Unique Solutions for Vector Calculus Problem with Boundary Conditions

[Gregg]

Homework Statement

$\gamma_1$ and $\gamma_2$ are both real continuous solutions of $\nabla^2 \gamma = \gamma$ in $V$ and $\gamma_1=\gamma_2$ on the boundary $\partial V$. We are looking at the function $g = \gamma_1 - \gamma_2$.

I have proved

$\nabla \cdot \left( g \nabla g) \right) = ||\nabla g||^2 + g\nabla^2 g$ already. I used this to show that

$\int_V ||\nabla g||^2 dV + \int_{\partial V} g^2 dV = 0$

The question is: what does this say about the value of $g=\gamma_1-\gamma_2$ in $V$ and are the solutions unique for $\nabla^2 \gamma = \gamma$?

Any help appreciated!

 

[Mindscrape]

If you can show g=constant then the solution is unique, within a constant (for dirichlet conditions it's obviously unique). Your first equation doesn't make sense, I think you mean to put g*laplacian(g). Also, if you managed to get rid of this term, your logic should probably also apply to the g^2 term on the boundary. Elaborate on what you've done at the boundary.