathbb{A}} \Phi(|h^{\circ}|) |h_N^{\circ}| |h_T^{\circ}|^{n-1}$$ The first term in the right hand side of (12.21) will be treated by Hölder's inequality in order to apply the lower bound at (7.17). This should be done in a way so one obtains equality for $h^{\circ}$ . Taking into an account the identities (12.23) $$\frac{|h_N^{\circ}|}{|h^{\circ}| \eta(|h^{\circ}|)} = \frac{\dot{H}}{H \eta_H} = \frac{1}{|x|}$$ We proceed as follows $$(12.24) \qquad \int_{\mathbb{A}} \left[ \frac{|h_N|}{|h| \ \eta(|h|)} \right]^n \geqslant \left( \int_{\mathbb{A}} \frac{|h|_N \ dx}{|h| \ \eta(|h|) \ |x|^{n-1}} \right)^n \left( \int_{\mathbb{A}} \frac{dx}{|x|^n} \right)^{1-n}$$ The first term in the right hand side is none other than a free Lagrangian at (7.9). This allows us to quickly evaluate the first term, without getting buried under massive computation. $$\int_{\mathbb{A}} \frac{|h|_N \ dx}{|h| \ \eta(|h|) \ |x|^{n-1}} = \int_{\mathbb{A}} \frac{|h^{\circ}|_N \ dx}{|h^{\circ}| \ \eta(|h^{\circ}|) \ |x|^{n-1}} = \int_{\mathbb{A}} \frac{\dot{H}(|x|) \ dx}{H \ \eta_H \ |x|^{n-1}} = \int_{\mathbb{A}} \frac{dx}{|x|^n}$$ Hence (12.25) $$\int_{\mathbb{A}} \left[ \frac{|h_N|}{|h| \ \eta(|h|)} \right]^n \geqslant \int_{\mathbb{A}} \frac{dx}{|x|^n} = \operatorname{Mod} \mathbb{A}$$ In conclusion, $$(12.26) \qquad \int_{\mathbb{A}} \|Dh\|^n \geqslant q \operatorname{Mod} \mathbb{A} + \omega_{n-1} \int_{r}^{R} \tau^{n-1} \Phi(\tau) d\tau$$ with equality occurring for $h^{\circ}$ , as claimed. The uniqueness results will be proven in more unified manner in Chapter 13. - 12.1.1. Proof of Theorem 1.10. Indeed, the condition $\operatorname{Mod} \mathbb{A}^* \leq \mathcal{N}^{\dagger}(\operatorname{Mod} \mathbb{A})$ is equivalent to (12.14), in view of the explicit formula for $\mathcal{N}^{\dagger}$ given in (12.13). - 12.1.2. Proof of Theorem 1.9. The proof is immediate from Proposition 12.1 once we recall that $\mathcal{N}^{\dagger} \equiv \infty$ , for n = 2, 3 posing no upper bound for Mod $\mathbb{A}^*$ .

Theorem 13.1. Below the upper Nitsche bound: that is, (13.4) $$\operatorname{Mod} \mathbb{A}^* \leqslant \mathcal{N}^{\dagger}(\operatorname{Mod} \mathbb{A})$$ every permissible minimizer $h: \mathbb{A} \to \mathbb{A}^*$ coincides with the radial extremal map modulo conformal automorphisms of $\mathbb{A}$ . Let $h \in \mathcal{P}(\mathbb{A}, \mathbb{A}^*)$ be any permissible extremal mapping. In all the preceding cases we came to the following equation as one of the necessary conditions for h to minimize the energy $$(13.5) |h_N| = \eta(|h|) |h_T|$$ see (9.17), (9.20), (11.5) and (12.21). Here $\eta = \eta(\tau)$ is a nonnegative function defined for $r_ \leqslant \tau \leqslant R$ by the rule (13.6) $$\tau^{n} \left( \eta^{2} + n - 1 \right)^{\frac{n-2}{2}} \left( \eta^{2} - 1 \right) \equiv c$$ Here the constant c comes from the characteristic equation for the radial extremal map $h^{\circ}(x) = H(|x|) \frac{x}{|x|}$ . Precisely, we have (13.7) $$c \equiv [H(t)]^n \left(\eta_H^2 + n - 1\right)^{\frac{n-2}{2}} \left(\eta_H^2 - 1\right)$$ , where $\eta_H(t) = \frac{t \dot{H(t)}}{H(t)}$ Note that $\eta$ is strictly positive if c>0. However, if c<0 then $\eta(\tau)$ is strictly increasing and, therefore, can vanish only at the endpoint $\tau=r_*$ . This latter situation arises when $\eta_H\equiv 0$ in the hammering part of $h^\circ$ . Another necessary condition for a mapping to be extremal takes the form (13.8) $$|h_N| |h_T|^{n-1} = J(x, h)$$ because in each case we used the estimate (7.16), which is sharp only when (13.8) holds. Consequently, the Cauchy-Green tensor of h becomes a diagonal matrix, (13.9) $$D^*h \cdot Dh = \begin{bmatrix} |h_N|^2 & 0 & \cdots & 0 \\ 0 & |h_T|^2 & \cdots & 0 \\ & & \ddots & \\ 0 & 0 & \cdots & |h_T|^2 \end{bmatrix} \stackrel{\text{def}}{=} \mathbf{C}(x)$$ see (6.12). Whenever Hölder's inequalities were used we always arrived at one of the following conditions for h to become extremal (13.10) $$|h_T| = \frac{C_1 |h|}{|x|}, \text{ or } |h_N| = \frac{C_2 |h|}{|x|} \eta(|h|)$$ where $C_1$ and $C_2$ are constants, see (11.6) and (12.24). No matter which case we look at, the conclusion is that (13.11) $$|h_T| = \frac{\lambda |h|}{|x|} \text{ and } |h_N| = \frac{\lambda |h|}{|x|} \eta(|h|)$$ for some constant $\lambda > 0$ , because of (13.5). This constant $\lambda$ is the same for all extremal solutions; in fact, $\lambda = 1$ . This can be observed as follows. Since equality holds in either (11.6) or (12.25), at least one of the integrals $$\int_{\mathbb{A}} \frac{|h_T|^{n-1}}{|x| |h|^{n-1}} \quad \text{and} \quad \int_{\mathbb{A}} \left[ \frac{|h_N|}{|x| \eta(|h|)} \right]^n$$ is equal to Mod A. In view of (13.11) this yields $\lambda = 1$ . Hence (13.12) $$D^*h \cdot Dh = \frac{\lambda^2 |h|^2}{|x|^2} \begin{bmatrix} \eta^2(|h|) & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 \\ & & \ddots & \\ 0 & 0 & \cdots & 1 \end{bmatrix} = \mathbf{G}(x, |h|)$$ We emphasize that the metric tensor $\mathbf{G} = \mathbf{G}(x,\tau)$ , viewed as a function of two variables $x \in \mathbb{A}$ and $\tau \in (r_, R_)$ , is the same for all extremals. A key step in establishing uniqueness is:

Lemma 13.2. (UNIQUENESS IN THE POINT CAUCHY PROBLEM) Suppose we are given two solutions $h^{\circ}$ and h to the Cauchy-Green equation (13.13) such that $|h(a)| = |h^{\circ}(a)|$ for some point $a \in \Omega$ . Then there is an isometry $T: \mathbb{R}^n \to \mathbb{R}^n$ such that (13.20) $$h(x) = T h^{\circ}(x) \quad \text{for all } x \in \Omega$$ PROOF. It follows from the equation (13.13) that (13.21) $$D^h(a) Dh(a) = D^h^{\circ}(a) Dh^{\circ}(a)$$ Therefore, there is an isometry $T: \mathbb{R}^n \to \mathbb{R}^n$ such that $$Dh(a) = T \circ Dh^{\circ}(a) = D(Th^{\circ})(a)$$ With a suitable choice of the isometry we may assume without loss of generality that the first order derivatives of h and $h^{\circ}$ also coincide at a; namely, (13.22) $$|h(a)| = |h^{\circ}(a)|$$ and $Dh(a) = Dh^{\circ}(a)$ We are going to show that these two equations hold in the entire domain $\Omega$ . Obviously, the set where (13.22) hold is relatively closed in $\Omega$ . Thus, we need only show that this set is also open. To this end we consider a small ball $$\mathbb{B} = \{ x \in \mathbb{R}^n ; |x - a| \leqslant \epsilon \}$$ center at a and contained in $\Omega$ . We estimate the supremum norms of $Dh - Dh^{\circ}$ over $\mathbb{B}$ with $\epsilon$ approaching zero. $$\|Dh - Dh^{\circ}\|_{\mathscr{L}^{\infty}(\mathbb{B})} \quad \leq \quad \epsilon \|D^{2}h - D^{2}h^{\circ}\|_{\mathscr{L}^{\infty}(\mathbb{B})}$$ $$= \quad \epsilon \|\Phi(x, |h|, Dh) - \Phi(x, |h^{\circ}|, Dh^{\circ})\|_{\mathscr{L}^{\infty}(\mathbb{B})}$$ $$\leq \quad \epsilon \||h| - |h^{\circ}|\|_{\mathscr{L}^{\infty}(\mathbb{B})} + \epsilon \|Dh - Dh^{\circ}\|_{\mathscr{L}^{\infty}(\mathbb{B})}$$ Here and subsequently, the symbol $\leq$ stands for the inequality with a constant independent of $\epsilon$ . This constant varies from line to line. Noting that $$(13.23) \epsilon \| |h| - |h^{\circ}| \|_{\mathscr{L}^{\infty}(\mathbb{B})} \preccurlyeq \epsilon \| Dh - Dh^{\circ} \|_{\mathscr{L}^{\infty}(\mathbb{B})},$$ we conclude that $Dh = Dh^{\circ}$ and $|h| = |h^{\circ}|$ in $\mathbb{B}$ , as desired. The equation $Dh(x) \equiv Dh^{\circ}(x)$ implies that $h^{\circ}(x) = h(x) - 2v$ , where v is a constant vector. This combined with the condition $|h^{\circ}(x)| \equiv |h(x)|$ yields that h(x) - v is orthogonal to v. If v was not zero the image of $\Omega$ under h would be an (n-1)-hyperplane and, consequently, $J(x,h) \equiv 0$ . This is impossible for any solution to the equation (13.13), because (13.24) $$J(x,h) = \sqrt{\det \mathbf{G}(x,|h|)} \neq 0$$

