Let
\({\mathbb{X} \subset \mathbb {R}^n}\) be a bounded Lipschitz domain and consider the energy functional
$${{\mathbb F}_{\sigma_2}}u; \mathbb{X}] := \int_\mathbb{X} {\mathbf F}(\nabla u) \, dx,$$
over the space of
admissible maps
$${{\mathcal {A}_\varphi}(\mathbb{X}) :=\{u \in W^{1,4}(\mathbb{X}, {\mathbb{R}^n}) : {\rm det}\, \nabla u > 0\, {\rm for}\, {\mathcal {L}^n}{\rm -a.e. in}\, \mathbb{X}, u|_{\partial \mathbb{X}} =\varphi \}},$$
where the integrand
\({{\mathbf F}\colon \mathbb M_{n\times n}\to \mathbb{R}}\) is
quasiconvex and sufficiently regular. Here our attention is paid to the prototypical case when
\({{\mathbf F}(\xi):=\frac{1}{2}\sigma_2(\xi)+\Phi(\det\xi)}\). The aim of this paper is to discuss the question of multiplicity
versus uniqueness for
extremals and
strong local minimizers of
\({\mathbb F_{\sigma_2}}\) and the relation it bares to the domain topology. In contrast, for constructing
explicitly and directly solutions to the system of Euler–Lagrange equations associated to
\({{\mathbb F}_{\sigma_2}}\), we use a topological class of maps referred to as
generalised twists and relate the problem to
extremising an associated energy on the compact Lie group
\({\mathbf {SO}(n)}\). 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.
