Regularity of minimizing $p$-harmonic maps - i.e., minimizers of the
Dirichlet $p$-energy among maps between two given manifolds - is known
to depend the topology of the target manifold. In particular, the
case of maps into spheres has been studied intensively, but still some
of the most basic questions concerning maps from $B^3$ into $S^3
$ remain open. Minimizing maps in this context were shown to be regular
when $p=2$ or $p\ge3$, and recently also when $2<p<2.13$, leaving a
peculiar gap in between. I will discuss known approaches to the problem
and how these can be extended to cover $2<p<2.36$ and $2.97<p<3$, thus
shrinking the gap. This is joint work with Katarzyna Mazowiecka
(University of Warsaw).