Energy-minimal Principles in Geometric Function Theory

Theorem 2.1. Let $h: \mathbb{X} \xrightarrow{\operatorname{onto}} \mathbb{Y}$ be a conformal map between bounded domains $\mathbb{X}, \mathbb{Y} \subset \mathbb{R}^2 \simeq \mathbb{C}$ . Then every orientation preserving homeomorphism $f: \mathbb{X} \xrightarrow{\operatorname{onto}} \mathbb{Y}$ of Sobolev class $\mathcal{W}^{1,2}(\mathbb{X},\mathbb{C})$ has Dirichlet energy at least that of h. Equality occurs if and only if f is conformal as well.

Theorem 4.1. Formula (4.3) represents all null Lagrangians of the form $$\mathbf{N}(x, f, Df) dx = \mathbf{N}(Df) dx$$ This result goes back to [33, 12, 17, 42]. Should it be required to appeal to a first order null Lagarangians of the general form E(x, f, Df) dx, we refer to the work by de Franchis [11]. The utility of null Lagrangians is best illustrated for polyconvex functionals discussed next. 4.2. Polyconvexity. In his mathematical models for nonlinear elasticity [4] J. Ball made the crucial observation that if the convexity of the stored energy integrand $\mathbf{E}(x,h,Dh)$ , with respect to the deformation gradient $Dh(x) \in \mathbb{R}^{m \times n}$ , must be ruled out, it could be replaced by a weaker requirement; namely, expressing the integrand as a convex function of subdeterminants of Dh. (4.4) $$\mathbf{E}(x, h, Dh) = \mathbf{E}_{\mathbf{x}}(x, h, \text{ subdeterminants of } Dh)$$ The number of all $\ell \times \ell$ -subdeterminants with $0 \le \ell \le \min\{m, n\}$ is equal to $\binom{m+n}{n}$ . Thus we are assuming that for every pair $(x, y) \in \mathbb{X} \times \mathbb{Y}$ the function $$\mathbf{E}_{\mathbf{x}}(x,y,\cdots); \mathbb{R}^{\binom{m+n}{n}} \to \mathbb{R},$$ is convex. The idea of minimizing polyconvex energy functionals is based on a quite far reaching extension of the direct method of the calculus of variations that we outlined earlier. It has turned out that so far this is the only practical idea that offers substantially more than that of just minimising convex energies, for instance the p-harmonic example. Remark 4.2. There is an extensive literature dealing with Morrey's notion of Quasiconvexity, [37]. However, from the point of view of mathematical challenges, this concept is not much more than a reformulation of the lower semi-continuity of the energy functionals and as such there remain only technical issues. One needs to develop this idea much further mathematically before it might be usefully applied. 4.3. Nearly Conformal Deformations. Given a bounded domain $\mathbb{X} \subset \mathbb{R}^n$ , we look at the mappings $h: \mathbb{X} \to \mathbb{R}^n$ in the reflexive Banach space $\mathfrak{B} \stackrel{\text{def}}{=} \mathscr{W}^{1,np}(\mathbb{X},\mathbb{R}^n)$ , $1 \leq p < \infty$ . Then the following nonlinear functional is well defined on this space. (4.5) $$\mathscr{E}_{n,p}[h] = \int_{\mathbb{X}} \left( |Dh(x)|^n - n^{n/2} \det Dh(x) \right)^p dx < \infty$$ Note that the integrand is non-negative and vanishes only if h satisfies the n-dimensional variant of the Cauchy-Riemann system. (4.6) $$\mathcal{K}(Dh) \equiv 0$$ , where $\mathcal{K}(X) \stackrel{\text{def}}{=} |X|^n - n^{n/2} \det X \geqslant 0$ , for $X \in \mathbb{R}^{n \times n}$ This motivates our calling $\mathscr{E}_{n,p}$ a nearly conformal energy functional. The Dirichlet boundary value problem consists of minimizing $\mathscr{E}_{n,p}[h]$ subject to mappings $h \in \mathfrak{B}_{\circ} \stackrel{\text{def}}{=} h_{\circ} + \mathscr{W}_{0}^{1,np}(\mathbb{X},\mathbb{R}^{n})$ , where $h_{\circ} \in \mathfrak{B}$ is given boundary data. Here are the essential steps in the Direct Method. • Coercivity in the mean The reason why the above example is approachable is that while we do not have point-wise coercivity in terms of the integrand, the energy functional $\mathcal{E}_{n,p}$ still exhibits coercivity in the sense of integral means; precisely, (4.7) $$\int_{\mathbb{Y}} |Dh(x)|^{np} dx \sim \mathscr{E}_{n,p}[h] + \int_{\mathbb{Y}} |Dh_{\circ}(x)|^{np} dx$$ To see this we appeal to the general (but not vary obvious) estimate (10.12) in [24] for mappings $f \in \mathcal{W}^{1,np}_{\circ}(\mathbb{X},\mathbb{R}^n)$ $$\int_{\mathbb{X}} |Df(x)|^{np} \, \mathrm{d}x \sim \mathscr{E}_{n,p}[f]$$ applied to the mapping $f = h - h_{\circ}$ . • Subgradient Estimate This is a fairly direct consequence of polyconvexity of the integrand. Precisely, for $n \times n$ -matrices $X, X_o \in \mathbb{R}^{n \times n}$ we have $$\mathcal{K}^{p}(X) - \mathcal{K}^{p}(X_{\circ}) \geqslant p \mathcal{K}^{p-1}(X_{\circ}) \left[ \mathcal{K}(X) - \mathcal{K}(X_{\circ}) \right] \geqslant n p \left\langle \mathcal{K}^{p-1}(X_{\circ}) |X_{\circ}|^{n-2} X_{\circ} \middle| X - X_{\circ} \right\rangle -n^{n/2} p \mathcal{K}^{p-1}(X_{\circ}) \left[ \det X - \det X_{\circ} \right]$$ • Lower semi-continuity The required inequality (3.3) can be achieved by applying the above subgradient estimate to $X = Df_i(x)$ and $X_o = Df(x)$ . We conclude, upon integrating over X, that $$\mathcal{E}_{n,p}[f_i] - \mathcal{E}_{n,p}[f]$$ $$\geqslant n p \int_{\mathbb{X}} \left\langle \mathcal{K}^{p-1}(Df) | Df|^{n-2} Df \, \Big| \, Df_i - Df \right\rangle \quad \text{(converging to 0)}$$ $$-n^{n/2} p \int_{\mathbb{X}} \mathcal{K}^{p-1}(Df) \left[ \det Df_i - \det Df \right] \quad \text{(converging to 0)}$$ The first limit is justified by the fact that $Df_i \rightharpoonup Df$ , weakly in the space $\mathscr{L}^{np}(\mathbb{X})$ , and the integration takes place against the factor $\mathcal{K}^{p-1}(Df) |Df|^{n-2}Df \approx |Df|^{np-1}$ which lies in the dual space $\mathscr{L}^{\frac{np}{np-1}}(\mathbb{X})$ . Similarly, for p>1, the null Lagrangians $\det Df_i$ converge to $\det Df$ weakly in $\mathscr{L}^p(\mathbb{X})$ and we integrate them against a function $\mathcal{K}^{p-1}(Df) \approx |Df|^{np-n}$ which belongs to the dual space $\mathscr{L}^{\frac{p}{p-1}}(\mathbb{X})$ . The case p=1 needs handling with greater care. It is not generally true that $\lim_{X} \int_{\mathbb{X}} \det Df_i(x) \, dx = \int_{\mathbb{X}} \det Df(x) \, dx$ , whenever $f_i \rightharpoonup f$ , weakly in $\mathscr{W}^{1,n}(\mathbb{X})$ . For instance the sequence of Möbius transformations $f_i:\mathbb{B} \xrightarrow{\mathrm{onto}} \mathbb{B}$ of the unit ball $\mathbb{B} \subset \mathbb{R}^n$ such that $f_i(0) \to a \in \partial \mathbb{B}$ . Their weak limit (indeed locally uniform in $\mathbb{B}$ ) is $f \equiv a$ , which has vanishing Jacobian. However $\int_{\mathbb{R}} \det Df_i(x) \, dx = |\mathbb{B}| > 0$ . The situation is quite different if we confine ourselves to the energy-minimising sequence of mappings $f_i \in \mathfrak{B}_{\circ}$ in which $f_i \in f_0 + \mathscr{W}_0^{1,n}(\mathbb{X})$ . Once the boundary values of $f_i$ are fixed, the weak limit enjoys the same boundary values; that is, $f \in f_0 + \mathscr{W}_0^{1,n}(\mathbb{X})$ . Thus, for all i = 1, 2, ..., we have the following identities $$\int_{\mathbb{X}} \det Df_i(x) dx = \int_{\mathbb{X}} \det Df_0(x) dx = \int_{\mathbb{X}} \det Df(x) dx,$$ by the very definition of null Lagrangians. Now, with these estimates at hand we may follow the principles of the direct method, and thereby obtain the following.

Proposition 4.3. The nearly conformal energy (4.5), subject to a given boundary data $h_o \in \mathcal{W}^{1,np}(\mathbb{X},\mathbb{R}^n)$ , attains its infimum. The example above shows how the over-arching strategy of the direct method can be nuanced in sophistication at each step. The following specific example further demonstrates this point.

