Suppose $u$ is $p$-harmonic, i.e., it solve $-\operatorname{div} |\nabla u|^{p-2} \,\nabla u = 0$ where $1<p<\infty$. Then is the following inequality true? $$ \int_{S_1} (u-k)^2|\nabla u|^{p-2} \,d\sigma \leq C \int_{S_1} |u_T|^2|\nabla u|^{p-2} \,d\sigma $$ where $S_1$ is the unit sphere in $\mathbb{R}^n$, $n$ is the dimension, $k = \frac{1}{|S_1|} \int_{S_1} u\, d\sigma$, $d\sigma$ is the surface measure on the sphere and $u_T$ denotes the tangential derivative of $u$.

I suspect this is true and that $C \leq \frac{1}{n-1}$.



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