Theorem 14.1. Suppose (14.54) $$\operatorname{Mod} \mathbb{A}^* > \sqrt{\frac{n-1}{n-3}} \operatorname{Mod} \mathbb{A}, \quad n \geqslant 4$$ Then (14.55) $$\inf_{h \in \mathscr{R}(\mathbb{A}, \mathbb{A}^)} \int_{\mathbb{A}} \frac{\|Dh\|^n}{|h|^n} > \inf_{h \in \mathscr{P}(\mathbb{A}, \mathbb{A}^)} \int_{\mathbb{A}} \frac{\|Dh\|^n}{|h|^n}$$ where $\mathscr{R}(\mathbb{A}, \mathbb{A}^)$ stands for the class of radial stretchings $h : \mathbb{A} \xrightarrow{\text{onto}} \mathbb{A}$ . PROOF. In Section 8.2 we have shown that the infimum in the left hand side is attained for (14.56) $$h = h^{\alpha}(x) = r_ r^{-\alpha} |x|^{\alpha - 1} x, \qquad \alpha = \frac{\operatorname{Mod} \mathbb{A}}{\operatorname{Mod} \mathbb{A}}$$ The $\mathscr{F}$ -energy of $h^{\alpha}$ is easily computed (14.57) $$\mathscr{F}_{h^{\alpha}} = (\alpha^2 + n - 1)^{\frac{n}{2}} \operatorname{Mod} A$$ We now test the infimum on the right hand side of (14.55) with the spherical mapping (which is not radial), (14.58) $$h(x) = r_* r^{-\alpha} |x|^{\alpha - 1} \Phi^{\lambda} \left(\frac{x}{|x|}\right)$$ where $\Phi^{\lambda}: \mathbb{S}^{n-1} \to \mathbb{S}^{n-1}$ is the spherical homothety defined at (14.28). If $\lambda$ is sufficiently close to 1, but different from 1, then (14.59) $$\int_{\mathbb{S}^{n-1}} \left[ \alpha^2 + (n-1) [D\Phi^{\lambda}]^2 \right]^{\frac{n}{2}} < (\alpha^2 + n - 1)^{\frac{n}{2}}$$ see (14.50). Now, the $\mathcal{F}$ -energy of h is computed as follows $$\mathscr{F}_{h} = \int_{\mathbb{A}} \left( \frac{|h_{N}|^{2}}{|h|^{2}} + (n-1) \frac{|h_{T}|^{2}}{|h|^{2}} \right)^{\frac{n}{2}}$$ $$= \int_{\mathbb{A}} \left( \alpha^{2} + (n-1) [D\Phi^{\lambda}]^{2} \right)^{\frac{n}{2}} \frac{dx}{|x|^{n}}$$ $$= \int_{\mathbb{S}^{n-1}} \left( \alpha^{2} + (n-1) [D\Phi^{\lambda}]^{2} \right)^{\frac{n}{2}} \cdot \operatorname{Mod} \mathbb{A}$$ $$< \left( \alpha^{2} + n - 1 \right)^{\frac{n}{2}} \operatorname{Mod} \mathbb{A} = \mathscr{F}_{h^{\alpha}}$$ $$(14.60)$$ as desired. The spherical homothety $\Phi^{\lambda}: \mathbb{S}^{n-1} \to \mathbb{S}^{n-1}$ works as well for the n-harmonic energy, though a computation is more involved. It results in the proof of Theorem 1.11.

Theorem 14.2. Let $n \ge 4$ and $$(14.61) \quad \delta_n = \left(\frac{\sqrt{n-1} + \sqrt{n-3}}{\sqrt{n-1} - \sqrt{n-3}}\right)^{\frac{1}{2}} \exp\left[\frac{n-2}{n\sqrt{n-1}} \tan^{-1} \sqrt{n-3}\right] \geqslant \sqrt{n}.$$ Consider the annuli $\mathbb{A} = \mathbb{A}(r,R)$ and $\mathbb{A}^ = \mathbb{A}(r_,R_*)$ , such that (14.62) $$1 < \frac{R}{r} < \delta_n \quad and \quad \frac{R}{r} > \frac{H_-(\delta_n)}{H_-(\delta_n \frac{r}{D})}.$$ Then (14.63) $$\inf_{h \in \mathcal{R}(\mathbb{A}, \mathbb{A}^)} \int_{\mathbb{A}} \|Dh\|^n > \inf_{h \in \mathcal{P}(\mathbb{A}, \mathbb{A}^)} \int_{\mathbb{A}} \|Dh\|^n$$ PROOF. First we show that the infimum on the left hand side is attained at the radial n-harmonic map (14.64) $$h(x) = H(|x|) \frac{x}{|x|}$$ of the form $H(t) = \lambda H_{-}(kt)$ , for suitable parameters $k > \frac{1}{r}$ and $\lambda > 0$ . Indeed, the energy of radial mappings is given by (14.65) $$\mathscr{E}_h = \omega_{n-1} \int_{-\pi}^{R} \left[ t^2 \dot{H}^2 + (n-1)H^2 \right]^{\frac{n}{2}} \frac{\mathrm{d}t}{t}$$ By standard convexity arguments the infimum is attained and the minimizer satisfies the Lagrange-Euler equation. This equation simply means that h is n-harmonic. Since $\operatorname{Mod} \mathbb{A} \leq \operatorname{Mod} \mathbb{A}$ we are in a situation in which h lies in the class generated by the principal solution $H_-$ ; that is, $H(t) = \lambda H_-(kt)$ , as claimed. These parameters are uniquely determined by the size of the annuli $\mathbb{A}$ and $\mathbb{A}$ . Precisely, we have (14.66) $$\lambda H_{-}(kr) = r_ \quad \text{and} \quad \lambda H_{-}(kR) = R$$ We eliminate the factor $\lambda$ by dividing the above equations, and then find k from the equation (14.67) $$\frac{H_{-}(kR)}{H_{-}(kr)} = \frac{R_{}}{r_{}} > \frac{R}{r}.