Differential and Integral Equations

$L^p$-gradient harmonic maps into spheres and SO(N)

Armin Schikorra

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We consider critical points of the energy $$ E(v) := \int_{\mathbb R^n} |\nabla^s v|^{\frac{n}{s}}, $$ where $v$ maps locally into the sphere or $SO(N)$, and $\nabla^s = (\partial_1^s,\ldots,\partial_n^s)$ is the formal fractional gradient, i.e., $\partial_\alpha^s$ is a composition of the fractional Laplacian with the $\alpha$-th Riesz transform. We show that critical points of this energy are Hölder continuous. As a special case, for $s = 1$, we obtain a new, more stable proof of Fuchs and Strzelecki's regularity result of $n$-harmonic maps into the sphere [22, 9], which is interesting on its own.

Article information

Differential Integral Equations, Volume 28, Number 3/4 (2015), 383-408.

First available in Project Euclid: 4 February 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 58E20: Harmonic maps [See also 53C43], etc. 35B65: Smoothness and regularity of solutions 35J60: Nonlinear elliptic equations 35S05: Pseudodifferential operators


Schikorra, Armin. $L^p$-gradient harmonic maps into spheres and SO(N). Differential Integral Equations 28 (2015), no. 3/4, 383--408. https://projecteuclid.org/euclid.die/1423055234

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