Axially symmetric harmonic maps. (English) Zbl 0738.58014
Nematics. Defects, singularities and patterns in nematic liquid crystals: mathematical and physical aspects, Proc. NATO Adv. Res. Workshop, Orsay/Fr. 1990, NATO ASI Ser., Ser. C 332, 179-187 (1991).
[For the entire collection see Zbl 0722.00043.]
A map \(u: B^ 3\to S^ 2\) is called axially symmetric if, in cylindrical coordinates, \(u(r,\vartheta,z)=(\cos\vartheta\sin\varphi, \sin\vartheta\sin\varphi, \cos\varphi)\) for some \(\varphi(r,z)\) defined on the half disk \(D=\{(r,z): 0\leq r^ 2\leq 1-z^ 2\}\). Analysis of these maps is important to get insight into possible structure of singularities of solutions to variational problems — in a geometric measure-theoretic setting. The author discusses here some recent results and displays some new classes of non-minimizing harmonic maps with unusual singular behaviour.
A map \(u: B^ 3\to S^ 2\) is called axially symmetric if, in cylindrical coordinates, \(u(r,\vartheta,z)=(\cos\vartheta\sin\varphi, \sin\vartheta\sin\varphi, \cos\varphi)\) for some \(\varphi(r,z)\) defined on the half disk \(D=\{(r,z): 0\leq r^ 2\leq 1-z^ 2\}\). Analysis of these maps is important to get insight into possible structure of singularities of solutions to variational problems — in a geometric measure-theoretic setting. The author discusses here some recent results and displays some new classes of non-minimizing harmonic maps with unusual singular behaviour.
Reviewer: A.Ratto (Brest)
