On regularity theory for n/p-harmonic maps into manifolds

Abstract

In this paper we continue the investigation of the regularity of the so-called weak $\frac{n}{p}$-harmonic maps in the critical case. These are critical points of the following nonlocal energy \[ {\mathcal{L}}_s(u)=\int_{\mathbb{R}^n}| ( {-Δ})^{\frac{s}{2}} u(x)|^p dx\,, \] where $u\in {\dot{H}}^{s,p}({\mathbb{R}}^n,\mathcal{N})$ and $\mathcal{N}\subset {\mathbb{R}}^N$ is a closed $k$ dimensional smooth manifold and $s=\frac{n}{p}$. We prove Hölder continuity for such critical points for $p\le 2$. For $p>2$ we obtain the same under an additional Lorentz-space assumption. The regularity theory is in the two cases based on regularity results for nonlocal Schrödinger systems with an antisymmetric potential.

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Cited by in corpus (5)

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• Minimal $W^{s,\frac{n}{s}}$-harmonic maps in homotopy classes
• Epsilon-regularity for p-harmonic maps at a free boundary on a sphere
• 3-Commutators Revisited