$$ To find k we observe that the function $Q(t) \stackrel{\text{def}}{=} \frac{H_{-}(tr)}{H_{-}(tR)}, \frac{1}{r} < t < \infty$ , is increasing from 0 to $\frac{r}{R}$ . Indeed $$\frac{t\dot{Q}(t)}{Q(t)} = \frac{tr\dot{H}_{-}(tr)}{H_{-}(tr)} - \frac{tR\dot{H}_{-}(tR)}{H_{-}(tR)} = \eta_{-}(tr) - \eta_{-}(tR) > 0$$ because the function $\eta_{-} = \eta_{-}(s)$ is decreasing, see Figure 8 in Chapter 5. This latter statement is immediate from equation (5.29); that is, $$\left(1 + \frac{\eta_-^2}{n-1}\right)^{\frac{n-2}{2}} (\eta_-^2 - 1) = \frac{-c}{|H(s)|^n}$$ , where $H(s) = H_-(s)$ , for simplicity Here -c > 0 and |H(s)| is increasing. Also note that $$\lim_{t \to \infty} Q(t) \ = \ \frac{r}{R} \cdot \lim_{t \to \infty} \frac{\frac{H(tr)}{tr}}{\frac{H(tR)}{tR}} = \frac{r}{R} \cdot \frac{\Theta}{\Theta} = \frac{r}{R}$$ see (5.59). Now the equation (14.67) has exactly one solution. Our hypothesis at (14.62) is equivalent to (14.68) $$Q(k_{\circ}) = \frac{r_{}}{R_{}} \le Q\left(\frac{\delta_{n}}{R}\right).$$ Since Q is increasing we infer that $k_{\circ}R \leq \delta_n$ . Next, since $\eta_{-}$ is decreasing then for r < t < R, we have $$\eta_{-}(k_{\circ}t) > \eta_{-}(k_{\circ}R) \geqslant \eta_{-}(\delta_n).$$ Now we appeal to formula (14.61) and (5.47). Accordingly, $$\Gamma_{-}\left(\frac{n-3}{n-1}\right) = \delta_n.$$ On the other hand, in view of (5.53) and (5.49) $$\eta_{-}(\delta_n) = \frac{1}{u(\delta_n)} = \frac{1}{\Gamma^{-1}(\delta_n)} = \sqrt{\frac{n-1}{n-3}}.$$ Therefore, $\eta_{-}(k_{\circ}t) \geqslant \sqrt{\frac{n-1}{n-3}}$ for every r < t < R. This means that (14.69) $$\eta_H(t) > \sqrt{\frac{n-1}{n-3}}, \quad \text{for all } r \leqslant t \leqslant R$$ Now, we return to the computation of the infimum in the left hand side of (14.63) to obtain $$\inf_{h \in \mathscr{R}(\mathbb{A}, \mathbb{A}^*)} \int_{\mathbb{A}} \|Dh\|^n = \omega_{n-1} \int_r^R \left[ t^2 \dot{H}^2 + (n-1)H^2 \right]^{\frac{n}{2}} \frac{dt}{t}$$ $$= \omega_{n-1} \int_r^R \left[ H(t) \right]^n \left[ \eta_H^2(t) + n - 1 \right]^{\frac{n}{2}} \frac{dt}{t}$$ Then we test the infimum in the right hand side of (14.63) with the mapping (14.71) $$h_{\lambda}(x) = H(|x|) \Phi^{\lambda}\left(\frac{x}{|x|}\right)$$ where, as in the previous case, $\Phi^{\lambda}: \mathbb{S}^{n-1} \to \mathbb{S}^{n-1}$ is the spherical homothety. We find that $$\inf_{h \in \mathscr{P}(\mathbb{A}, \mathbb{A}^*)} \int_{\mathbb{A}} \|Dh\|^n \leqslant \int_{\mathbb{A}} \|Dh_{\lambda}\|^n$$ $$= \omega_{n-1} \int_r^R [H(t)]^n \int_{\mathbb{S}^n} \left[\eta_H^2(t) + (n-1)|D\Phi^{\lambda}|^2\right]^{\frac{n}{2}} \frac{dt}{t}$$ $$< \omega_{n-1} \int_r^R [H(t)]^n \left[\eta_H^2(t) + n - 1\right]^{\frac{n}{2}} \frac{dt}{t}$$ $$= \inf_{h \in \mathcal{R}(\mathbb{A}, \mathbb{A}^*)} \int_{\mathbb{A}} \|Dh\|^n$$ The above strict inequality follows from (14.50) applied to $\alpha = \eta_H(t) > \sqrt{\frac{n-1}{n-3}}$ , where we have chosen $\lambda \neq 1$ sufficiently close to 1.

Theorem 15.1. Suppose $h:\mathbb{A}\xrightarrow{onto}\mathbb{A}^*$ is a quasiconformal map between annuli. Then (15.5) $$\frac{1}{K_I} \leqslant \left(\frac{\operatorname{Mod} \mathbb{A}^*}{\operatorname{Mod} \mathbb{A}}\right)^{n-1} \leqslant K_O$$ This estimate is classic in the theory of quasiconformal mappings, see the pioneering work by F.W. Gehring [16]. In the proof below we shall not appeal to any advances in Quasiconformal Theory or PDEs. In fact, our proof provides a method to tackle the uniqueness problem in the borderline cases of (15.5), which seems to be unknown in higher dimensions.

