# A constant ratio of integrals? Part I

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

For $$0, consider the average of its Dirichlet integral $$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 $$r$$.

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

## 1 Answer

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

• Ouch! I know what I missed in the assumptions. Jun 21 at 21:52
• @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. Jun 21 at 22:07
• Done, Iosif Pinelis, now there is a Part II. Jun 21 at 22:16
• Why the downvote? Can you explain? Jun 22 at 13:07