Proposition 5.2. Under the coercivity condition above, the determinants $\det Dh_{\kappa}$ are bounded in the Hardy space $\mathscr{H}^{1}(\mathbb{X}) \subset \mathscr{L}^{1}(\mathbb{X})$ . As such, they converge to $\det Dh_{\infty} \in \mathscr{L}^{1}(\mathbb{X})$ in the sense of distributions, (5.3) $$\int_{\mathbb{X}} \eta(x) \det Dh_{\kappa}(x) dx \to \int_{\mathbb{X}} \eta(x) \det Dh_{\infty}(x) dx , \text{ for } \eta \in \mathscr{C}_{0}^{\infty}(\mathbb{X})$$ In fact a stronger statement holds; namely, since the Hardy space $\mathscr{H}^1(\mathbb{R}^n)$ is the dual of $VMO(\mathbb{R}^n)$ (functions of vanishing mean oscillation), we conclude that formula (5.3) remains valid for the test functions $\eta \in VMO(\mathbb{R}^n)$ with compact support in $\mathbb{X}$ . For a discussion and results concerning biting convergence see [18]. Let us demonstrate the lines of reasoning for (5.3) by using the results of [28].

Theorem 5.3. The energy functional (5.1) assumes its infimum in $\mathfrak{B}_{\circ}$ . In effect, the weak $\mathscr{W}^{1,n-1}$ -limit map $h_{\kappa} \rightharpoonup h_{\infty}$ minimizes the energy.

Proposition 6.2. The following differential n-form $\mathbf{F}(x, h, Dh) dx$ , : (6.5) $$\frac{(d|h|) \wedge \star d|x|}{|h||x|^{n-1}} \stackrel{\text{def}}{=} \sum_{i=1}^{n} \frac{x_i dx_1 \wedge \ldots \wedge dx_{i-1} \wedge d|h| \wedge dx_{i+1} \wedge \ldots \wedge dx_n}{|h||x|^n}$$ is a free Lagrangian in the homotopy class of orientation preserving homeomorphisms that preserve the order of the boundary components of the annuli $\mathbb{A}$ and $\mathbb{A}$ . Precisely, for such homeomorphisms of Sobolev class $\mathcal{W}^{1,1}(\mathbb{A}, \mathbb{A}^)$ , we have (6.6) $$\int_{\mathbb{A}} \mathbf{F}(x, h, Dh) \, \mathrm{d}x = \int_{\mathbb{A}} \frac{d|h| \wedge \star d|x|}{|h| |x|^{n-1}} = \operatorname{Mod} \mathbb{A}^ \stackrel{\text{def}}{=} \log \frac{R}{r^*}$$ Here we have used the Hodge star duality operator in the exterior algebra; namely, $*: \Lambda^1(\mathbb{R}^n) \xrightarrow{\text{onto}} \Lambda^{n-1}(\mathbb{R}^n)$ . Observe that the function $|h|: \mathbb{A} \to (r_, R_)$ extends continuously to the closure of $\mathbb{A}$ . That h preserves the order of the spherical boundary components simply means that $|h(x)| = r$ for |x| = r and $|h(x)| = R$ for |x| = R. Accordingly, $$\mathbf{F} dx = \frac{d|h| \wedge \star d|x|}{|h| |x|^{n-1}} \in \mathscr{F}_L(\mathbb{A}, \mathbb{A}^*)$$ $$\mathscr{F}_L^4$$ ) (Spherical derivatives of $h$ ; $\mathscr{F}_L(\mathbb{A}, \mathbb{A}^*)$ ) Another free Lagrangian in $\mathbf{F}(\cdot,\cdot,\cdot)\in\mathscr{F}_L(\mathbb{A},\mathbb{A}^*)$ , dual to that in Proposition 6.2, exploits topological degree of the mappings $h:\mathbb{S}^{n-1}_t\to\mathbb{R}^n\setminus\{0\}$ restricted to the concentric spheres of radii $t\in(r,R)$ . The degree is equal to 1 on every sphere, which yields

