Vol. 13, No. 5, 2020

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

Katarzyna Mazowiecka, Rémy Rodiac and Armin Schikorra

Vol. 13 (2020), No. 5, 1301–1331

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 (np)-dimensional Hausdorff measure.

$p$-harmonic maps, free boundary regularity, epsilon-regularity
Mathematical Subject Classification 2010
Primary: 58E20, 35B65, 35R35, 35J58, 35J66
Received: 6 September 2017
Revised: 31 January 2019
Accepted: 29 June 2019
Published: 27 July 2020
Katarzyna Mazowiecka
Institut de Recherche en Mathématique et Physique
Université Catholique de Louvain
Rémy Rodiac
Institut de Recherche en Mathématique et Physique
Université Catholique de Louvain
Armin Schikorra
Department of Mathematics
University of Pittsburgh
Pittsburgh, PA
United States