Theorem 15.2. If one of the two estimates at (15.5) becomes equality, then it is attained only on the corresponding extremal mappings $h: \mathbb{A} \xrightarrow{onto} \mathbb{A}^*$ of the form (15.6) $$h(x) = \sqrt{r_ R} \left( \frac{\sqrt{rR}}{|x|} \right)^{\pm \alpha} \Phi\left( \frac{x}{|x|} \right), \quad r \leqslant |x| \leqslant R$$ where $$(15.7) \qquad \alpha = \frac{\operatorname{Mod} \mathbb{A}}{\operatorname{Mod} \mathbb{A}} = \left\{ \begin{array}{ll} K_O^{\frac{1}{n-1}} & \quad if \ \operatorname{Mod} \mathbb{A}^ \geqslant \operatorname{Mod} \mathbb{A} \\ \\ K_I^{\frac{1}{1-n}} & \quad if \ \operatorname{Mod} \mathbb{A}^* \leqslant \operatorname{Mod} \mathbb{A} \end{array} \right.$$ Here the spherical part $\Phi: \mathbb{S}^{n-1} \xrightarrow{onto} \mathbb{S}^{n-1}$ can be any homeomorphism satisfying - (i) volume condition: $J(\omega, \Phi) \equiv \pm 1$ , for a.e. $\omega \in \mathbb{S}^{n-1}$ - (ii) $\alpha$ -contraction condition: meaning that - $|D\Phi(\omega)| \leq \alpha$ , for the equality in the upper bound of (15.5) - $|(D\Phi(\omega))^{-1}| \leq 1/\alpha$ , for the equality in the lower bound of (15.5) Note, that the only volume preserving homeomorphisms $\Phi: \mathbb{S}^1 \xrightarrow{\text{onto}} \mathbb{S}^1$ are isometries. Thus, in dimension n=2, the extremal quasiconformal mappings take the form $$(15.8) h(z) = \lambda |z|^{\alpha - 1} z$$ where $\lambda$ is a complex number of modulus $r r_*^{-\alpha}$ . Example 15.4 shows that in higher dimensions there exist non-isometric homeomorphisms $\Phi: \mathbb{S}^{n-1} \xrightarrow{\text{onto}} \mathbb{S}^{n-1}$ which are volume preserving and satisfy the $\alpha$ -contraction condition, see also [42]. The exception is the conformal case of $\alpha = 1$ for which the contraction condition $|D\Phi(\omega)| \leq 1$ together with $J(\omega, \Phi) \equiv \pm 1$ imply that $\Phi$ is an isometry. As a corollary to these observations we obtain Schottky's theorem (1877), [45], in $\mathbb{R}^n$ .

Theorem 15.3. An annulus $\mathbb{A} = \mathbb{A}(r,R)$ can be mapped conformally onto $\mathbb{A}^ = \mathbb{A}(r_,R_)$ , if and only if $\frac{R}{r} = \frac{R}{r_*}$ . Moreover, modulo isometry and rescaling, every conformal mapping takes the form (15.9) $$h(x) = \begin{cases} x & \text{the identity} \\ \frac{x}{|x|^2} & \text{the inversion} \end{cases}$$ For both Theorems it involves no loss of generality in assuming that h preserves orientation and the order of boundary components of the annuli. And we do so from now on. 15.0.1. Proof of Theorems 15.1. Let us first prove the inequality at the right hand side of (15.5). We begin with the identity (7.6) and, after Hölder's inequality, use the free Lagrangian (7.4) $$\operatorname{Mod} \mathbb{A}^{} = \int_{\mathbb{A}} \frac{\mathrm{d}|h| \wedge \mathrm{d}|x|}{|h| |x|^{n-1}} = \int_{\mathbb{A}} \left\langle [Dh]^{*} \frac{h}{|h|} , \frac{x}{|x|} \right\rangle \frac{\mathrm{d}x}{|h| |x|^{n-1}}$$ $$\leq \int_{\mathbb{A}} \frac{|Dh| \, \mathrm{d}x}{|h| |x|^{n-1}} \leq \left( \int_{\mathbb{A}} \frac{|Dh|^{n}}{|h|^{n}} \right)^{\frac{1}{n}} \left( \int_{\mathbb{A}} \frac{\mathrm{d}x}{|x|^{n}} \right)^{\frac{n-1}{n}}$$ $$\leq K_{O}^{\frac{1}{n}} \left( \int_{\mathbb{A}} \frac{J(x,h) \, \mathrm{d}x}{|h(x)|^{n}} \right)^{\frac{1}{n}} \left( \operatorname{Mod} \mathbb{A} \right)^{\frac{n-1}{n}}$$ $$= K_{O}^{\frac{1}{n}} \left( \operatorname{Mod} \mathbb{A}^{*} \right)^{\frac{1}{n}} \left( \operatorname{Mod} \mathbb{A} \right)^{\frac{n-1}{n}}$$ as desired. For the left hand side at (15.5) we begin with the identity (7.12) to compute in the similar fashion that $$\operatorname{Mod} A = \int_{\mathbb{A}} \frac{\mathrm{d}|x|}{|x|} \wedge h^{\sharp} \omega = \int_{\mathbb{A}} \left\langle [D^{\sharp}h] \frac{x}{|x|}, \frac{h}{|h|} \right\rangle \frac{\mathrm{d}x}{|x| |h|^{n-1}}$$ $$\leq \int_{\mathbb{A}} \frac{|D^{\sharp}h| \, \mathrm{d}x}{|x| |h|^{n-1}} \leq \left( \int_{\mathbb{A}} \frac{|D^{\sharp}h|^{\frac{n}{n-1}}}{|h|^{n}} \right)^{\frac{n-1}{n}} \left( \int_{\mathbb{A}} \frac{\mathrm{d}x}{|x|^{n}} \right)^{\frac{1}{n}}$$ $$\leq K_{I}^{\frac{1}{n}} \left( \int_{\mathbb{A}} \frac{J(x,h) \, \mathrm{d}x}{|h|^{n}} \right)^{\frac{n-1}{n}} (\operatorname{Mod} A)^{\frac{1}{n}}$$ $$= K_{I}^{\frac{1}{n}} (\operatorname{Mod} A^{*})^{\frac{n-1}{n}} (\operatorname{Mod} A)^{\frac{1}{n}}$$ as desired. 