Whirl Mappings on Generalised Annuli and the Incompressible Symmetric Equilibria of the Dirichlet Energy
Tóm tắt
In this paper we show a striking contrast in the symmetries of equilibria and extremisers of the total elastic energy of a hyperelastic incompressible annulus subject to pure displacement boundary conditions. Indeed upon considering the equilibrium equations, here, the nonlinear second order elliptic system formulated for the deformation
$u=(u_{1}, \ldots, u_{N})$
:
$$ {\mathbb{E}} {\mathbb{L}}[u, {\mathbf {X}}] = \left \{ \textstyle\begin{array}{l@{\quad}l} \Delta u = \operatorname{div}(\mathscr{P} (x) \operatorname{cof} \nabla u) & \textrm{in }{\mathbf {X}},\\ \det\nabla u = 1 & \textrm{in }{\mathbf {X}},\\ u \equiv\varphi& \textrm{on }\partial{\mathbf {X}}, \end{array}\displaystyle \right . $$
where
${\mathbf {X}}$
is a finite, open, symmetric
$N$
-annulus (with
$N \ge2$
),
$\mathscr{P}=\mathscr{P}(x)$
is an unknown hydrostatic pressure field and
$\varphi$
is the identity mapping, we prove that, despite the inherent rotational symmetry in the system, when
$N=3$
, the problem possesses no non-trivial symmetric equilibria whereas in sharp contrast, when
$N=2$
, the problem possesses an infinite family of symmetric and topologically distinct equilibria. We extend and prove the counterparts of these results in higher dimensions by way of showing that a similar dichotomy persists between all odd vs. even dimensions
$N \ge4$
and discuss a number of closely related issues.
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