Proposition 6.3. The following differential n-form $\mathbf{F}(x,h,Dh) dx$ : (6.7) $$\frac{d|x|}{|x|} \wedge h^{\sharp}\omega = \sum_{i=1}^{n} \frac{h^{i} dh^{1} \wedge \dots \wedge dh^{i-1} \wedge d|x| \wedge dh^{i+1} \wedge \dots \wedge dh^{n}}{|x| |h|^{n}}$$ is a free Lagrangian in the class of all orientation preserving homeomorphisms $h \in \mathcal{W}^{1,n-1}(\mathbb{A},\mathbb{A}^*)$ . Precisely, we have (6.8) $$\int_{\mathbb{A}} \mathbf{F}(x, h, Dh) \, \mathrm{d}x = \int_{\mathbb{A}} \frac{d|x|}{|x|} \wedge h^{\sharp} \omega = \operatorname{Mod} \mathbb{A} \stackrel{\text{def}}{=} \log \frac{R}{r}$$ Here $h^{\sharp}\omega$ stands for the pullback of the (n-1)-area form defined in $\mathbb{A}^*$ ; namely, $$\omega(y) = \sum_{i=1}^n (-1)^i \, \frac{y^i \, dy^1 \wedge \ldots \wedge dy^{i-1} \wedge dy^{i+1} \wedge \ldots \wedge dy^n}{|y|^n}$$ This is none other than the (n-1) area form on any (n-1)-closed surface in $\mathbb{A}^*$ that is homologous to $\mathbb{S}^{n-1}$ . A far more detailed exposition, suited to this example, is presented in [29, Chapters 6 and 7]. 6.2. The Nitsche frictionless problem. Given a pair planar annuli $\mathbb{A} = \{x \in \mathbb{C} : r < |x| < R\}$ and $\mathbb{A}^ = \{y \in \mathbb{C} : r_ < |y| < R_\}$ , the objective is to minimize the Dirichlet energy subject to Sobolev homeomorphisms $h : \mathbb{A} \xrightarrow{\text{onto}} \mathbb{A}$ in $\mathcal{W}^{1,2}(\mathbb{A}, \mathbb{C})$ . $$\mathscr{E}_2[h] = \int_{\mathbb{A}} |Dh(x)|^2 \, \mathrm{d}x$$ Note that no boundary values of these homeomorphisms are prescribed, whence the name frictionless is given to this problem. We denote this class of mappings by $\mathcal{H}^{1,2}(\mathbb{A},\mathbb{A}^*)$ . Naturally, polar coordinates (6.9) $$x = t e^{i\theta}, \quad r < t < R \quad \text{and} \quad 0 \leqslant \theta < 2\pi$$ are best suited. The radial (normal) and angular (tangential) derivatives of h are defined by $$(6.10) h_{\scriptscriptstyle N}(x) = \frac{\partial h(te^{i\theta})}{\partial t}, t = |x|$$ and $$(6.11) h_{\scriptscriptstyle T}(x) = \frac{1}{t} \frac{\partial h(te^{i\theta})}{\partial \theta} \,, t = |x|$$ The stored energy integrand takes the form $$|Dh(x)|^2 = |h_N(x)|^2 + |h_T(x)|^2$$ This also provides an effective formula for the Jacobian determinant $$J_h(x) = \det Dh(x) = \operatorname{Im}(\overline{h_N}h_T) \leqslant |h_N| |h_T|.$$ Our free Lagrangians $\mathscr{F}_L^1$ ), ..., $\mathscr{F}_L^4$ ) in $\mathscr{F}_L(\mathbb{A}, \mathbb{A}^)$ can easily be stated using polar coordinates in even slightly greater generality. The main players and their energy integrals for $h \in \mathscr{H}^{1,2}(\mathbb{A}, \mathbb{A}^)$ are: $$\begin{split} \mathscr{F}_{1}) & \int_{\mathbb{A}} M(x) \, dx \,, \qquad M \in \mathscr{L}^{1}(\mathbb{A}) \\ \mathscr{F}_{2}) & \int_{\mathbb{A}} N(|h|) J_{h}(x) \, \mathrm{d}x \, = \, 2 \, \pi \int_{r_{}}^{R_{}} N(s) \, s \, \mathrm{d}s \\ \mathscr{F}_{3}) & \int_{\mathbb{A}} A(|h|) \frac{|h|_{N}}{|x|} \, \mathrm{d}x \, = \, 2 \pi \int_{r_{}}^{R_{}} A(s) \, \mathrm{d}s \qquad A \in \mathscr{L}^{1}(r_{}, R_{}) \\ \mathscr{F}_{4}) & \int_{\mathbb{A}} B(|x|) \operatorname{Im} \frac{h_{T}}{h} \, \mathrm{d}x \, = \, 2 \pi \int_{r_{*}}^{R} B(t) \, dt, \qquad B \in \mathscr{L}^{1}(r, R). \end{split}$$ 6.3. Energy minimizers among radial mappings. It is natural to first look at the radial mappings as candidates for energy-minimizers. However, this expectation is far from being guaranteed. In spite of the radial symmetry of the annuli and the invariance of the Dirichlet energy under rotations of $\mathbb A$ and $\mathbb A^*$ , such a lack of symmetry of the energy-minimizers has, quite surprisingly, been confirmed already in the analogous Nitsche problem in dimensions $n \geqslant 3$ . Nevertheless, the extremals within the radial mappings give us the pinpoint of free-Lagrangian to solve the minimization problem in full generality. The radial mapping (6.12) $$h_{\circ}(x) = H(|x|) \frac{x}{|x|}, \quad \text{where } H \colon [r, R] \xrightarrow{\text{onto}} [r_, R_]$$ If one seeks harmonic radial mappings then $$\Delta h_{\circ}(x) = h_{tt} + \frac{1}{t}h_{t} + t^{-2}h_{\theta\theta} = 0,$$ where $x=t\,e^{i\theta}$ . Then H=H(t) must satisfy the Euler's ordinary differential equation $$t^2 \ddot{H}(t) + t \dot{H}(t) - H(t) = 0 \text{ for } r < t < R$$ Its two fundamental solutions t and $\frac{1}{t}$ generate all solutions $$H(t) = at + b/t.$$ To ensure the boundary constraints, $H(r) = r$ and $H(R) = R$ , we must set, (6.13) $$H(t) = at + b/t$$ , where $a = \frac{RR_ - rr}{R^2 - r^2}$ and $b = \frac{R^2 rr_ - r^2 RR}{R^2 - r^2}$ It is advantageous to transfer the above equation to the first order ODE by simply multiply by the integrating factor -2H(t). We obtain the characteristic equation (6.14) $$\mathcal{L}[H] \stackrel{\text{def}}{=} H^2 - t^2 \dot{H}^2 \equiv c, \text{ where } c \in \mathbb{R} \text{ is a constant.}$$ Note that, as opposed to the second order Laplace equation, the first order characteristic equation admits an additional solution, namely, a constant function. We shall be concerned with monotone $\mathscr{C}^1$ -solutions $H: [r, R] \xrightarrow{\text{onto}} [r_, R_]$ , thus having $\dot{H}(t) \ge 0$ . This includes solutions that are partially constant. In particular, they may squeeze but not fold subintervals. Under these assumptions the respective radial mappings $h_{\circ}$ become uniform limits of homeomorphisms, as desired in the weak formulation of the principle of noninterpenetration of matter discussed earlier. Remark 6.4. Equation (6.14), for such solutions, is none other than the variational equation of the minimization problem when confined to the radial mappings. This fact, though natural to expect, is not automatic and we will exploit it. We will not use the explicit formulas (6.13), but only the characteristic equation (6.14). This means to our arguments have can be developed for frictionless problems in which one can predict the PDEs for the energy-minimisers, if not their explicit solutions. There is a useful quantity associated with the radial mappings called the elasticity of stretching $$\eta_H(t) \stackrel{\text{def}}{=} \frac{t\dot{H}(t)}{H(t)}$$ All $\mathscr{C}^1$ -solutions fall into three categories. If c is the constant at (6.14) we say that: - H is conformal if c=0, equivalently $\eta_H(t)=1$ iff $\frac{R}{r}=\frac{R}{r}$ - H is expanding if c<0, equivalently $\eta_H(t)>1$ iff $\frac{R}{r}>\frac{R}{r}$ H is contracting if c>0, equivalently $\eta_H(t)<1$ iff $\frac{R}{r}<\frac{R}{r}$ The characteristic equation (6.14) has an injective solution if and only if $$(6.15) \frac{1}{2} \left( \frac{R}{r} + \frac{r}{R} \right) \leqslant \frac{R}{r}$$ We call the set of values for whiich (6.15) holds the Nitsche range, it includes the expanding case. We reserve the notation $F(\tau) \stackrel{\text{def}}{=} H^{-1}(\tau)$ for $r_ < \tau < R$ . Thus the characteristic equation reads as (6.16) $$\left(\frac{F}{\dot{F}}\right)^2 - \tau^2 \equiv c \qquad F(\tau) = \frac{\tau + \sqrt{\tau^2 - c}}{2a}.$$ When the reverse inequality to (6.15) holds, we say the data lies beyond the Nitsche range. We consider a $\mathscr{C}^{1,1}$ -solution $H: [r,R] \xrightarrow{\text{onto}} [r_,R_]$ of (6.14) defined by the rule. $$(6.17) \ \ H(t) = \begin{cases} r_ & t \in [r,\rho] \\ H_\rho(t) & t \in [\rho,R] \end{cases}, \qquad \rho \ \text{is determined by} \quad \frac{1}{2} \left( \frac{R}{\rho} + \frac{\rho}{R} \right) = \frac{R}{r_*}$$ Here, $H_{\rho}(t) = \frac{r}{2} \left(t + \frac{\rho}{t}\right)$ , is an increasing solution $H_{\rho}: [\rho, R] \xrightarrow{\text{onto}} [r_, R_*]$ , of the characteristic equation (6.14). 6.4. The conformal case. In this case the pullback of the area form alone is sufficient to identify the energy-minimal mappings. Precisely, using $\mathscr{F}_2$ ), we obtain. $$\int_{\mathbb{A}} |Dh(x)|^2 dx = \int_{\mathbb{A}} |h_N|^2 + |h_T|^2 \ge 2 \int_{\mathbb{A}} |h_N| |h_T| \ge 2 \int_{\mathbb{A}} J_h(x) dx = 2|\mathbb{A}^*|$$ Equality occurs only for a similarity transformation of $\mathbb{A}$ onto $\mathbb{A}^*$ (scalar multiple of a rotation) . 6.5. The expanding case. Choose and fix a radial mapping $h_{\circ} = H(|x|) \frac{x}{|x|}$ , where H solves Equation (6.14) with c < 0. Explicitly, in complex notation, we have the formula $$(6.18) h_{\circ}(z) = az + \frac{b}{\bar{z}}.$$ It is true that $h_{\circ}$ turns out to be the energy-minimal solution among radial mappings, but we shall not exploit this property; equation (6.14) is sufficient. Now suppose we are given an arbitrary mapping $h \in \mathcal{H}^{1,2}(\mathbb{A}, \mathbb{A}^)$ . We shall derive a series of sharp estimates involving h and Dh, each of which becomes an equality if $h = h_{\circ}$ - the radial Nitche map. Let us introduce the following functions; first defined for $r_ \leq \tau \leq R_*$ by the rule $$p(\tau) = \eta_F(\tau) = \frac{\tau \dot{F}(\tau)}{F(\tau)} < 1, \qquad \left( p(\tau) = \frac{\tau \sqrt{\tau^2 - c} + \tau^2}{\tau \sqrt{\tau^2 - c} + \tau^2 - c} \right)$$ We note that $p(|h(x)|) = \frac{|h_N(x)|}{|h_T(x)|}$ for $h \stackrel{\text{def}}{=} h_o$ . The second function of two variables is defined by $$A(t,\tau) \stackrel{\text{def}}{=} \frac{F(\tau)}{t \, \dot{F}(\tau)}, \qquad \text{for } r \leqslant t \leqslant R \text{ and } r_ \leqslant \tau \leqslant R$$ We note that the function $x \mapsto A(|x|, |h(x)|)$ is equal to $|h|_N(x)$ in case that $h = h_{\circ}(x)$ . To proceed we make identify algebraic inequalities which lead to the lower bounds of the integrand by means of free-Lagrangians. In our case, the following point-wise inequality holds whenever $0 \le p \le 1$ and $A \ge 0$ . $$|Dh|^{2} = |h_{N}|^{2} + |h_{T}|^{2} \ge (1 - p^{2}) |h_{N}|^{2} + 2p |h_{N}| |h_{T}|$$ $$\ge (1 - p^{2}) 2 A |h|_{N} - (1 - p^{2}) A^{2} + 2 p J_{h}$$ Now, according to (6.16) we have $(1-p^2) A^2 = c|x|^{-2}$ and so $$|Dh|^2 \geqslant 2\left(\frac{F(|h|)}{\dot{F}(|h|)} - \frac{|h|^2 \dot{F}(|h|)}{F(|h|)}\right) \frac{|h|_N}{|x|} - \frac{c}{|x|^2} + 2p(|h|) J_h$$ Here the right hand side consists of free-Lagrangians and equality occurs for the radial Nitsche map $h_{\circ}$ . Hence (6.19) $$\int_{\mathbb{A}} |Dh(x)|^2 dx \geqslant \int_{\mathbb{A}} |Dh_{\circ}(x)|^2 dx , \text{ as desired.}$$ - 6.6. The contracting case. In contrast to the expanding case, in the contracting case apply spherical free-Lagrangians $\mathscr{F}_4$ ) in place of the radial free-Lagrangian $\mathscr{F}_3$ ). We discuss two subcases. - 6.6.1. Annuli still within the Nitsche range at (6.15). In this subcase the radial solution $h_{\circ}$ still remains injective. The first step is to choose a good point-wise inequality. Precisely, for all parameters $0 \leqslant q \leqslant 1$ and $B \geqslant 0$ it holds, as is easily verified, that $$|Dh|^{2} = |h_{N}|^{2} + |h_{T}|^{2} \ge (1 - q^{2}) |h_{T}|^{2} + 2q |h_{N}| |h_{T}|$$ $$\ge (1 - q^{2}) 2B |h_{T}| - (1 - q^{2}) B^{2} + 2q J_{h}$$ $$\ge (1 - q^{2}) 2B |h| \operatorname{Im} \frac{h_{T}}{h} - (1 - q^{2}) B^{2} + 2q J_{h}$$ Then, in the second step, we take for q and B the following functions $$(6.20) q = q(|h|) = \frac{1}{\eta_F(|h|)} = \frac{F(|h|)}{|h| \dot{F}(|h|)} < 1,$$ (6.21) $$B = B(|x|, |h|) = \left(\frac{H^2(|x|)}{|x|} - |x| \dot{H}^2(|x|)\right) \frac{1}{|h| (1 - q^2(|h|))}$$ Now since $|h|^2 (1 - q^2(|h|) \equiv c$ by (6.16) we obtain the desired lower bound on $|Dh|^2$ by free-Lagrangians. (6.22) $$|Dh|^2 \geqslant 2\left(\frac{H^2(|x|)}{|x|} - |x|\dot{H}^2(|x|)\right) \operatorname{Im} \frac{h_T}{h} - \frac{1}{c}\left(\frac{H^2(|x|)}{|x|} - |x|\dot{H}^2(|x|)\right)^2 + \frac{2J_h}{\eta_F(|h|)}$$ Here again the right hand side consists of free-Lagrangians and equality occurs for the radial Nitsche map $h_\circ$ . Hence (6.23) $$\int_{\mathbb{A}} |Dh(x)|^2 dx \geqslant \int_{\mathbb{A}} |Dh_{\circ}(x)|^2 dx$$ 6.6.2. Annuli beyond the Nitsche bound. This means that $$\frac{1}{2}\left(\frac{R}{r} + \frac{r}{R}\right) > \frac{R}{r}$$ It is exactly in this subcase that the radial solution $h_{\circ}$ fails to be injective. A plausible candidate for the energy-minimal deformation is the squeezing/stretching radial mapping $h_{\circ}(x) = H(|x|) \frac{x}{|x|}$ , where $H \colon [r, R] \xrightarrow{\text{onto}} [r_, R_]$ is given by (6.17). For $\rho < |x| < R$ we apply (6.22) and obtain a lower bound in terms of free-Lagrangians $$|Dh|^2 \geqslant 2 \left( \frac{H_{\rho}^2(|x|)}{|x|} - |x| \dot{H}_{\rho}^2(|x|) \right) \operatorname{Im} \frac{h_T}{h} - \frac{1}{c} \left( \frac{H_{\rho}^2(|x|)}{|x|} - |x| \dot{H}_{\rho}^2(|x|) \right)^2 + 2 \frac{J_h}{\eta_{F_{\rho}}(|h|)}$$ The remaining free-Lagrangians lower bound for $r < |x| \leqslant \rho$ is just as easy to verify. $$|Dh|^2 = |h_N|^2 + |h_T|^2 \ge 2B r_ \operatorname{Im} \frac{h_T}{h} - B^2 \text{ with } B(|x|) = r_ |x|^{-1}$$ The point is that both lower bounds also become equalities for $h_{\circ}$ . In conclusion, (6.24) $$\int_{\mathbb{A}} |Dh(x)|^2 dx \geqslant \int_{\mathbb{A}} |Dh_{\circ}(x)|^2 dx , \text{ as well}$$ All the above cases summarize as follows

