We prove an -regularity theorem for vector-valued -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 ): the reflected equation can be interpreted as a -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 , imply Hölder regularity of solutions. In the supercritical regime, , 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 , for stationary -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 -dimensional Hausdorff measure.
"Epsilon-regularity for $p$-harmonic maps at a free boundary on a sphere." Anal. PDE 13 (5) 1301 - 1331, 2020. https://doi.org/10.2140/apde.2020.13.1301