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
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 (np)-dimensional Hausdorff measure.

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