A question in studying Maximum-Minimum principle?

Question

Let $ u $ be a continuous function on closed ball $ B[x,r_0] \subseteq \mathbb{R}^n, n = 2,3 $ and satisfy \begin{equation*} \int_{B[x,r]} u(y) dy = 0, \forall r \leq r_0 \end{equation*} How can we deduce that $ u(y) = 0, \forall y \in B[x,r_0] $?

Answer 1

If $n=1$,

$$2a\min\{u(y): x-a < y < x+a\} \le \int_{x-a}^{x+a}u(y)dy\, \le\, 2a\max\{u(y): x-a < y < x+a\}$$

If $u$ is continuous, ask yourself what this means for $a\rightarrow 0$ and how to generalize this to higher dimensions.

Answer 2

You can't deduce that. For instance, you could have $u=r\sin\phi$ in polar coordinates for $n=2$.