Theorem 6.5. The frictionless deformations $h : \mathbb{A} \xrightarrow{\text{onto}} \mathbb{A}^*$ between annuli assume their minimum Dirichlet energy among radial mappings. Actually, our proofs easily give an even more precise statement; namely, the energy-minimal mappings are unique up to a rotation of the variable $x \in \mathbb{A}$ or, equivalently, a rotation of $y \in \mathbb{A}^*$ .

Theorem 7.1. (p=1). Let $\Omega$ be a doubly connected planar domain with $s = mod(\Omega)$ and $\mathbb{A} = \mathbb{A}(1,R)$ , $r = mod(\mathbb{A})$ . Then there is a homeomorphism $f : \mathbb{A} \xrightarrow{\text{onto}} \Omega$ of Sobolev class $\mathcal{W}^{1,1}(\mathbb{A},\Omega)$ minimising $\int_{\mathbb{A}} \mathbb{K}(z,f) dz$ if and only if $\cosh(s) \leq e^r$ . This minimiser is a diffeomorphism and is unique up to conformal automorphisms of $\Omega$ . This is the Nitsche bound (6.15). There some subtlety here regarding the regularity of $h = f^{-1}$ . In particular is $h \in \mathcal{W}^{1,2}(\Omega, \mathbb{A})$ ? This is discussed in [1, §21] and [21]. Despite being of considerable interest, we set aside discussion of these issues in this article. In contrast to Theorem 7.1 we have the following theorem concerning extremal quasiconformal mappings.

Theorem 7.2. (p= $\infty$ ). Let $\Omega$ be a doubly connected planar domain with $0 < mod(\Omega) < \infty$ . Then for every $0 < mod(\mathbb{A}) < \infty$ , there is a homeomorphism $f : \mathbb{A} \xrightarrow{\text{onto}} \Omega$ of Sobolev class $\mathcal{W}^{1,2}(\mathbb{A},\Omega)$ minimising $\|\mathbb{K}(z,f)\|_{L^{\infty}(\mathbb{A})}$ This minimiser is a diffeomorphism and is unique up to conformal automorphisms of $\Omega$ . One can reduce this to the case $\Omega = \mathbb{A}(1, e^s)$ using (7.2) and $\mathbb{A} = \mathbb{A}(1, e^r)$ . Then identify the extremal quasiconformal mapping as the radial map $z \mapsto z|z|^{\alpha-1}$ , $\alpha = \log(s)/\log(r)$ . There is a surprising difference here between Theorem 7.1 and Theorem 7.2. Namely there is always and extremal quasiconformal mapping, but a minimiser of mean distortion exists only within a range of moduli – precisely the Nitsche range. This leads naturally to the question of what happens for 1 when we minimise $$\int_{\mathbb{A}} \mathbb{K}^p(z,f) \, dz$$ The following answer is a little surprising, [35], (this also was recently developed in the context of free-Lagrangians in [30] by the first and third authors, but here we give a different, but actually equivalent, approach). The result suggests an interesting critical phase type phenomena. When p < 1, apart from the identity map, minimizers never exist. When p = 1 we observe Nitsche type phenomena; minimisers exist within a range of conformal moduli determined by properties of the weight function and not otherwise. When p > 1 minimisers always exist. We will now go through the proof of this. In [35] rather more is proved, namely there the weighted $L^p$ -mean distortion is considered. As noted above the problem is reduced to considering deformations $f : \mathbb{A}_1 \to \mathbb{A}_2$ between round annuli. 7.1. Mappings of finite distortion. A homeomorphism $f: \Omega \to \Omega'$ between planar domains of Sobolev class $\mathcal{W}^{1,1}_{loc}(\Omega,\Omega')$ has finite distortion if the Jacobian determinant J(z,f) is nonnegative and there is a function $\mathbb{K}(z,f)$ finite almost everywhere such that $$|Df(z)|^2 \leqslant \mathbb{K}(z,f) \ J(z,f).$$ Recall function $\mathbb{K}(z, f)$ is the distortion of the mapping f. We saw at (2.6) that $\mathbb{K}$ is a measure of the anisotropic local stretching. Mappings of finite distortion are generalisations of quasiconformal homeomorphisms and have found considerable recent application in geometric function theory and nonlinear PDEs, [1]. We define annuli $$A_1 = \{1 \le |z| \le R\}, \qquad A_2 = \{1 \le |z| \le S\}$$ with moduli $\sigma_1 = \log(R)$ and $\sigma_2 = \log(S)$ . We consider homeomorphisms of finite distortion $f: \mathbb{A}_1 \to \mathbb{A}_2$ mapping the boundary components to each other, $$f(\{|z|=1\}) = \{|z|=1\},$$ and $f(\{|z|=R\}) = \{|z|=S\}.$ On the annulus $\mathbb{A}_1$ place a positive weight $\eta: \mathbb{A}_1 \to \mathbb{R}_+$ . In polar coordinates (7.4) $$f_z = \frac{1}{2} e^{-i\theta} \left( f_\rho - \frac{i}{\rho} f_\theta \right), \quad f_{\bar{z}} = \frac{1}{2} e^{i\theta} \left( f_\rho + \frac{i}{\rho} f_\theta \right)$$ and $|f_z|^2 + |f_{\bar{z}}|^2 = \frac{1}{2}(|f_\rho|^2 + \rho^{-2}|f_\theta|^2)$ , $J(z, f) = |f_z|^2 - |f_{\bar{z}}|^2 = \frac{1}{\rho} \operatorname{Im}(f_\theta \overline{f_\rho})$ which together yield (7.5) $$\mathbb{K}(z,f) = \frac{|f_z|^2 + |f_{\bar{z}}|^2}{|f_z|^2 - |f_{\bar{z}}|^2} = \frac{\rho |f_\rho|^2 + \rho^{-1} |f_\theta|^2}{2\operatorname{Im}(f_\theta \overline{f_\rho})}.$$ Given a convex function $\varphi:[1,\infty)\to[0,\infty)$ a Nitsche type problem asks us to establish the existence or otherwise of a minimizer (or perhaps stationary point) of the functional (7.6) $$f \mapsto \int_{\mathbb{A}_1} \varphi(\mathbb{K}(z, f)) \; \eta(z) \; |dz|^2.$$ Thus we seek a deformation of the annulus $\mathbb{A}_1$ to $\mathbb{A}_2$ which minimises some weighted $L^{\varphi}$ average of the distortion. §6.2 deals with the case $\varphi(t) = t$ and $\eta(x) \equiv 1$ ; minimisers of mean distortion. 7.2. Grötzsch type problems. The classical Grötzsch problem asks one to identify the linear mapping as the homeomorphism of least maximal distortion between two rectangles. Put $$\mathbf{Q}_1 = [0, \ell] \times [0, 1], \qquad \mathbf{Q}_2 = [0, L] \times [0, 1]$$ and suppose we have a deformation of finite distortion $f: \mathbf{Q}_1 \to \mathbf{Q}_2$ with (7.7) $$\operatorname{Re} f(0, y) = 0$$ , $\operatorname{Re} f(\ell, y) = L$ , $\operatorname{Im} f(x, 0) = 0$ , $\operatorname{Im} f(x, 1) = 1$ (so f is orientation preserving and maps edges to edges). This Sobolev map is absolutely continuous on lines and so $\int_0^\ell \mathrm{Re}(f_x) \ dx = L$ and $\int_0^1 \mathrm{Im}(f_y) \ dy = 1$ for almost all y and x respectively, and hence (7.8) $$\operatorname{Re} \int_{\mathbf{Q}_1} f_x(z) |dz|^2 = L, \quad \operatorname{Im} \int_{\mathbf{Q}_1} f_y(z) |dz|^2 = \ell.$$ The distortion function is (7.9) $$\mathbb{K}(z,f) = \frac{|f_x|^2 + |f_y|^2}{J(z,f)} \geqslant 1.$$ A Grötzsch problem seeks a minimiser, satisfying the boundary conditions (7.7), to the functional (7.10) $$f \mapsto \int_{\mathbf{Q}_1} \varphi(\mathbb{K}(z, f)) \ \lambda(z) \ |dz|^2$$ for some positive weight function $\lambda$ . 7.3. Equivalence between Nitsche and Grötzsch problems. The universal cover of an annulus is effected by the exponential map, so $z \mapsto \exp(2\pi z)$ takes $z = x + iy \in [0, L] \times [0, 1]$ to $\mathbb{A}_2$ if $\sigma_2 = \log(S) = 2\pi L$ . A branch of logarithm must be chosen to define an "inverse" map $[0, \ell] \times [0, 1] \to \mathbb{A}_1$ . If $f : \mathbb{A}_1 \to \mathbb{A}_2$ is given, then we can define $\tilde{f}(z) = \frac{1}{2\pi} \log(f(\exp 2\pi z))$ . A particular point here is that log is conformal (in fact we only really need log to define a univalent conformal mapping from $\mathbb{A}_1 \setminus ([1, S] \times \{0\})$ to $\mathbb{Q}_2$ with edges matching up) so (7.11) $$\mathbb{K}(z,\tilde{f}) = \mathbb{K}\left(z, \frac{1}{2\pi}\log\left(f(e^{2\pi z})\right)\right) = \mathbb{K}\left(z, f(e^{2\pi z})\right),$$ and hence a change of variables yields $$\begin{split} \int_{\mathbf{Q}_1} \varphi \big( \mathbb{K}(z,\tilde{f}) \big) \lambda(z) \; |dz|^2 &= \int_{\mathbf{Q}_1} \varphi \big( \mathbb{K}(z,f(e^{2\pi z}) \big) \lambda(z) \; |dz|^2 \\ &= 4\pi^2 \, \int_{\mathbb{A}_1} \varphi \big( \mathbb{K}(w,f) \big) \; \lambda(z) e^{-4\pi \mathrm{Re}(z)} \; |dw|^2. \end{split}$$ With the choice (7.12) $$\eta(w) = 4\pi^2 \lambda(z)e^{-4\pi \text{Re}(z)}, \qquad e^z = w,$$ the equivalence between the two problems (with related weight) is seen. Again, the exact branch of log here will be immaterial to our considerations. Remark 7.3. In fact the equivalence between Nitsche and Grötzsch problems is only when one assumes periodic boundary behaviour in the Grötzsch problem. We will be fortunate in that the absolute minimisers for the Grötzsch problem in the situations we consider do exhibit this periodicity and so can be lifted. 7.4. Sublinear distortion functionals; p < 1. The purpose of this brief section is to show that minimisers never exist for the $L^p$ -minimisation problem when p < 1. We recall from [3, Theorem 5.3] (actually the proof of this result)

