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  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 $p>1$.
"A compactness result for $p$-harmonic maps." Differential Integral Equations 14 (1) 75 - 84, 2001. https://doi.org/10.57262/die/1356123376