15.0.2. Proof of Theorem 15.2. Concerning the borderline cases, let $h: \mathbb{A} \xrightarrow{\text{onto}} \mathbb{A}$ denote the extremal mapping for the right hand side of (15.5). We simplify the matters by assuming that $r = r_ = 1$ . Note that the power mapping (15.12) $$h^{\alpha}(x) = |x|^{\alpha - 1}x, \qquad \alpha = \frac{\operatorname{Mod} \mathbb{A}^*}{\operatorname{Mod} \mathbb{A}}$$ is among the extremals. The borderline equation $$(15.13) \qquad (\operatorname{Mod} \mathbb{A}^*)^{n-1} = K_O (\operatorname{Mod} \mathbb{A})^{n-1}$$ implies that we have equalities everywhere in (15.10). This amounts to the following two conditions. (i) $$\left\langle D^h \cdot \frac{h}{|h|}, \frac{x}{|x|} \right\rangle \equiv \left| D^h \cdot \frac{h}{|h|} \right| \equiv |Dh| \equiv \alpha \frac{|h|}{|x|}$$ , $\alpha$ - a positive constant and (ii) $$|Dh|^n \equiv K_O J(x,h)$$ The first set of equations is fulfilled if and only if (15.14) $$D^*h \cdot \frac{h}{|h|} \equiv |Dh(x)| \frac{x}{|x|} \equiv \alpha \frac{|h|}{|x|} \frac{x}{|x|}, \quad \alpha$$ - a positive constant We identify $\alpha$ from the following equations (15.15) $$\frac{\alpha^n}{|x|^n} \equiv \frac{|Dh|^n}{|h|^n} \equiv K_O \frac{J(x,h)}{|h|^n}$$ Upon integration over the annulus $\mathbb{A}$ we obtain (15.16) $$\alpha^n \operatorname{Mod} A = K_O \operatorname{Mod} A^*$$ which in view of (15.13) yields (15.17) $$\alpha = \frac{\operatorname{Mod} A}{\operatorname{Mod} A} = \sqrt[n-1]{K_O} \geqslant 1$$ Now, an interesting nonlinear PDE arises from (i) and, in a more direct manner, from (15.14) (15.18) $$D^*h \cdot \frac{h}{|h|^2} = \alpha \frac{x}{|x|^2}$$ Rather unexpectedly we can easily solve this equation for |h|. Let us express it as $$(15.19) \nabla \log |h|^2 = \alpha \nabla \log |x|^2$$ Hence, in view of the normalization |h(x)| = 1 for |x| = 1, we find that $$(15.20) |h(x)| = |x|^{\alpha}$$ Now, h takes the form $$h(x) = |x|^{\alpha} \Psi(x)$$ , with $\Psi : \mathbb{A} \to \mathbb{S}^{n-1}$ The map h, being a homeomorphism in $\mathcal{W}^{1,n}(\mathbb{A},\mathbb{A}^*)$ , is differentiable almost everywhere, then so is $\Psi$ . From (15.14) we also find the operator norm of the differential (15.21) $$|Dh(x)| = \alpha \frac{|h|}{|x|} = \alpha |x|^{\alpha - 1} = |Dh^{\alpha}(x)|$$ This, in view of (ii), yields (15.22) $$J(x,h) = \alpha |x|^{n\alpha - n} = J(x,h^{\alpha})$$ Further examination of the mapping $\Psi: \mathbb{A} \to \mathbb{S}^{n-1}$ will reveal that its normal derivative vanishes almost everywhere. To this end, let $x_o \in \mathbb{A}$ be a point of differentiability of $\Psi$ . For small real numbers $\epsilon$ we consider Taylor's expansions of order one, (15.23) $$\Psi(x_0 + \epsilon x_0) = \Psi(x_0) + \epsilon \Psi_N(x_0) + o(\epsilon)$$ and (15.24) $$h(x_{\circ} + \epsilon x_{\circ}) - h(x_{\circ}) = \epsilon \left[ Dh(x_{\circ}) \right] x_{\circ} + o(\epsilon)$$ Upon elementary computation, letting $\epsilon$ go to zero, we arrive at the following estimate $$(15.25) |\alpha \Psi(x_\circ) + \Psi_N(x_\circ)| \leqslant \alpha$$ We square it to obtain (15.26) $$\alpha^2 + 2\alpha \langle \Psi, \Psi_N \rangle + |\Psi_N|^2 \leqslant \alpha^2$$ It is important to observe that $\Psi$ is orthogonal to its directional derivatives, because $\langle \Psi, \Psi \rangle = |\Psi|^2 \equiv 1$ . Thus, in particular, $\Psi_N(x_\circ) = 0$ . This simply means that (15.27) $$\Psi(x) = \Phi\left(\frac{x}{|x|}\right), \quad \text{where } \Phi: \mathbb{S}^{n-1} \to \mathbb{S}^{n-1}$$ Obviously, $\Phi \in \mathcal{W}^{1,n}(\mathbb{S}^{n-1},\mathbb{S}^{n-1})$ is a homeomorphism of $\mathbb{S}^{n-1}$ onto itself. It induces the linear tangent map (15.28) $$D\Phi(\omega): T_{\omega}\mathbb{S}^{n-1} \to T_{\sigma}\mathbb{S}^{n-1}, \quad \sigma = \Phi(\omega)$$ The Jacobian determinant of $\Phi$ , with respect to the standard volume form on $\mathbb{S}^{n-1}$ , will be denoted by (15.29) $$J(\omega, \Phi) = \det D\Phi$$ It relates to the Jacobian determinant of h by the rule $$(15.30) J(x,h) = \alpha |x|^{n\alpha-n} \langle \Phi, \Phi_{T_2} \times \dots \times \Phi_{T_n} \rangle$$ (15.31) $$= \alpha |x|^{n\alpha - n} J(\omega, \Phi), \quad \omega = \frac{x}{|x|}$$ Thus, in view of (15.22) we see that $\Phi: \mathbb{S}^{n-1} \to \mathbb{S}^{n-1}$ is volume preserving. Here we can take the cross product of directional derivatives with respect to an arbitrary positively oriented orthonormal frame $T_2, ..., T_n$ in $\mathbf{T}_{\omega} \mathbb{S}^{n-1}$ . Thus, in