Examples of $p$-harmonic maps

TL;DR

The paper addresses the regularity of vector-valued $p$-harmonic maps $u:\mathbb{R}^n\to\mathbb{R}^N$ and constructs explicit examples that are more irregular than previously known, providing new upper bounds on regularity. The authors develop a unifying construction $u(x)=|x|^{\gamma-k}h(x)$ with a harmonic homogeneous polynomial $h$ of degree $k$ and determine $\gamma$ from the equation $\gamma^2(p-1)+\gamma(n-p)-k(k+n-2)=0$, linking homogeneity to regularity. By employing the Hurwitz problem to optimize the target dimension $N$ for given $n$, they obtain admissible pairs $(n,N)$ and show $N_{\min}(n)=O(n^2/\log n)$; they also derive sharp regularity observations via $\tau=(\gamma-1)/p'$ and $\theta=1/p$, revealing ellipsoidal constraints on possible regularity. The results include explicit $p$-harmonic maps in all dimensions $n\ge2$, disproving conjectures about $V(\nabla u)$-regularity for certain ranges and illustrating that in two dimensions vectorial $p$-harmonic maps can be far less regular than scalar ones; the work also connects PDE regularity questions to the Hurwitz problem and extends to infinity-harmonic limits. Overall, the framework provides systematic, explicit examples that establish practical upper bounds on regularity and deepen the understanding of irregular behavior in vector-valued $p$-harmonic maps.

Abstract

We construct explicit examples of $p$-harmonic maps $u:\mathbb{R}^n \to \mathbb{R}^N$. These are more irregular than the previously known examples and thus provide new upper bounds for the regularity of $p$-harmonic maps, including the case of $\infty$-harmonic maps. To optimize our approach, we utilize solutions of the Hurwitz problem from algebra.

Figures (4)

• Figure 1: Optimal regularity of $p$-harmonic functions for $n=2$, $N=1$, as given in \ref{['eq:alpha2d']}. Left: the regularity $\alpha$ of $\nabla u$. Right: $\tau=\alpha /p'$ forms an ellipse.
• Figure 2: The regularity of $\nabla u$, $A(\nabla u)=\abs{\nabla u}^{p-1}\nabla u$ and $V(\nabla u)=\abs{\nabla u}^{\frac{p-2}{2}}\nabla u$ for $n=2$ and $N=1$.
• Figure 3: Left: Regularity for $n=N=2$ expressed in $\tau=(\gamma -1)/p'$. It is more irregular than any scalar $p$-harmonic function on $\setR^2$. Right: Regularity of our examples for $n=2,3,4$. This disproves the $V\in C^1$-conjecture for $n\geq 3$.
• Figure 4: Regularity of our examples dimensions $2,8,32,128,512,2048$. The regularity is expressed in $\tau =(\gamma -1)/p'$ via $\tau ^2 + \tau (1+\theta (n-2))-\theta(1-\theta)(n+1)=0$. For $n\to \infty$ the functions approach $\tau = (1-\theta)$ (dashed).

Theorems & Definitions (18)

• definition 1: $p$-harmonic maps
• theorem 1
• proof : Proof of Theorem \ref{['thm:cond_for_h']}
• definition 2
• theorem 2
• proof
• theorem 3
• proof
• remark 1
• remark 2