Help solving Green's identities question.

[yungman]

Homework Statement

Suppose $u$ is harmonic ($\nabla^2 u = 0$) and $v=0 \;\hbox{ on } \;\Gamma$ where $\Gamma$ is the boundary of a simple or multiply connected region and $\Omega$ is the region bounded by $\Gamma$.

Using Green's identities, show:

$$\int \int_{\Omega} \nabla u \cdot \nabla v \; dx dy = 0$$

Homework Equations

Green's identities:

$$\int \int_{\Omega}\; (u\nabla^2 v \;+\; \nabla u \cdot \nabla v) \; dx dy \;= \;\int_{\Gamma} \; u\frac{\partial v}{\partial n} \; ds$$

$$\int \int_{\Omega} \;(u\nabla^2 v \;-\; v\nabla^2 u )\; dx dy \;=\; \int_{\Gamma} \;(u\frac{\partial v}{\partial n} - v\frac{\partial u}{\partial n}) \;ds$$

$$\frac{\partial u}{\partial n} = \nabla u \; \cdot \; \widehat{n}$$

The Attempt at a Solution

I use

$$\int \int_{\Omega}\; (u\nabla^2 v \;+\; \nabla u \cdot \nabla v) \; dx dy \;= \;\int_{\Gamma} \; u\frac{\partial v}{\partial n} \; ds$$

If $v=0 \hbox { on } \Gamma \;\;\Rightarrow\;\; v \hbox { is a constant = 0 } \;\;\Rightarrow\;\; v= 0 \;\hbox{ on } \;\Omega$.

$$\int \int_{\Omega}\; (u\nabla^2 v \;+\; \nabla u \cdot \nabla v) \; dx dy \;= \int \int_{\Omega}\; \nabla u \cdot \nabla v) \; dx dy \;= \;\int_{\Gamma} \; u\frac{\partial v}{\partial n} \; ds = 0$$

Is this the right way?

 

[yungman]

I think I have the answer, I edit the original post already. Can anyone verify my work?

 

[yungman]

Anyone please?