Is $\|x\|^6 \sin^6 \|x\|^6$ harmonic?

Question

Suppose the function $$ u(x)=\|x\|^6 \sin^6 \|x\|^6$$ for $x \in \mathbb{R}^d$, where $$\|x\| = \sqrt{x_1^2 + \ldots +x_d^2}.$$ How can I decide if the function $u$ is harmonic in the unit ball $B_1(0)$ without calculating $\Delta u$?

Answer 1

Harmonic functions have the mean value property: For all small enough $r > 0$

$$u(x) = \frac{1}{\omega_{d-1}\cdot r^{d-1}} \int_{\lVert y-x\rVert = r} u(y)\,dS(y),$$

where $\omega_k$ is the $k$-dimensional volume of the $k$-dimensional unit sphere, and $dS$ is the surface measure on the sphere.

Here, we have

$$u(0) = 0 < \frac{1}{\omega_{d-1}\cdot r^{d-1}}\int_{\lVert y\rVert = r} u(y)\,dS(y)$$

for $0 < r < \pi^{1/6}$, since $u(y) > 0$ on that sphere.