By Tadeusz Iwaniec

The significant topic of this paper is the variational research of homeomorphisms $h: {\mathbb X} \overset{\textnormal{\tiny{onto}}}{\longrightarrow} {\mathbb Y}$ among given domain names ${\mathbb X}, {\mathbb Y} \subset {\mathbb R}^n$. The authors search for the extremal mappings within the Sobolev area ${\mathscr W}^{1,n}({\mathbb X},{\mathbb Y})$ which reduce the strength fundamental ${\mathscr E}_h=\int_{{\mathbb X}} \,|\!|\, Dh(x) \,|\!|\,^n\, \textrm{d}x$. a result of common connections with quasiconformal mappings this $n$-harmonic substitute to the classical Dirichlet quintessential (for planar domain names) has drawn the eye of researchers in Geometric functionality conception. specific research is made right here for a couple of concentric round annuli the place many unforeseen phenomena approximately minimum $n$-harmonic mappings are saw. The underlying integration of nonlinear differential varieties, referred to as loose Lagrangians, turns into actually a piece of paintings

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25) ω(x) = dt tn−1 n (−1)i = i=1 x1 dx1 ∧ ... ∧ dxi−1 ∧ dxi+1 ∧ ... ∧ dxn |x|n 48 6. VECTOR CALCULUS ON ANNULI Viewing ω as a diﬀerential form on punctured space Rn◦ , we ﬁnd that dω = 0. 27) (−1)i hω= i=1 hi dh1 ∧ ... ∧ dhi−1 ∧ dhi+1 ∧ ... ∧ dhn |h|n 1,n−1 Under suitable regularity hypothesis, for instance if h ∈ Wloc (Ω, Rn ), and |h(x)| const > 0, this form is also closed, meaning that d (h ω) = h (dω) = 0. , dhn . With such a view dh ∧ dt becomes an n-tuple of 2-forms. Further notation is self explanatory.

42) Θ = Θ+ = 1 1− n n−2 2n 4− n exp 1 n−2 1 √ tan−1 √ n n−1 n−1 Indeed, under the notation above, we have the following equation for t = Γ(s) 1 H (t) − Θ t = 1 − s2 n n n s2 1+ n−1 1− n 2 def − Θn Γn+ (s) == A(s) s−1 Note that A = A(s) is C ∞ -smooth on (−1, ∞). It vanishes at s = 1. 45) lim t→∞ H(t) = Θ = lim t H(t) t→0 t 5. 46) t2 H˙ 2 LH = H + n−1 2 n−2 2 H 2 − t2 H˙ 2 ≡ −1 , H(1) = 0 40 5. RADIAL n-HARMONICS Obviously H˙ cannot vanish. 47) 1−s n n−1 where −1 < s < 1. 48) This shows that Γ− : (−1, 1) → (0, ∞) is strictly increasing.

This is justiﬁed under an appropriate assumption on the degree of integrability of Λ, DivΛ, η and Dη. 12) is possible if and only if the vector ﬁeld 1 || Dh || n−2 D∗ h · Dh − || Dh || n I n(x) n is orthogonal to ∂X. In other words, at each x ∈ ∂X the linear mapping || Dh || n−2 D∗ h·Dh− n1 || Dh || n I takes the (one-dimensional) space of normal vectors into itself. Of course, the same holds for the mapping D∗ h · Dh. Since D∗ h · Dh is symmetric one can say, equivalently, that D∗ h · Dh preserves the tangent space.