Lemma 7.4. Let $\Psi(t)$ be a positive strictly increasing function of sublinear growth: $$\lim_{t \to \infty} \frac{\Psi(t)}{t} = 0$$ Let $\mathbb{B} = \mathbb{D}(z_0, r)$ be a round disk and suppose that $f_0 : \mathbb{B} \to \mathbb{C}$ is a homeomorphism of finite distortion with $\int_{\mathbb{B}} \Psi(\mathbb{K}(z, f_0)) < \infty$ . Then there is a sequence of mappings of finite distortion $f_n : \mathbb{B} \to f_0(\mathbb{B})$ with $f_n(\zeta) = f_0(\zeta)$ near $\partial \mathbb{B}$ and with - $\mathbb{K}(z, f_n) \to 1$ uniformly on compact subsets of $\mathbb{B}$ - $\int_{\mathbb{R}} \Psi(\mathbb{K}(z, f_n)) \to \int_{\mathbb{R}} \Psi(1) \ as \ n \to \infty.$ We now prove the following theorem.

Theorem 7.5. Let $\Psi(t)$ be a positive strictly increasing function of sublinear growth, let $\Omega$ be a domain and let $\lambda(z) \in L^{\infty}(\Omega)$ be a positive weight. Suppose that $g_0: \Omega \to \mathbb{C}$ is a homeomorphism of finite distortion with $$\int_{\Omega} \Psi(\mathbb{K}(z, g_0)) < \infty$$ Then there is a sequence of mappings of finite distortion $g_n: \Omega \to g_0(\Omega)$ with $g_n(\zeta) = g_0(\zeta), \ \zeta \in \partial \Omega$ with (7.13) $$\int_{\Omega} \Psi(\mathbb{K}(z, g_n)) \lambda(z) \to \Psi(1) \int_{\Omega} \lambda(z) \quad \text{as } n \to \infty$$ Proof. Let $\epsilon > 0$ . Since $\left(\Psi(\mathbb{K}(z, g_0)) - \Psi(1)\right)\lambda(z) \in L^1(\Omega)$ we can choose a finite collection of disjoint disks contained in $\Omega$ , say $\{B_i\}_{i=1}^N$ , so that (7.14) $$\left| \int_{\Omega \setminus 1 \setminus B_{\epsilon}} \left( \Psi(\mathbb{K}(z, g_0)) - \Psi(1) \right) \lambda(z) \, dz \right| < \epsilon/2$$ Next, for each i we use Lemma 7.4 in the obvious way to find $h_i: B_i \to \mathbb{C}$ with $h_i = g_0$ in a neighbourhood of $\partial B_i$ and $$\Big| \int_{B_i} \Psi(\mathbb{K}(z, h_i)) \lambda(z) - \Psi(1) \int_{B_i} \lambda(z) \Big| < \frac{\epsilon}{2N}$$ Then the map $$g_{\epsilon}(z) = \begin{cases} g_0(z) & z \in \Omega \setminus \bigcup B_i \\ h_i(z) & z \in B_i \end{cases}$$ is of finite distortion and $$\begin{split} \Big| \int_{\Omega} \Psi(\mathbb{K}(z, g_{\epsilon})) \, \lambda(z) - \Psi(1) \int_{\Omega} \lambda(z) \Big| &= \Big| \int_{\Omega \backslash \bigcup B_{i}} \Big( \Psi(\mathbb{K}(z, g_{\epsilon})) - \Psi(1) \Big) \, \lambda(z) \, dz \\ &+ \sum_{i=1}^{N} \int_{B_{i}} \Big( \Psi(\mathbb{K}(z, h_{i})) - \Psi(1) \Big) \, \lambda(z) \, dz \Big| \\ &< \epsilon \end{split}$$ The result follows. And the next corollary is what we seek. Corollary 7.6. Let $\Psi(t)$ be a positive strictly increasing function of sublinear growth, let $\Omega$ be a domain and let $\lambda(z) \in L^{\infty}(\Omega)$ be a positive weight. Suppose that $g_0: \Omega \to \mathbb{C}$ is a homeomorphism of finite distortion with $$\int_{\Omega} \Psi(\mathbb{K}(z, g_0)) < \infty$$ Then $$\min_{\mathcal{F}} \int_{\Omega} \Psi(\mathbb{K}(z,g)) \, \lambda(z) = \Psi(1) \int_{\Omega} \lambda(z) \, dz$$ with equality achieved by a mapping of finite distortion if and only if the boundary values of $g_0$ are shared by a conformal mapping. Here $\mathcal{F}$ consists of homeomorphisms of finite distortion g with $g|\partial\Omega=g_0$ . 7.5. Minimisers of convex distortion functionals. A natural class of homeomorphic mappings between rectangles satisfying the Grötzsch boundary conditions (7.7) are those of the form $$(7.15) f_0(z) = u(x) + iy,$$ which will correspond to the lifts of the radial stretchings. For these mappings we have $(f_0)_x = u_x$ and $(f_0)_y = i$ . We will show these mappings are the extremals for our mapping problems, but we will have to deal with degenerate situations as well - in particular where $f_0$ is not well defined, but has a well defined inverse. These are topologically monotone mappings we have talked about earlier as local uniform limits of homeomorphisms, and so for us as limits of minimising sequences. In order to avoid excess technical complications we make the following assumptions: Let $$w = a + ib \in [0, L] \times [0, 1]$$ and set $$(7.16) g_0(w) = v(a) + ib$$ where $v:[0,L] \to [0,\ell]$ is an absolutely continuous, increasing (but not necessarily strictly increasing) surjection. The derivative of $v_a$ of v is a non-negative $\mathcal{L}^1([0,\ell])$ function which if it is positive almost everywhere makes v strictly increasing and we may set $$(7.17) f_0 = g_0^{-1} : [0, \ell] \times [0, 1] \to [0, L] \times [0, 1]$$ We now proceed as follows.

