Let \X ⊂ \R^{n} be a bounded Lipschitz domain and consider the σ_{2}energy functional
\mathbb F_{σ2}[u; \X] : =  ⌠ ⌡

\X

 ⎢ ⎢

∧^{2} ∇u  ⎢ ⎢

2

dx, 

over the space of admissible Sobolev maps
A(\X) :=  ⎧ ⎨
⎩

u ∈ W^{1,4}(\X, \Sp^{n−1}) : u_{∂\X} = xx^{−1}  ⎫ ⎬
⎭

. 

In this article we address the question of multiplicity versus uniqueness for extremals and
strong local minimizers of the σ_{2}energy funcional \mathbb F_{σ2}[·, \X]
in A(\X) where the domain \mathbb X is ndimensional annuli.
We consider a topological class of maps referred to as spherical twists
and examine them in connection with the EulerLagrange equations associated with σ_{2}energy
functional over A(\X), the socalled σ_{2}harmonic map equation on \X.
The main result is a surprising discrepancy between even and odd dimensions. In
even dimensions the latter system of equations admits infinitely many smooth solutions
amongst such maps whereas in odd dimensions this number reduces to one.
The result relies on a careful analysis of the full versus the restricted
EulerLagrange equations.
Download TeX format
