Let $u(x)$ be a harmonic polynomial in the unit ball $B_1(0)\subset\mathbb{R}^n$ with $u(0)=0$.

For $0<r\leq1$, consider the average of its Dirichlet integral $$A(r):=\frac1{\vert B_r(0)\vert}\int_{B_r(0)}\vert\nabla u\vert^2dx,$$ and the average of the square function on the boundary $$B(r):=\frac1{\vert \partial B_r(0)\vert}\int_{\partial B_r(0)}u^2d\sigma.$$

I would like to ask:

QUESTION. Is this true? The ratio $\frac{r^2A(r)}{B(r)}$ is a constant in $r$.

  • $\begingroup$ Is $u$ real-valued, or did you mean $|u|^2$ in place of $u^2$? $\endgroup$ Jun 21 at 17:06
  • $\begingroup$ Yes, $u$ is real-valued. $\endgroup$ Jun 21 at 17:15

1 Answer 1


No. E.g., if $n=2$ and $u(x,y)=x + x^2 - y^2$ for $(x,y)\in\mathbb R^2$, then $\dfrac{r^2A(r)}{B(r)}=2\dfrac{1+2r^2}{1+r^2}$.

  • $\begingroup$ Ouch! I know what I missed in the assumptions. $\endgroup$ Jun 21 at 21:52
  • 2
    $\begingroup$ @T.Amdeberhan : Despite the simplicity of the example, I spent quite a bit of time to find it. So, It would be great if you would reverse your latest edit. $\endgroup$ Jun 21 at 22:07
  • $\begingroup$ Done, Iosif Pinelis, now there is a Part II. $\endgroup$ Jun 21 at 22:16
  • $\begingroup$ Why the downvote? Can you explain? $\endgroup$ Jun 22 at 13:07

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.