A compactness result for $p$-harmonic maps

Abstract

For $p>1$ we prove a compactness result for $p$-harmonic maps with values in $S^k$, the $(k+1)$-dimensional sphere. We generalize a lemma from [12] to vector-valued functions with assumptions on the $p$-Laplacian. We obtain the existence of weak solutions of the $p$-harmonic flow with values in $S^k$ for each $k\geq 1$ and $p>1$.