view of (15.22), we obtain (15.32) $$J(\omega, \Phi) = |\Phi_{T_2} \times ... \times \Phi_{T_n}| \leqslant |\Phi_{T_2}| \cdots |\Phi_{T_n}| \leqslant |D\Phi|^{n-1}$$ Next, we look at the Cauchy-Green tensor of h in terms of the orthonormal frame $N, T_2, ..., T_n$ at $T_x \mathbb{A}$ $$D^*h Dh = \begin{bmatrix} ---h_N & --- \\ ---h_{T_2} & --- \\ \vdots \\ ---h_{T_n} & --- \end{bmatrix} \begin{bmatrix} ---h_N & h_{T_2} & h_{T_n} \\ ---h_{T_n} & --- \end{bmatrix}$$ $$(15.33) = |x|^{2\alpha-2} \begin{bmatrix} ---\alpha\Phi & --- \\ ---\Phi_{T_2} & --- \\ \vdots \\ ---\Phi_{T_n} & --- \end{bmatrix} \begin{bmatrix} ----h_N & h_{T_2} & h_{T_n} \\ ----h_{T_n} & --- & h_{T_n} \\ -----------------------------------$$ and (15.34) $$D^*\Phi D\Phi = \begin{bmatrix} |\Phi_{T_2}|^2 & \dots & \langle \Phi_{T_2}, \Phi_{T_n} \rangle \\ \vdots & \vdots & \\ \langle \Phi_{T_n}, \Phi_{T_2} \rangle & \dots & |\Phi_{T_n}|^2 \end{bmatrix}$$ Let $0 \le \lambda_2(x) \le ... \le \lambda_n(x)$ denote the singular values of $D\Phi$ , meaning that $\lambda_2^2(x) \le ... \le \lambda_n^2(x)$ are eigenvalues of $D^*\Phi D\Phi$ . It follows from (15.10) that the numbers $$\lambda_2(x)|x|^{\alpha-1}, \dots, \lambda_n(x)|x|^{\alpha-1}$$ and $\alpha|x|^{\alpha-1}$ are the singular values of Dh. Since $\alpha |x|^{\alpha-1} = |Dh|$ , this latter number is the largest singular value, $$(15.35) 0 \leqslant \lambda_2(x) \leqslant \dots \leqslant \lambda_n(x) \leqslant \alpha$$ In particular, $$[D\Phi(x)] \equiv \lambda_n(x) \leqslant \alpha$$ as desired. This computation also shows that the inequality $|D\Phi(x)| \leq \alpha$ is both sufficient and necessary for the equation $|Dh(x)| = \alpha |x|^{\alpha-1}$ . A backwards inspection of the above arguments reveals that $h(x) = |x|^{\alpha} \Phi\left(\frac{x}{|x|}\right)$ is an extremal map if and only if $|D\Phi(x)| \leq \alpha$ and $J(\omega, \Phi) \equiv 1$ . It is rewarding to look at the inverse map $f = h^{-1} : \mathbb{A}^* \xrightarrow{\text{onto}} \mathbb{A}$ , which takes the form (15.37) $$f(y) = |y|^{\frac{1}{\alpha}} \Phi^{-1} \left( \frac{y}{|y|} \right)$$ We observe that for every quasiconformal map h and its inverse $h^{-1} = f$ it holds (15.38) $$K_O(x,h) = K_I(y,f), \quad y = h(x)$$ Now in much the same way we find all extremals for the left hand side of (15.5). Theorem 15.2 follows. Let us finish this subsection with an example of the volume preserving mappings of $\mathbb{S}^2 \subset \mathbb{R}^3$ . EXAMPLE 15.4. For every $\epsilon > 0$ there exists a homeomorphism $\Phi \in \mathcal{W}^{1,\infty}(\mathbb{S}^2,\mathbb{S}^2)$ , not rotation, such that $|\Phi(\omega) - \omega| \leq \epsilon$ and $J(\omega,\Phi) \equiv 1$ for almost every $\omega \in \mathbb{S}^2$ . 15.0.3. Construction of Φ. We shall view Φ as a small perturbation of id: $\mathbb{S}^2 \to \mathbb{S}^2$ . The construction of such perturbation is made in three steps. In the first step we project the sphere $x^2 + y^2 + z^2 = 1$ with two poles removed, $z \neq \pm 1$ onto the cylinder $x^2 + y^2 = 1$ , -1 < z < 1. The horizontal rays from the axis of the cylinder project the points of the sphere towards the surface of the cylinder. This projection, sometimes attributed to Archimedes, is well known as Lambert's Cylindrical Projection (Johann H. Lambert 1772). In the second step we cut the cylinder along a path from the top to the bottom circles and unroll it flat. The third map is a piece-wise linear perturbation of the triangle ABC inside the flat region, see Figure 1. It keeps A, B, C fixed while permuting X, Y, Z in the cyclic way, $X \to Y \to Z \to X$ . It is geometrically clear that such piece-wise linear deformation preserves the area. Moreover, choosing X, Y, Z close to the barycenter makes the deformation arbitrarily close to the identity. Finally we roll it back onto the cylinder and use inverse of Lambert's projection to end up with the desired area preserving perturbation of the identity. REMARK 15.5. As pointed out by the referee, a simpler example can be given : $(\theta, \varphi) \mapsto (\theta + g(\theta), \varphi)$ , where $\theta, \varphi$ are the longitude and latitude, and g is any Lipschitz function. Acknowledgements. We thank Leonid Kovalev for valuable discussions on the subject of this paper and the referee for very careful reading of the paper and many useful suggestions.  Figure 1. Constructing an area-preserving Lipschitz homeomorphism.