Lemma 7.7. Set $g_0(w) = v(a) + ib$ , where $v : [0, L_0] \to [0, \ell]$ is an absolutely continuous, increasing surjection. Let $f : [0, \ell] \times [0, 1] \to [0, L] \times [0, 1]$ be a homeomorphism of finite distortion satisfying the boundary conditions (7.7). Then $$(7.18) |v_a(a)f_x(g_0(w)) + if_y(g_0(w))|^2 \ge 0$$ Equality holds for f and almost every w if and only if v is strictly increasing $L = L_0$ and $f = g_0^{-1}$ . Proof. We consider $$h(w) = (f \circ g_0)(w)$$ The mapping $h \in \mathcal{W}_{loc}^{1,1}$ and maps $[0, L_0] \times [0, 1] \to [0, L] \times [0, 1]$ respecting the sides. We compute the $\bar{w}$ -derivative of h; $$2h_{\bar{w}}(w) = f_z(v(a) + ib)v_a(a) + f_{\bar{z}}(v(a) + ib)v_a(a) + if_z(v(a) + ib) - if_{\bar{z}}(v(a) + ib)$$ $$= f_x(g_0(w))v_a(a) + if_y(g_0(w)) = 0$$ as an $\mathcal{L}^1$ -function. Thus h is analytic by the Looman-Menchoff theorem. The boundary conditions and analyticity imply that h is a homeomorphism of the boundary which must therefore be a homeomorphism of the rectangles. Then $L = L_0$ and h must be the identity since the two rectangles have moduli $L_0$ and L. The result follows For a suitable function v as above, let us write z = x + iy where (7.19) $$z = g_0(w) \text{ and } \omega(x) = v_a(a).$$ We note that $\omega$ is well defined. First, $g_0$ is a surjection and if $g_0(w_1) = g_0(w_2)$ , then $w_1$ and $w_2$ lie in a common interval on which v is constant, whereupon $v_a(a_1) = v_a(a_2) = 0$ . However if $\omega(x) > 0$ , then $$(7.20) |\omega(x)f_x(z) + if_y(z)|^2 \geqslant 0$$ with equality almost everywhere if and only if $g_0$ is a homeomorphism and $f_0 = g_0^{-1}$ . Also, when $\omega > 0$ , v is strictly increasing, (7.21) $$g_0^{-1}(z) = f_0(z) = u(x) + iy, \qquad \omega(x) = 1/u_x(x).$$ We now suppose that $\omega > 0$ and expand out (7.20). $$0 \leqslant |\omega(x)f_x + if_y|^2 = (\omega(x)f_x + if_y)(\omega(x)\overline{f_x} - i\overline{f_y})$$ $$= \omega^2(x)|f_x|^2 + |f_y|^2 - 2\operatorname{Im}(\omega(x)f_y\overline{f_x})$$ which yields (7.22) $$\omega^2(x)|f_x|^2 + |f_y|^2 \geqslant 2\omega(x)\operatorname{Im}(f_y\overline{f_x}).$$ Notice that if we write f = U + iV, then $$\operatorname{Im}(f_y \overline{f_x}) = \operatorname{Im}(U_x(z) - iV_x(z))(U_y(z) + iV_y(z)) = J(z, f),$$ so (7.22) gives us (7.23) $$\omega^{2}(x)|f_{x}|^{2} + |f_{y}|^{2} \geqslant 2\omega(x)J(z,f)$$ with equality almost everywhere if and only if $f = f_0$ (with the implication that $f_0$ is a homeomorphism). We can rewrite (7.23) in two different ways. Namely $$|f_x|^2 + |f_y|^2 \ge (1 - \omega^{-2}(x))|f_y|^2 + 2\omega^{-1}(x)J(z,f),$$ $|f_x|^2 + |f_y|^2 \ge (1 - \omega^2(x))|f_x|^2 + 2\omega(x)J(z,f),$ which gives us two estimates on the distortion function (writing J = J(z, f)), $$\mathbb{K}(z, f) \geqslant (1 - \omega^{-2}(x)) \frac{|f_y|^2}{J} + 2\omega^{-1}(x),$$ $\mathbb{K}(z, f) \geqslant (1 - \omega^2(x)) \frac{|f_x|^2}{J} + 2\omega(x),$ Next, when $\omega > 0$ almost everywhere we can define $f_0$ by (8.1) with (??). Then $$\mathbb{K}(z, f_0) = (1 - \omega^{-2}(x)) \frac{|(f_0)_y|^2}{J_0} + 2\omega^{-1}(x),$$ $$\mathbb{K}(z, f_0) = (1 - \omega^2(x)) \frac{|(f_0)_x|^2}{J_0} + 2\omega(x),$$ and thus we have our first useful inequalities

Lemma 7.8. If $\omega(x) > 0$ , then (7.24) $$\mathbb{K}(z,f) - \mathbb{K}(z,f_0) \geqslant (1 - \omega^{-2}(x)) \left[ \frac{|f_y|^2}{J} - \frac{|(f_0)_y|^2}{J_0} \right],$$ and (7.25) $$\mathbb{K}(z,f) - \mathbb{K}(z,f_0) \geqslant (1 - \omega^2(x)) \left[ \frac{|f_x|^2}{J} - \frac{|(f_0)_x|^2}{J_0} \right]$$ with equality holding almost everywhere in either inequality if and only if $f = f_0$ . We leave it to the reader to establish the elementary inequality for complex numbers X, $X_0$ and real J, $J_0$ , (7.26) $$\frac{|X|^2}{J} - \frac{|X_0|^2}{J_0} \geqslant 2 \operatorname{Re}\left(\frac{\overline{X_0}}{J_0} (X - X_0)\right) - \frac{|X_0|^2}{J_0^2} (J - J_0),$$ with equality holding if and only if $X/X_0 = J/J_0$ is a positive real number (expand $|X/X_0 - J/J_0|^2 \ge 0$ ). This shows $$(X,Y,J) \mapsto \frac{|X|^2 + |Y|^2}{I}$$ to be convex on $\mathbb{C} \times \mathbb{C} \times \mathbb{R}_+$ . We apply (7.26) and this requires that the coefficient $(1-\omega^{-2}(x)) > 0$ in the first case or $(1-\omega^2(x)) > 0$ in the second. Since this depends on $u_x$ for the candidate extremal mapping, we carry along the two inequalities and write $\mathbb{K}_0 = \mathbb{K}(z, f_0)$ . First note that if $\varphi : \mathbb{R} \to \mathbb{R}$ is convex, then its graph lies above any tangent line: $$\varphi(\mathbb{K}) - \varphi(\mathbb{K}_0) \geqslant \varphi'(\mathbb{K}_0) \big( \mathbb{K} - \mathbb{K}_0 \big)$$ Notice that if $\varphi'' > 0$ , equality quality holds here if and only if $\mathbb{K} = \mathbb{K}_0$ . This therefore yields the following two inequalities: $$\varphi(\mathbb{K}(z,f)) - \varphi(\mathbb{K}(z,f_0)) \geqslant (1 - \omega^{-2}(x))\varphi'(\mathbb{K}_0) \\ \left[ 2\operatorname{Re}\left(\frac{\overline{(f_0)_y}}{J_0} \left(f_y - (f_0)_y\right)\right) - \frac{|(f_0)_y|^2}{J_0^2} \left(J - J_0\right) \right],$$ $$\varphi(\mathbb{K}(z,f)) - \varphi(\mathbb{K}(z,f_0)) \geqslant (1 - \omega^2(x))\varphi'(\mathbb{K}_0)$$ $$\left[2\operatorname{Re}\left(\frac{\overline{(f_0)_x}}{J_0}\left(f_x - (f_0)_x\right)\right) - \frac{|(f_0)_x|^2}{J_0^2}\left(J - J_0\right)\right].$$ Now $(f_0)_y = i$ and $(f_0)_x = 1/\omega(x) = J_0$ so these equations read as $$\varphi(\mathbb{K}(z,f)) - \varphi(\mathbb{K}(z,f_0))$$ $$\geqslant \left(1 - \frac{1}{\omega^2(x)}\right) \varphi'(\mathbb{K}_0) \left[\frac{2}{J_0} \operatorname{Im}(f_y - 1) - \frac{J - J_0}{J_0^2}\right]$$ $$= 2\left(\omega(x) - \frac{1}{\omega(x)}\right) \varphi'(\mathbb{K}_0) \operatorname{Im}(f_y - 1)$$ $$+ \left(\omega^2(x) - 1\right) \varphi'(\mathbb{K}_0) (J_0 - J),$$ (7.27) (7.28) $$\varphi(\mathbb{K}(z,f)) - \varphi(\mathbb{K}(z,f_0))$$ $$\geqslant (1 - \omega^2(x))\varphi'(\mathbb{K}_0) \left[ 2\operatorname{Re}(f_x - (f_0)_x) - (J - J_0) \right].$$ Now we want to multiply these two inequalities by a weight function $\lambda(x)$ and integrate. We are naturally led to consider the Euler-Lagrange equation for the variational problem minimising $$\int_{\mathbf{Q}} \varphi(\mathbb{K}(z,f))\lambda(x) |dz|^2$$ among functions of the form (7.15). This equation reduces to the next equation in one real variable (7.29) $$\frac{d}{dx} \left[ \lambda(x) \left( 1 - \frac{1}{u_x^2} \right) \varphi' \left( u_x + \frac{1}{u_x} \right) \right] = 0$$ We would therefore like $\omega(x)$ to be chosen chosen so that (7.30) $$\lambda(x)(1-\omega^2(x))\varphi'\left(\omega(x)+\frac{1}{\omega(x)}\right)=\alpha\neq0$$ for a real constant $\alpha$ . This equation implicitly defines $\omega$ directly, and does not involve any of its derivatives. Remark 7.9. We postpone a discussion of boundary values for the solution $f_0$ (really $g_0$ ) that we seek. Set (7.31) $$\int_0^\ell \frac{dx}{\omega(x)} = L_0$$ The boundary conditions we want are that $L = L_0$ to identify the minimum. However, if $L_0 < L$ , then Lemma 7.7 still applies - and we obtain strict inequality. Also, we note that from (7.30), with an assumption that $\lambda > 0$ and $\varphi'$ are continuous, that ω = 0 implies that λ(x)ϕ ′ (∞) = α. In particular, we cannot have ω(x) = 0 unless ϕ ′ is bounded - a condition we will see again. We now suppose that we have (7.30) holding almost everywhere and L⁰ < L. Then (7.30) forces 0 ≤ ω(x) < 1 for all x or ω(x) > 1 for all x. The case ω ≡ 1, α = 0 yielding g⁰ = f⁰ = identity. The first case (where we will ultimately have to deal with degeneration as we cannot guarantee the boundary conditions) has u^x > 1 and so must correspond to stretching L > ℓ. In the other case ℓ < L. We proceed as follows. $$\int_{\mathbf{Q}_{1}} \varphi(\mathbb{K}(z,f))\lambda(x) |dz|^{2} \geqslant \int_{\mathbf{Q}_{1}} \varphi(\mathbb{K}(z,f_{0}))\lambda(x) |dz|^{2} - \alpha \int_{\mathbf{Q}_{1}} (J_{0} - J) |dz|^{2} + 2 \int_{\mathbf{Q}_{1}} \lambda(x) \left(\omega(x) - \frac{1}{\omega(x)}\right) \varphi'(\mathbb{K}_{0}) \operatorname{Im}(f_{y} - 1) |dz|^{2},$$ $$\int_{\mathbf{Q}_{1}} \varphi(\mathbb{K}(z,f))\lambda(x) |dz|^{2} \geqslant \int_{\mathbf{Q}_{1}} \varphi(\mathbb{K}(z,f_{0}))\lambda(x) |dz|^{2} + \alpha \int_{\mathbf{Q}_{1}} (J_{0} - J) |dz|^{2} + 2\alpha \int_{\mathbf{Q}_{1}} \operatorname{Re}(f_{x} - (f_{0})_{x}) |dz|^{2}.$$ For an arbitrary Sobolev homeomorphism it is well known that $$\int_{\mathbf{Q}_1} J |dz|^2 \leqslant |\mathbf{Q}_2| = L = \int_0^\ell u_x(x) dx = \int_{\mathbf{Q}_1} J_0 |dz|^2$$ We will use the first inequality above when α < 0 and the second when α > 0. Thus, for α < 0 $$\int_{\mathbf{Q}_{1}} \varphi(\mathbb{K}(z,f))\lambda(x) |dz|^{2}$$ $$\geqslant \int_{\mathbf{Q}_{1}} \varphi(\mathbb{K}(z,f_{0}))\lambda(x) |dz|^{2} + 2 \int_{\mathbf{Q}_{1}} \lambda\left(\omega - \frac{1}{\omega}\right)\varphi'(\mathbb{K}_{0})\operatorname{Im}(f_{y} - 1) |dz|^{2}$$ , while for α > 0 we have $$\int_{\mathbf{Q}_{1}} \varphi(\mathbb{K}(z, f))\lambda(x) |dz|^{2}$$ $$\geqslant \int_{\mathbf{Q}_{1}} \varphi(\mathbb{K}(z, f_{0}))\lambda(x) |dz|^{2} + 2\alpha \int_{\mathbf{Q}_{1}} \operatorname{Re}(f_{x} - (f_{0})_{x}) |dz|^{2}$$ Next, from (7.8) we see that $$\int_{\mathbf{Q}_{1}} \lambda(x) \left( \omega(x) - \frac{1}{\omega(x)} \right) \varphi'(\mathbb{K}_{0}) \operatorname{Im}(f_{y} - 1) |dz|^{2}$$ $$= \int_{0}^{\ell} \lambda(x) \left( \omega(x) - \frac{1}{\omega(x)} \right) \varphi'(\mathbb{K}_{0}) \left[ \int_{0}^{1} \operatorname{Im}(f_{y} - 1) dy \right] dx = 0$$ and $$\int_{\mathbf{Q}_1} \text{Re}(f_x - (f_0)_x) |dz|^2 = \int_0^1 \left[ \int_0^\ell \text{Re}(f_x - (f_0)_x) dx \right] dy = 0.$$ Thus we have established

