Suppose $U \in \mathbb{R}^n$ is an open domain, and $u\in C^2(U) \cap C(\bar{U})$ such that $\Delta u = 0$ in $U$.
I'm working on a couple of problems pertaining to the mean value formula/harmonic functions. Although I have a feeling that a maximum principle is lurking in the second problem due to the hypotheses.
- Suppose $u \geq 0$ in $U$. Use the mean value property to show either $u>0$ in $U$ or $u \equiv 0$ in $U$.
We have $u(x) = \displaystyle\frac{1}{|\partial B(x,r)}\int_{\partial B(x,r)} u(y)\,dS(y)$ for each $B(x,r) \subset U$. I'm not sure where to go from here, the conclusion seems like it would follow obviously from the hypotheses, especially if considering if $u=0$ and $u>0$ as separate cases.
- Let $u$ and $v$ be harmonic and $u \geq v$ on $\partial U$. Assume $u$ is not identically equal to $v$ on $\partial U$, Prove that $u > v$ in $U$.
I'm not sure what to do with this one either, other than to maybe examine $u - v \geq 0$ and try to apply a maximum principle. I don't think we can apply the first problem.