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2020 Epsilon-regularity for $p$-harmonic maps at a free boundary on a sphere
Katarzyna Mazowiecka, Rémy Rodiac, Armin Schikorra
Anal. PDE 13(5): 1301-1331 (2020). DOI: 10.2140/apde.2020.13.1301

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.

Citation

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Katarzyna Mazowiecka. Rémy Rodiac. Armin Schikorra. "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

Information

Received: 6 September 2017; Revised: 31 January 2019; Accepted: 29 June 2019; Published: 2020
First available in Project Euclid: 17 September 2020

zbMATH: 07271831
MathSciNet: MR4149063
Digital Object Identifier: 10.2140/apde.2020.13.1301

Subjects:
Primary: 35B65 , 35J58 , 35J66 , 35R35 , 58E20

Keywords: $p$-harmonic maps , epsilon-regularity , free boundary regularity

Rights: Copyright © 2020 Mathematical Sciences Publishers

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