Theorem 7.10. Let $\lambda(x) > 0$ be a positive weight and $\varphi : [1, \infty) \to [0, \infty)$ be convex increasing. Let the function $u : [0, \ell] \to [0, L]$ $$(7.32) u(0) = 0, u(\ell) = L_0 < L$$ be a solution to the ordinary differential equation (7.33) $$\lambda(x) \left( 1 - \frac{1}{u_x^2(x)} \right) \varphi' \left( u_x(x) + \frac{1}{u_x(x)} \right) = \alpha$$ where $\alpha$ is a nonzero constant. Set $$(7.34) f_0(z) = u(x) + iy, f_0: [0, \ell] \times [0, 1] \to [0, L_0] \times [0, 1].$$ Let $f:[0,\ell]\times[0,1]\to[0,L]\times[0,1]$ be a surjective homeomorphism of finite distortion with $$Ref(0, y) = 0$$ , $Ref(\ell, y) = L$ , $Imf(x, 0) = 0$ , $Imf(x, 1) = 1$ . Then (7.35) $$\int_{\mathbf{Q}_1} \varphi(\mathbb{K}(z,f))\lambda(x) |dz|^2 \geqslant \int_{\mathbf{Q}_1} \varphi(\mathbb{K}(z,f_0))\lambda(x) |dz|^2.$$ Equality holds if and only if $f = f_0$ . In particular, if $L_0 < L$ , then this inequality is strict. Notice $\alpha=0$ gives the identity mapping - clearly always an absolute minimiser when it is a candidate. 7.6. Degenerate Cases. Theorem 8.1 identifies the extremal homeomorphism of finite distortion when we can find $\alpha$ so that $L_0 = L$ . We will see later that this is not always possible and then Theorem 8.1 provides us with the unattainable lower bound $\int_{\mathbf{Q}_1} \varphi(\mathbb{K}(z, f_0))\lambda(x)$ - since the inequality is strict. When $L_0 < L$ of course $f_0$ is not a candidate mapping for the minimisation problem - so it might not be surprising the bound is unattainable. However it might be possible that this value is the limit of a minimising sequence of candidates. What we want to do here is to find circumstances in which this happens.

Theorem 7.11. Suppose that $f_0$ is defined as in Theorem 8.1 and for no choice of $\alpha$ is it possible that $L = L_0$ . Suppose that $\varphi'$ is bounded. Then there is a sequence of surjective homeomorphism of finite distortion $f_j : [0, \ell] \times [0, 1] \to [0, L] \times [0, 1]$ such that (7.36) $$\int_{\mathbf{Q}_1} \varphi(\mathbb{K}(z, f_j)) \lambda(x) |dz|^2 = \int_{\mathbf{Q}_1} \varphi(\mathbb{K}(z, f_0)) \lambda(x) |dz|^2$$ In particular, under these circumstances there is no extremal homeomorphism of finite distortion whatsoever. Remark. We will see in the next few sections the condition $\phi'$ bounded is necessary for nonexistence of minimisers, but not sufficient. The behaviour of the weight $\lambda$ near its minimum determines whether we can solve the boundary problem for arbitrary L. Proof. Our assumption is that $\varphi$ is convex increasing and thus $\varphi'$ is positive and increasing, not necessarily strictly. We may also assume $\lim_{t\to\infty} \varphi'(t) = 1$ . The function $t\mapsto (1-t^{-2})\varphi'(t+1/t)$ is strictly increasing and our solution $u^{\alpha}$ is obtained by the rule $u_x^{\alpha}(x) = t_x$ where $(1-t_x^{-2})\varphi'(t_x+1/t_x) = \alpha/\lambda(x)$ . This implies that $\alpha \leq \alpha_0 = \min_x \lambda(x)$ . It is easy to see that $u^{\alpha}(\ell) \nearrow u^{\alpha_0}(\ell)$ and our hypothesis is that last value is $L_0 < L$ . Thus for $\alpha \le \alpha_0$ , the family $u^{\alpha} \in \mathcal{W}^{1,1}([0,\ell])$ , with a uniform bound. Further $u_0 = u^{\alpha_0}$ is strictly increasing with derivative tending to $\infty$ as x approaches a minimum, say $x_0$ , of $\lambda$ (which may be an endpoint of $[0,\ell]$ ). Let $$g_0(w) = v_0(a) + ib, v = u_0^{-1}$$ Then $(v_0)_a(a) = 1/(u_0)_x(x)$ with $u_0(x) = a \in [0, L_0]$ . With $u_0(x_0) = a_0$ we have $(v_0)_a(a_0) = 0$ . We now define a new function $g: [0, L] \times [0, 1] \to [0, \ell] \times [0, 1]$ by simply defining g(w) = v(a) + ib to be constant near $x_0$ . That is (with appropriate modification should $x_0$ , the minimum of $\lambda$ be an endpoint) (7.37) $$v(a) = \begin{cases} v_0(a) & a \le a_0 \\ v_0(a_0) & a_0 \le a \le a_0 + L - L_0 \\ v_0(a + L_0 - L) & a_0 + L - L_0 \le a \le L \end{cases}$$ Then $v_a$ is a non-negative $L^1$ function, vanishing on $[a_0, a_0 + L - L_0]$ and with $||v_a||_1 = \ell$ . It is routine to approximate $v_a$ by positive $v_a^j$ in $L^1$ and with $||v_a^j||_1 = \ell$ . Define $v(a) = \int_0^a v_a^j$ to get a homeomorphic mapping of finite distortion $g^j(w) = v^j(a) + ib$ . Notice that $g^j \to g$ uniformly in $\mathcal{W}^{1,1}([0,L] \times [0,1])$ . Set $$f^j = (g^j)^{-1}: [0,\ell] \times [0,1] \to [0,L] \times [0,1]$$ The mappings $f^j$ are surjective homeomorphisms of finite distortion. We calculate, with the change of variables $g^j(w) = z$ , $$\int_{\mathbf{Q}_{1}} \varphi(\mathbb{K}(z, f^{j})) \lambda(z) dz = \int_{\mathbf{Q}_{2}} \varphi\left(\frac{\|Df^{j}(g^{j})\|^{2}}{J(g^{j}, f^{j})}\right) \lambda(g^{j}(w)) J(w, g^{j}) dw$$ $$= \int_{\mathbf{Q}_{2}} \varphi\left(\|(Dg^{j}(w))^{-1}\|^{2} J(w, g_{j})\right) \lambda(g^{j}(w)) J(w, g^{j}) dw$$ $$= \int_{\mathbf{Q}_{2}} \varphi\left(v_{a}^{j}(a) + \frac{1}{v_{a}^{j}(a)}\right) \lambda(v^{j}(a)) v_{a}^{j}(a) da$$ $$\rightarrow \int_{\mathbf{Q}_{2}} \varphi\left(v_{a}(a) + \frac{1}{v_{a}(a)}\right) \lambda(v(a)) v_{a}(a) da$$ $$= \int_{[0, L_{0}] \times [0, 1]} \varphi\left((v_{0})_{a}(a) + \frac{1}{(v_{0})_{a}(a)}\right) \lambda((v_{0})(a)) (v_{0})_{a}(a) da$$ $$= \int_{[0, \ell] \times [0, 1]} \varphi\left((u_{0})_{x}(x) + \frac{1}{(u_{0})_{x}(x)}\right) \lambda(x) dx$$ $$= \int_{\mathbf{Q}_{1}} \varphi\left(\mathbb{K}(z, f_{0})\right) \lambda(z) dz$$

