Some Problems On Harmonic Maps from $\mathbb{B}^3$ to $\mathbb{S}^2$

Harmonic map equations are an elliptic PDE system 
arising from the minimisation of Dirichlet energies between two 
manifolds. In this talk we present 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 the 
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 
$p≥2$.

 (Joint work with Prof. Robert Hardt.)