n-Harmonic mappings between annuli

Proposition 1.3. (Transition to the inverse map) Let f ∈W 1,n−1 loc (Y, X) be a homeomorphism of finite outer distortion between bounded domains, with KI(y, f) ∈L 1(Y). Then the inverse map h = f −1 : X onto −→Y belongs to the Sobolev class W 1,n(X, Y) and we have (1.11) n n 2 Z Y KI(y, f)dy

Theorem 1.4. Let X ⊂Rn be a ball with a k-dimensional closed disk removed, and let Y ⊂Rn be a ball with a (k + 1)-dimensional closed disk removed, 1 ⩽k < n −1. Then every homeomorphism h : X onto −→Y has infinite n-harmonic energy. Note that both X and Y are topological annuli; that is, homeomorphic images of a spherical annulus A = {x: 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

Theorem 1.5. [30, Theorem 1.1] For the above-mentioned pair of do- mains X and Y, there exists a nonnegative continuous function η = η(x) defined on X such that (1.16) dist hj(x), ∂Y
⩽η(x) || Dhj || L n(X), η ≡0 on ∂X. 1Homeomorphisms converging weakly in W 1,n(X, Y) also converge c-uniformly, so their limits are still continuous, taking X into Y .

Theorem 1.6. [30, Theorem 1.4] The mapping h is continuous and Y ⊂h(X) ⊂Y. Furthermore, there exists a measurable mapping f : Y →X, such that h ◦f = id : Y →Y, everywhere on Y. This right inverse mapping has bounded variation, || f || BV(Y) ⩽ Z 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 K ⊂Y its preimage h−1(K) ⊂X is also a continuum; that is compact and connected. The proof of this theorem is presented in Section 3.3. 1.6. Annuli The first nontrivial case is that of doubly connected domains. Thus we consider mappings h : A →A∗between concentric spherical annuli in Rn. A = A(r, R) = {x ∈Rn ; r < |x| < R} , 0 ⩽r < R < ∞ A∗= A(r∗, R∗) = {y ∈Rn ; r∗< |y| < R∗} , 0 ⩽r∗< R∗< ∞ Such domains are of different confor

Theorem 1.8. Let A = A(r, R) and A∗= A(r∗, R∗) be planar annuli, 0 < r < R < ∞and 0 < r∗< R∗< ∞. We have:

Theorem 1.9. Let Mod A∗> Mod A. Then for n = 2, 3, the n-harmonic radial map h◦= λℵ−(kx) assumes the minimum conformal energy within all homeomorphisms. Such a minimizer is unique up to a conformal auto- morphism of A. Surprisingly, for n ⩾4 the answer will depend on how wide is the target annulus A∗, relatively to A.

Theorem 1.10. For dimensions n ⩾4, there exists a function N † = N †(t), t < N †(t) < ∞for t > 0, see Figure 2, such that: if (1.32) Mod A ⩽Mod A∗⩽N †(Mod A) -upper Nitsche bound for n ⩾4, then the map h◦: A →A∗is a unique (up to an automorphism of A) minimizer of the conformal energy among all homeomorphisms.

Theorem 1.11. In dimensions n ⩾4, there are annuli A and A∗such that no radial stretching from A onto A∗minimizes the conformal energy. 1.11. Conformally contracting pair In this case we obtain the minimizers from the principal solution ℵ+ = H+ |x|
x |x|. As before, we observe that the mappings (1.33) h◦(x) = λ H+(kx) , k > 0 , λ > 0 are radial n-harmonics. Recall that A∗is conformally thinner than A. But it is not enough. In contrast to the previous case, such n-harmonic mappings take the an

Theorem 1.12. Under the condition at (1.34) there exist unique k > 0 and λ > 0 such that h◦(x) = λH+(kx) takes A homeomorphically onto A∗. This map is a unique (up to a conformal automorphism of A) minimizer of the conformal energy among all homeomorphisms of A onto A∗.

