n-harmonic mappings between annuli: the art of integrating by Tadeusz Iwaniec

By Tadeusz Iwaniec

The principal subject 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 power fundamental ${\mathscr E}_h=\int_{{\mathbb X}} \,|\!|\, Dh(x) \,|\!|\,^n\, \textrm{d}x$. as a result of usual connections with quasiconformal mappings this $n$-harmonic substitute to the classical Dirichlet imperative (for planar domain names) has drawn the eye of researchers in Geometric functionality concept. 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 kinds, known as loose Lagrangians, turns into really a piece of artwork

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n-harmonic mappings between annuli: the art of integrating free Lagrangians

The principal subject matter 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 house ${\mathscr W}^{1,n}({\mathbb X},{\mathbb Y})$ which reduce the power quintessential ${\mathscr E}_h=\int_{{\mathbb X}} \,|\!

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24) dt = 1 |x| n (−1)i xi dx1 ∧ ... ∧ dxi−1 ∧ dxi+1 ∧ ... ∧ dxn i=1 This form integrates naturally on spheres centered at the origin. We wish to normalize this form in such a way that the integrals will be independent of the sphere. 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.

34) is more involved. 36) dh ∧ dt n−1 n−1 n (−1)i i=1 hi 1 dh ∧ ... ∧ dhi−1 ∧ dhi+1 ∧ ... ∧ dhn ∧ dt |h| 2. 37) 1 2 n n−1 dh ∧ ... ∧ dh 1 i−1 ∧ dh i+1 ∧ ... ∧ dh ∧ dt n 2 i=1 Note that here in both sides only spherical derivatives of h are significant; the terms containing hN dt vanish after wedging them with dt. This observation permits us to replace dhi by the covectors ai = dhi − hiN dt. We view ai as elements of the space 1 (Rn−1 ). Once this interpretation is accepted, the proof continues via an algebraic inequality.

34 5. RADIAL n-HARMONICS suitable n-harmonic map on the rest of the interval. Such solutions, however, cannot be C 2 -smooth; C 1,1 -regular at best. 6). 1. A solution which is constant on an interval. Solutions H = H(t) to the characteristic equation LH ≡ c, that are constant in a proper subinterval like in the above figure, cannot be even C 1,1 -regular if c 0. 2. 21) LH◦ ≡ 0 LH∞ ≡ 0 H+ (1) = 1 H− (1) = 0 We shall solve these Cauchy problems and show that both H+ and H− are actually C ∞ -smooth.

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n-harmonic mappings between annuli: the art of integrating by Tadeusz Iwaniec
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