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Harmonic diffeomorphisms with rotational symmetry. (English) Zbl 0726.58017
Harmonic diffeomorphisms of a surface are energy minimizing according to a result of {\it J. M. Coron} and the author [Compos. Math. 69, No.2, 175-228 (1989; Zbl 0686.58012)]. Here the author shows that, for $m\ge 3$, the identity map of the unit ball $B\sp m$ onto itself with canonical metric on the domain and `SO(m)-equivariant metric’ h on the range is energy minimizing (provided it is harmonic) in the Sobolev space $H'$ of $B\sp m$ to $B\sp m.$ SO(m)-equivariance means that $$ h\sb{ij}(x)=h\sb{\Vert}x\sp ix\sp jr\sp{-2}+h\sb{\perp}(r)[\delta\sb{ij}-x\sp ix\sp jr\sp{-2}],\quad r=\vert x\vert, $$ where $h\sb{\Vert},h\sb{\perp}: [0,1]\to (0,\infty)$ are continuous. The proof is based on a comparison with a null Lagrangian.
Reviewer: G.Tóth (Camden)

58E20Harmonic maps between infinite-dimensional spaces
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