Theorem 8.1. Let $\lambda(x)$ be a piecewise continuous positive weight bounded and bounded away from 0. Let $\varphi: [1, \infty) \to [0, \infty)$ be smooth and convex increasing with $\varphi'(s)$ unbounded as $s \to \infty$ . Then the minimisation problem (8.5) $$\min_{f \in \mathcal{F}} \int_{\mathbf{Q}_1} \varphi(\mathbb{K}(z, f)) \lambda(x) |dz|^2$$ has a unique solution of the form f(z) = u(x) + iy. Here $\mathcal{F}$ is the family of all mappings of finite distortion satisfying the boundary conditions described in 7.2 We then have the following corollary about the weighted $L^p$ -norms of distortion functions. Corollary 8.2. Let $\lambda(x)$ be a piecewise continuous positive weight bounded and bounded away from 0. Then the minimisation problem (8.6) $$\min_{f \in \mathcal{F}} \int_{\mathbf{Q}_1} \mathbb{K}^p(z, f) \lambda(x) |dz|^2$$ has a unique solution of the form f(z) = u(x) + iy. Here $\mathcal{F}$ is the family of all mappings of finite distortion satisfying the boundary conditions described above. 8.3. Critical case: $\varphi'$ bounded. Examining the above argument we see that in this case we can always find a solution to the minimisation problem of the given form if $L < \ell$ by varying $\alpha$ among negative values, $\alpha = 0$ produces the identity mapping. However, there are further subtleties. The reader will quickly get to a condition on the integrability of $\psi(\lambda_0/\lambda(x))$ where $\psi$ is the inverse of the bounded increasing function $t \mapsto \varphi'(t+t^{-1})(1-t^{-2})$ with $\lambda_0 = \min_{[0,\ell]} \lambda$ . Let us give two illustrative examples in the standard (Nitsche) case with $\ell = 1$ , $\lambda(x) = e^{-4\pi x}$ . We may assume that $\varphi'(t) \nearrow 1$ and the limiting case $\alpha = e^{4\pi}$ : Case: $$\varphi(t) = t - \log(t), \ \varphi'(t) = 1 - \frac{1}{t}, \ a = a(x) = e^{4\pi(x-1)} \le 1.$$ We choose $u_x$ to be the largest real root of the polynomial: $$\left(1 - \frac{1}{t + t^{-1}}\right) \left(1 - \frac{1}{t^2}\right) = a$$ $$p(t) = -1 + t - at^2 - t^3 + (1 - a)t^4 = 0.$$ Since $$p(\frac{1}{1-a}) = -1 + \frac{1}{1-a} - \frac{a}{(1-a)^2} - \frac{1}{(1-a)^3} + \frac{1}{(1-a)^3} = -\frac{a^2}{(1-a)^2} < 0$$ the largest real root $u_x(x) > 1/(1 - a(x))$ and $$\int_0^x u_y(y) > \int_0^x \frac{1}{1 - e^{4\pi(y-1)}} \approx \frac{1}{4\pi} \log\left(\frac{1}{1 - x}\right)$$ and this diverges as $x \to 1$ . Therefore with appropriate choice of $\alpha$ we can always solve u(0) = 0 and u(1) = L. Hence there is no Nitsche phenomena. Case: $$\varphi(t) = t + \frac{1}{(p-1)t^{p-1}}, p > 0, p \neq 1.$$ We have $\varphi'(t) = 1 - \frac{1}{t^p}$ , $0 < a = a(x) = e^{-4\pi x} < 1$ for 0 < x < 1, and hence $u_x$ is the largest real root of the polynomial (8.7) $$P(t) = \left(1 - \frac{1}{(t+t^{-1})^p}\right) \left(1 - \frac{1}{t^2}\right) - a = 0.$$ Note that when t > 0, P(t) is a continuous monotonically increasing function of t. Also note that P(1) = -a < 0, and $\lim_{t\to\infty} P(t) = 1 - a > 0$ , so that P has exactly one real positive root $u_x > 1$ . First let us deal with 0 . Observe that $$(1 - (1 - a)^2)((1 + (1 - a)^2) - (1 - a)) - a(1 + (1 - a)^2) = -a^2(1 - a)^2 < 0.$$ This may be rewritten as $$\left(1 - \left(\frac{1}{1-a}\right)^{-2}\right)\left(1 - \frac{1}{\frac{1}{1-a} + \frac{1-a}{1}}\right) - a < 0$$ Now using the fact that 0 , we see that $$P\left(\frac{1}{1-a}\right) = \left(1 - \frac{1}{\left(\frac{1}{1-a}\right)^2}\right) \left(1 - \frac{1}{\left(\frac{1}{1-a} + \frac{1-a}{1}\right)^p}\right) - a < 0$$ and hence the largest real root $u_x > 1/(1-a)$ . The integral of the right hand side diverges (see the reasoning for the case $\varphi' = 1 - t^{-1}$ ). Thus with appropriate choice for $\alpha$ we can always solve u(0) = 0, u(1) = L and therefore we see no Nitsche phenomena for p < 1. Next, take $p\geqslant 2$ . Recall (8.7). Note that $\left(t+\frac{1}{t}\right)^p>\left(t+\frac{1}{t}\right)^2>t^2$ . Set Q(t) as $$P(t) = \left(1 - \frac{1}{(t+t^{-1})^p}\right) \left(1 - \frac{1}{t^2}\right) - a > \left(1 - \frac{1}{t^2}\right)^2 - a = Q(t), \quad t > 1.$$ The largest real root of P(t) is therefore dominated by the largest real root of Q(t). Solving Q(t) = 0 gives $$\int_0^1 u_x \, dx < \int_0^1 \frac{1}{\sqrt{1 - e^{-2\pi x}}} \, dx = \log\left(e^{\pi} + \sqrt{e^{2\pi} - 1}\right),$$ a finite number. Therefore, when $p \ge 2$ , $u_x(x)$ is dominated by an integrable function and we must see the Nitsche phenomenon. It is no coincidence that the value of the integral here is strongly reminiscent of that for the Nitsche case (8.1); the integrands for that case and the estimate here are very similar. It remains to cover the case where 1 . Note that for <math>p > 1, $1 - \frac{1}{(t+t^{-1})^p} > 1 - \frac{1}{t^p}$ , and for p < 2, $1 - \frac{1}{t^2} > 1 - \frac{1}{t^p}$ . Therefore the polynomial $$P(t) = \left(1 - \frac{1}{(t+t^{-1})^p}\right) \left(1 - \frac{1}{t^2}\right) - a > \left(1 - \frac{1}{t^p}\right)^2 - a = Q(t),$$ and the largest real root of P(t) is again dominated by the largest real root of Q(t). Solving Q(t)=0 yields $u_x<\left(1-\sqrt{a(x)}\right)^{-1/p}$ . Near $x=0,\,\sqrt{a(x)}=e^{-2\pi x}\approx 1-2\pi x$ and so $$\int_0^1 \frac{1}{\left(1 - \sqrt{a(x)}\right)^{1/p}} dx \approx \left(\frac{1}{2\pi}\right)^{1/p} \int_0^1 \frac{1}{x^{1/p}} dx,$$ which converges if and only if p > 1. Therefore in this case, too, $u_x$ is dominated by an integrable function and we must see a critical Nitsche-type phenomenon.

Definition 6.1. Suppose that a given energy integral (6.1) $$\mathscr{F}[h] \stackrel{\text{def}}{=} \int_{\mathbb{X}} \mathbf{F}(x, h, Dh) dx$$ , where $\mathbf{F} : \mathbb{X} \times \mathbb{Y} \times \mathbb{R}^{n \times n} \to \mathbb{R}$ converges for all Sobolev mappings of class $\mathcal{W}^{1,p}(\mathbb{X},\mathbb{Y})$ . We say that the differential n-form $\mathbf{F}(\cdot,\cdot,\cdot)\,dx$ is a Free Lagrangian if $$\mathscr{F}[h_1] = \mathscr{F}[h_2]$$ , whenever $h_1 \simeq h_2$ That is, whenever the Sobolev homeomorphisms $h_1, h_2 : \mathbb{X} \xrightarrow{\text{onto}} \mathbb{Y}$ , in $\mathcal{W}^{1,p}(\mathbb{X}, \mathbb{R}^n)$ , are homotopy equivalent. It is important to note here that we are only considering homotopy equivalence between surjections (sometimes perhaps with other restrictions such as homeomorphisms). If $\mathbb{X}$ , $\mathbb{Y}$ are balls, then any continuous map $\mathbb{X} \to \mathbb{Y}$ is certainly homotopic to a constant map. We shall now make this concept clear with selected examples and illustrate how this leads to questions of the existence of frictionless energy minimal deformations.