Epsilon-regularity for $p$-harmonic maps at a free boundary on a sphere

Abstract

We prove an $𝜖$-regularity theorem for vector-valued $p$-harmonic maps, which are critical with respect to a partially free boundary condition, namely that they map the boundary into a round sphere.

This does not seem to follow from the reflection method that Scheven used for harmonic maps with free boundary (i.e., the case $p=2$): the reflected equation can be interpreted as a $p$-harmonic map equation into a manifold, but the regularity theory for such equations is only known for round targets.

Instead, we follow the spirit of Schikorra’s recent work on free boundary harmonic maps and choose a good frame directly at the free boundary. This leads to growth estimates, which, in the critical regime $p=n$, imply Hölder regularity of solutions. In the supercritical regime, $p<n$, we combine the growth estimate with the geometric reflection argument: the reflected equation is supercritical, but, under the assumption of growth estimates, solutions are regular.

In the case $p<n$, for stationary $p$-harmonic maps with free boundary, as a consequence of a monotonicity formula we obtain partial regularity up to the boundary away from a set of $\left(n-p\right)$-dimensional Hausdorff measure.