Theorem 1.13. The following deformation (1.37) h◦(x) def ==  x |x| r < |x| ⩽1 , hammering part ℵ+(x) 1 ⩽|x| ⩽R , n-harmonic part is a W 1,n-limit of homeomorphisms h: A onto

Theorem 1.14. Let A and A∗be spherical annuli in Rn, n = 2, 3. Then for every homeomorphism h: A onto −→A∗we have (1.47) Fh def == Z A || Dh || n |h|n ⩾ n −1 + α2
n 2 Mod A , where α = Mod A∗

Theorem 1.15. For each n ⩾4, there exists αn > 1 such that (1.47) holds whenever (1.48) α def == Mod A∗ Mod A < αn The power stretching h(x) = r∗r−α|x|α−1x is the only minimizer of Fh modulo conformal automorphism of A. Examples will be given to show that the extremals are no longer power stretchings if (1.49) Mod A∗ Mod A ⩾ r n −1

Theorem 1.16. Under the Nitsche bounds (1.63) ℵ†(Mod A) ⩽Mod A∗⩽ℵ†(Mod A) the L 1(Y)-norm of the inner distortion KI(y, f) assumes its minimum value on a mapping f : A∗onto −→A whose is inverse is a radial n-harmonic mapping h◦: A onto −→A∗. Such an extremal mapping f is unique up to a conformal automorphism of A.

Theorem 1.17. If the domain annulus A∗is too thin relative to the target annulus A; precisely, under the condition (1.64) Mod A∗< ℵ†(Mod A) -below the Nitsche bound. then the infimum of the L 1(A∗)-norm of KI(y, f) is not attained among homeomorphisms f : A∗onto −→A. Nevertheless, we were able to find the infimum of the 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. Conclusio

Proposition 2.1. If ǫ is small enough then there is no n-harmonic homeomorphism h: A into −→Vǫ such that S ⊂h(A).

Lemma 3.1. Every n-harmonic mapping h ∈W 1,n loc (X, Rn) solves the generalized n-harmonic equation (3.6).

Proposition 3.2. The equilibrium solution for mappings that are slip- ping along the boundaries satisfies, in addition to (3.6), the following con- dition (3.13) D∗h · Dh: Tx∂X →Tx∂X equivalently, (3.14) D∗h · Dh: Nx∂X →Nx∂X. where Tx∂X and Nx∂X designate the tangent and normal spaces at x ∈∂X.

Lemma 5.1. Let h(x) = H(|x|) x |x| be a radial stretching of class C 2(A, Rn), where A = {x ; a < |x| < b}. Then, with the notation |x| = t , we have n ˙H n −1 div || Dh || n−2Dh = − x tn+1 d dt n h (n −1)H2 + t2 ˙H2i n−2 2  H2 −t2 ˙H2 o

Proposition 5.3. Every radial n-harmonic mapping in the annulus A = {x ; a < |x| < b} takes the form (5.24) g(x) = λ · h(kx) , λ ∈R , k > 0 where h ∈C ∞(Rn ◦), is one of the four principal n-harmonics. It should be observed, as a corollary, that radial n-harmonics are C ∞- smooth in the entire space Rn ◦. 5.3. The elasticity function We shall distinguish four classes of the radial n-harmonics h(x) = H(|x|) x |x|. The concept of so-called conformal elasticity underlines this distinction. In

Proposition 6.1. The following point-wise estimates hold (6.33) |hN| ⩾| d|h| ∧⋆dt | (6.34) |hT |n−1 ⩾|h|n−1 dt ∧h♯ω

Lemma 6.2. For every n-tuple of covectors in Rn−1, a1, ..., an ∈V1(Rn−1), we have (6.38) |a1|2+...+|an|2 ⩾(n−1) n X i=1 |a1 ∧... ∧ai−1 ∧ai+1 ∧... ∧an|2 ! 1 n−1

Lemma 7.1. Let Φ : [r∗, R∗] →R be any integrable function. Then the n-form (7.3) Φ(|h|) dh1 ∧... ∧dhn is a free Lagrangian. Precisely, we have (7.4) Z A Φ(|h|)J(x, h) dx = ωn−1 Z R∗ r∗ τ n−1 Φ(τ) dτ for every orientation preserving homeomorphism h ∈W 1,n(A, A∗). This is none other than a general formula of integration by substitution.

Lemma 7.2. The following differential n-form (7.5) n X i=1 xi dx1 ∧... ∧dxi−1 ∧d|h| ∧dxi+1 ∧... ∧dxn |h| |x|n = (d|h|) ∧⋆dt |h| tn−1 is a free Lagrangian in the class of all homeomorphisms h ∈W 1,1(A, A∗) that preserve the order of the boundary components of the annuli A and A∗. 57

Lemma 7.2, is obtained as follows.