Definition 1.1. A homeomorphism $h: \mathbb{X} \to \mathbb{R}^n$ of Sobolev space $\mathcal{W}^{1,1}_{loc}(\mathbb{X},\mathbb{R}^n)$ is said to have finite outer distortion if $$\|\,Dh(x)\,\|^{\,n}\leqslant n^{\frac{n}{2}}\,K(x)\,J(x,h)$$ for some measurable function $1 \leq K(x) < \infty$ . The smallest such K(x) is called the outer distortion, denoted by $\mathbb{K}_{\mathcal{O}}(x,h)$ . Then h is K-quasiconformal if $\mathbb{K}_{\mathcal{O}}(x,h) \leq K$ for some constant K. A concept somewhat dual to outer distortion is the inner distortion. For this, consider the cofactor matrix $D^{\sharp}h$ (it represents infinitesimal deformations of (n-1)-dimensional entities) defined for invertible Dh via Cramer's rule $$D^{\sharp}h \cdot D^*h = J(x,h)\mathbf{I}$$ For a map $h \in \mathcal{W}^{1,1}_{loc}(\mathbb{X}, \mathbb{R}^n)$ , not necessarily a homeomorphism, but with nonnegative Jacobian, we introduce the inner distortion function (1.8) $$\mathbb{K}_{I}(x,h) = \begin{cases} \frac{\|D^{\sharp}h(x)\|^{n}}{n^{n/2} [J(x,h)]^{n-1}} & \text{if } J(x,h) > 0\\ 1 & \text{if } J(x,h) = 0 \end{cases}$$ Remark 1.2. Any map of finite outer distortion has finite inner distortion and $\mathbb{K}_I(x) \leq \mathbb{K}_O(x)^{n-1}$ , but not vice versa. It is again interesting to find the place for such mappings in continuum mechanics. The latter deals with the positive definite matrix $\mathbf{C}(x) = D^*h(x)\,Dh(x)$ as the right Cauchy-Green deformation tensor. While on the other hand, there is a fundamental interplay between mappings of finite distortion and the Beltrami equation (1.9) $$D^*h(x) Dh(x) = J(x,h)^{\frac{2}{n}} \mathbf{G}(x), \quad \det \mathbf{G}(x) \equiv 1$$ Thus $\mathbf{G} = \mathbf{G}(x)$ , called the distortion tensor of h, is none other than the Cauchy-Green tensor renormalized so as to have determinant identically equal to one. The symmetric positive definite matrix function $\mathbf{G} = \mathbf{G}(x) = [G_{ij}(x)] \in \mathbb{R}^{n \times n}$ can be viewed as a Riemann metric tensor on $\mathbb{X}$ . In this way h becomes conformal with respect to this, usually only measurable, metric structure on $\mathbb{X}$ . Thus $\mathbf{G}(x)$ is uniformly elliptic in case of K-quasiconformal mappings. It is in this Riemannian manifold framework that variational interpretations of quasiconformal mappings really crystalize. For example, the solutions to the Beltrami equation (1.9) are none other then the absolute minimizers of their own energy integrals. Indeed, a homeomorphism $h: \mathbb{X} \to \mathbb{Y}$ of Sobolev class $\mathcal{W}^{1,n}(\mathbb{X}, \mathbb{Y})$ solves the Beltrami equation (1.9) if and only if (1.10) $$\mathscr{E}_h \stackrel{\text{def}}{=} \int_{\mathbb{X}} \mathbf{E}(x, Dh) \, \mathrm{d}x = n^{\frac{n}{2}} \int_{\mathbb{X}} J(x, h) \, \mathrm{d}x = n^{\frac{n}{2}} |\mathbb{Y}|$$ where the integrand is defined on $\mathbb{X} \times \mathbb{R}^{n \times n}$ by the rule $$\mathbf{E}(x,\xi) = \left( \operatorname{Tr} \left[ \xi \mathbf{G}^{-1}(x) \xi^* \right] \right)^{\frac{n}{2}}, \quad \xi \in \mathbb{R}^{n \times n}$$ As in the conformal case, for all other homeomorphisms $g: \mathbb{X} \xrightarrow{\text{onto}} \mathbb{Y}$ , in the Sobolev space $\mathcal{W}^{1,n}(\mathbb{X},\mathbb{Y})$ , we have the lower bound $\mathscr{E}_g \geqslant n^{\frac{n}{2}}[\mathbb{Y}]$ . The most appealing conclusion is a connection between the n-harmonic energy of $h: \mathbb{X} \to \mathbb{Y}$ and the inner distortion function of the inverse mapping $f = h^{-1}: \mathbb{Y} \to \mathbb{X}$ .

Definition 5.2. The term principal solution pertains to each of the following four functions of class $\mathscr{C}^2(0,\infty)$ which solve the equation $\mathcal{L}H = constant$ ; $$(5.20) H_{\circ}(t) = t, \mathcal{L}H_{\circ} \equiv 0$$ (5.21) $$H_{\infty}(t) = \frac{1}{t}, \qquad \mathcal{L}H_{\infty} \equiv 0$$ (5.22) $$\mathcal{L}H_{+} \equiv 1, \qquad H_{+}(1) = 1$$ (5.23) $$\mathcal{L}H_{-} \equiv -1, \qquad H_{-}(1) = 0$$ We shall solve these Cauchy problems and show that both $H_+$ and $H_-$ are actually $\mathscr{C}^{\infty}$ -smooth. $^{^2}$ This noninjective solution will play an important role in the sequel.  FIGURE 2. The principal solutions for n = 2. Similar ones for $n \ge 3$ are illustrated in Figures 7 and 10. The name "principal solutions" is given to $H_{\circ}$ , $H_{\infty}$ , $H_{+}$ and $H_{-}$ because they generate all radial n-harmonics. We denote the corresponding radial mapping by $h_{\circ}$ , $h_{\infty}$ , $h_{+}$ and $h_{-}$ , and call them principal n-harmonics. Precisely, we have