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Epsilon-regularity
for $p$-harmonic maps at a free boundary on a sphere
Katarzyna Mazowiecka, Rémy Rodiac and Armin Schikorra |
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Vol. 13 (2020), No. 5, 1301–1331
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Abstract |
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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. |
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Keywords
$p$-harmonic maps, free boundary regularity,
epsilon-regularity
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Mathematical Subject Classification 2010
Primary: 58E20, 35B65, 35R35, 35J58, 35J66
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Milestones
Received: 6 September 2017
Revised: 31 January 2019
Accepted: 29 June 2019
Published: 27 July 2020
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Authors |
| Katarzyna Mazowiecka | |
| Institut de Recherche en
Mathématique et Physique Université Catholique de Louvain Louvain-la-Neuve Belgium |
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| Rémy Rodiac | |
| Institut de Recherche en
Mathématique et Physique Université Catholique de Louvain Louvain-la-Neuve Belgium |
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| Armin Schikorra | |
| Department of Mathematics University of Pittsburgh Pittsburgh, PA United States |
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