Lemma 7.3. The following differential n-form (7.11) n X i=1 hi dh1 ∧... ∧dhi−1 ∧d|x| ∧dhi+1 ∧... ∧dhn |x| |h|n

Corollary 7.5. Let h be a homeomorphism between spherical rings A and A∗in the Sobolev class W 1,n(A, A∗). Then (7.16) Z A Φ |h|
|hN| |hT |n−1 ⩾ωn−1 Z R∗ r∗ τ n−1Φ(τ) dτ whenever Φ is integrable in [r∗, R∗]. We have the equality in (7.16) if and only if |hN| |hT |n−1 = J(x, h). Furthermore, (7.17)

Theorem 8.1. Let A and A∗be spherical rings in Rn, n ⩾2. Then for every h ∈P(A, A∗) we have (8.7) Z A |Dh|n |h|n ⩾max {1, αn} Mod A As for the sharpness of this estimate we note that equality holds for the power stretching h(x) = |x|α−1x. There are, however, other cases of equality in (8.7) if Mod A∗̸= Mod A. 8.2. Radial symmetry Suppose that the radial stretching (8.8) h(x) = H(|x|) x

Proposition 8.2. For each radial stretching we have (8.9) Fh = Z A || Dh || n |h|n ⩾ Z A || Dhα || n |hα|n = α2 + n −1
n

Lemma 9.1. Let X, Y ⩾0 and 1 ⩽α < αn. Then (9.3) a = a(α) def == (α2 + n −1) n−2 2 (α2 −1) αn < 1 and, we have (9.4)  X2 + (n −1)Y 2 n 2 ⩾a Xn + b XY n−1 where (9.5)

Lemma 9.1.

Lemma 9.2. Let X, Y ⩾0 and 0 ⩽α ⩽1. Then (9.10)  X2 + (n −1)Y 2 n 2 ⩾a Y n + b XY n−1 where (9.11) a = (n −1) α2 + n −1
n−2 2 1 −α2
and (9.12) b = nα α2 + n −1

Proposition 12.1. Let h : A →A∗be a permissible map in P(A, A∗) where (12.14) R r < R∗ r∗ < H− γn R r
H−(γn) Then (12.15) Z A

Theorem 13.1. Below the upper Nitsche bound; that is, (13.4) Mod A∗⩽N †(Mod A) every permissible minimizer h : A →A∗coincides with the radial extremal map modulo conformal automorphisms of A. Let h ∈P(A, 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) |hN| = η |h|
|hT | see (9.17), (9.20), (11.5) and (12.21). Here η = η(τ) is a nonnegative function

Lemma 13.2. (Uniqueness in the point Cauchy problem) Suppose we are given two solutions h◦and h to the Cauchy-Green equation (13.13) such that |h(a)| = |h◦(a)| for some point a ∈Ω. Then there is an isometry T : Rn →Rn such that (13.20) h(x) = T h◦(x) for all x ∈Ω

Lemma 9.1 in its borderline case when a = a(αn) = 1. By formula (9.4) we obtain (14.26) h α2 + (n −1)X 2 n−1 i n 2 ⩾αn + b αX for every random variable X : S →R+. Equality holds if and only if X assumes exactly two values 0 and  α αn n−1

Theorem 14.1. Suppose (14.54) Mod A∗> r n −1 n −3 Mod A , n ⩾4 Then (14.55) inf h∈R(A , A∗) Z A || Dh || n |h|n

Theorem 14.2. Let n ⩾4 and (14.61) δn = √n −1 + √n −3 √n −1 −√n −3  1 2 exp  n −2 n√n −1 tan−1 √ n −3  ⩾√n. Consider the annuli A = A(r, R) and A∗= A(r∗, R∗), such that

Theorem 15.1. Suppose h : A onto −→A∗is a quasiconformal map between annuli. Then (15.5) 1 KI ⩽ Mod A∗ ModA n−1 ⩽KO 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

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 : A onto −→ A∗of the form (15.6) h(x) = p r∗R∗ √ rR |x| !±α Φ  x

Theorem 15.3. An annulus A = A(r, R) can be mapped conformally onto A∗= A(r∗, R∗), if and only if R r = R∗ r∗. Moreover, modulo isometry and rescaling, every conformal mapping takes the form (15.9) h(x) =  x the identity x |x|2 the inversion 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.

Definition 1.1. A homeomorphism h : X →Rn of Sobolev space W 1,1 loc (X, Rn) is said to have finite outer distortion if || Dh(x) || n ⩽n n 2 K(x) J(x, h) for some measurable function 1 ⩽K(x) < ∞. The smallest such K(x) is called the outer distortion, denoted by KO(x, h) . Then h is K -quasiconformal if KO(x, h) ⩽K for some constant K.

Definition 5.2. The term principal solution pertains to each of the following four functions of class C 2(0, ∞) which solve the equation LH = constant; (5.20) H◦(t) = t , LH◦≡0 (5.21) H∞(t) = 1 t , LH∞≡0