Applying Green's Formula in 2D

[Somefantastik]

for

$$\Delta w = \frac{ \partial^{2} w }{x_{1}^{2}} + \frac{ \partial^{2} w }{x_{2}^{2}}$$

and $$\nabla = \left(\frac{\partial}{\partial x_{1}},\frac{\partial}{\partial x_{2}}\right)$$

$$\int_{\Omega} \nabla v \nabla w d \Omega = \int_{\Gamma} v \frac{\partial w}{\partial n} d \Gamma - \int_{\Omega}v \Delta w d \Omega$$

where $$\Gamma$$ is the boundary of $$\Omega$$

so if I have

$$- \int_{\Omega}v \Delta w d \Omega$$

I can apply Green's thm to get

$$- \int_{\Omega}v \Delta w d \Omega = \int_{\Omega} \nabla v \nabla w d \Omega - \int_{\Gamma} v \frac{\partial w}{\partial n} d \Gamma$$

But what if I'm starting with

$$\int_{\Omega} v \left( \frac{\partial^{2} w }{\partial x_{1}^{2}} + k \frac{\partial^{2} w }{\partial x_{2}^{2}} \right) d \Omega$$

where k is some scalar? I'm thrown off by only one of the dimensions being scaled.

 

[jvicens]

it is important to approach problems with a systematic and analytical mindset. In this case, we can break down the given equation into smaller components to better understand how to handle the scaling in one dimension.

First, let's consider the integral of the second derivative term with respect to x1. We can rewrite this term as:

\int_{\Omega} v \frac{\partial^{2} w }{\partial x_{1}^{2}} d \Omega

Using integration by parts, we can expand this term as:

\int_{\Omega} \frac{\partial v}{\partial x_{1}} \frac{\partial w}{\partial x_{1}} d \Omega - \int_{\Gamma} v \frac{\partial w}{\partial x_{1}} \frac{\partial n}{\partial x_{1}} d \Gamma

where n is the unit outward normal vector on the boundary, and we have assumed that v and w are smooth enough for these derivatives to exist.

Similarly, we can rewrite the second derivative term with respect to x2 as:

\int_{\Omega} k v \frac{\partial^{2} w }{\partial x_{2}^{2}} d \Omega

Using integration by parts again, we get:

\int_{\Omega} k \frac{\partial v}{\partial x_{2}} \frac{\partial w}{\partial x_{2}} d \Omega - \int_{\Gamma} k v \frac{\partial w}{\partial x_{2}} \frac{\partial n}{\partial x_{2}} d \Gamma

Now, putting these two terms together, we can rewrite the original equation as:

\int_{\Omega} \left( \frac{\partial v}{\partial x_{1}} \frac{\partial w}{\partial x_{1}} + k \frac{\partial v}{\partial x_{2}} \frac{\partial w}{\partial x_{2}} \right) d \Omega - \int_{\Gamma} \left( v \frac{\partial w}{\partial x_{1}} \frac{\partial n}{\partial x_{1}} + k v \frac{\partial w}{\partial x_{2}} \frac{\partial n}{\partial x_{2}} \right) d \Gamma

Notice that the scaling factor k now appears in both terms on the boundary