Applying Green's Formula in 2D

  • #1
for

[tex] \Delta w = \frac{ \partial^{2} w }{x_{1}^{2}} + \frac{ \partial^{2} w }{x_{2}^{2}} [/tex]

and [tex] \nabla = \left(\frac{\partial}{\partial x_{1}},\frac{\partial}{\partial x_{2}}\right) [/tex]

[tex]\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[/tex]

where [tex] \Gamma [/tex] is the boundary of [tex]\Omega [/tex]

so if I have

[tex] - \int_{\Omega}v \Delta w d \Omega [/tex]

I can apply Green's thm to get

[tex] - \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 [/tex]

But what if I'm starting with

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

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

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