We establish small energy Hölder bounds, uniform with respect to
, for
minimizers
of
where
is a positive definite quadratic form and the potential
constrains
to be close to a given manifold
. This implies that, up to
subsequence,
converges
locally uniformly to an
-valued
-harmonic
map, away from its singular set. This is the first result of its kind for
general anisotropic energies covering in particular the previously open
case of three-dimensional Landau–de Gennes model for liquid crystals,
with three distinct elastic constants. Similar results in the isotropic case
rely
on three ingredients: a monotonicity formula for the scale-invariant energy on
small balls, a uniform pointwise bound, and a Bochner equation for the
energy density; all of these ingredients are absent for general anisotropic
’s. In particular,
the lack of monotonicity formula is an important reason why optimal estimates on the singular
set of
-harmonic
maps constitute an open problem. To circumvent these difficulties we
devise an argument that relies on showing appropriate decay for the
energy on small balls, separately at scales smaller and larger than
:
the former is obtained from the regularity of solutions to elliptic
systems, while the latter is inherited from the regularity of
-harmonic
maps. This also allows us to handle physically relevant boundary conditions for
which, even in the isotropic case, uniform convergence up to the boundary was
open.
