Originally Posted by

**OliviaB** Hi,

I am really struggling with a subject at my university which is all about partial differential equations. It takes me a long time to understand even the most basic concepts, and I really need some help understanding and completing the following questions.

Let $\displaystyle \mathcal{R}$ be a bounded region in $\displaystyle \mathbb{R}^3$, and suppose $\displaystyle p(x) > 0 $ on $\displaystyle \mathcal{R}$.

(i) If $\displaystyle u$ is a solution of

$\displaystyle \bigtriangledown^2 u = p(x) u \ \ x \in \mathcal{R} \ \ \bigtriangledown \cdot n = 0 \ \x \in \partial \mathcal{R}$

show that $\displaystyle u \equiv 0$ on $\displaystyle \mathcal{R}$

(ii) If $\displaystyle u$ is a solution of

$\displaystyle \bigtriangledown^2 u = p(x) u \ \ x \in \mathcal{R} \ \ \bigtriangledown \cdot n = g(x) \ \x \in \partial \mathcal{R}$

show that $\displaystyle u$ is unique (It can be assumed that part (i) is true).