This is a follow up question I had while reading through this question and "Representing Homotopy Groups and Spaces of Maps by $p$-harmonic maps" by Shihshu Walter Wei.
We know that continuous weakly harmonic maps from two Riemannian manifolds $M$ and $N$ are smooth, i.e. $C^2(M,N)$ or in that case even $C^\infty(M,N)$. Generally, the best we can hope for regarding $p$-harmonic maps is $C^{1,\alpha}(M,N)$. Since I am mostly interested in the case $p > \dim M$ the linked paper gives an answer about the existence and regularity of $p$-harmonic maps:
2.2 Theorem. For any compact manifolds $M$ and $N$, and for any $p > \dim M$, each component of $L^p_1(M,N)$ or $C^1(M,N)$ contains a $C^{1,\alpha}$, $p$-harmonic representative which is an $E_p(u)( = \frac{1}{p} \int_M \vert \nabla u \vert^p)$-minimum in its component.
My question is now: Do we know anything about specific targets (like spherical targets) where the regularity could be improved to get $C^2(M,N)$ or $C^\infty(M,N)$? Any references in that direction are also highly appreciated.
Right now, I know of the paper "Regularity of p-harmonic functions on the plane" by Tadeusz Iwaniec and Juan J. Manfredi where they show that, generally, $C^{1,\alpha}(M,N)$ holds for maps between Euclidean spaces and $p > \dim M$.
