Harmonic map equations are an elliptic PDE system arising from the
minimisation problem of Dirichlet energies between two manifolds. In
this talk we present some some recent works concerning the symmetry
and stability of harmonic maps. We construct a new family of
''twisting'' examples of harmonic maps and discuss the existence,
uniqueness and regularity issues. In particular, we characterise of
singularities of minimising general axially symmetric harmonic maps,
and construct non-minimising general axially symmetric harmonic maps
with arbitrary 0- or 1-dimensional singular sets on the symmetry axis.
Moreover, we prove the stability of harmonic maps from $\mathbb{B}^3$
to $\mathbb{S}^2$ under $W^{1,p}$-perturbations of boundary data, for
$p \geq 2$. The stability fails for $p <2$ due to Almgren--Lieb and
Mazowiecka--Strzelecki.
(Joint work with Prof. Robert M. Hardt.)