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$$\int_{\Omega} \nabla u \cdot \mathbf{n}\, v \, d\Omega,$$

where $\Omega \subset \mathbb{R}^2$ is a bounded domain with Lipschitz continuous and piecewise smooth boundary $\Gamma:=\partial \Omega$, $u, v \in H^1(\Omega)$ and $\mathbb{n}$ is the unit normal vector. In others words, is it possible to apply the Divergence theorem to this integral? What is the solution?

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  • $\begingroup$ Are you sure about the integral you mentioned? The normal derivative $\nabla u\cdot n$ is defined on the boundary. Green's theorem tells you that $\endgroup$ – PierreCarre May 8 at 10:38
  • $\begingroup$ Yes. I obtained $-\int_{\Omega} \nabla^2 u v= \int_{\Omega} \nabla u \nabla v -\int_{\Gamma} \nabla u \cdot \mathbb{n} v$. Then, I need to integrate by parts $\int_{\Gamma} \nabla u\cdot \mathbb{n} v$ again to obatin something in the boundary of $\Gamma$. But I did not find any results on that. $\endgroup$ – Neo May 8 at 12:35
  • $\begingroup$ How is your normal vector defined? The normal vector is normal to what? $\endgroup$ – SprocketsAreNotGears May 8 at 15:20
  • $\begingroup$ $\mathbf{n}$ stands for the exterior unit normal vector field. $\endgroup$ – Neo May 8 at 16:24
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Are you sure about your integral? The normal derivative is defined on the boundary. Green's theorem would give you (with some extra regularity): $$ \int_{\Gamma} (\nabla u \cdot n) v = \int_{\Omega} \nabla u \cdot \nabla v - \int_{\Omega} \Delta u v. $$

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  • $\begingroup$ Unfortunately, the integral is in $\Omega$. I'd like to know if it is possible to integrate by part the following equation $\int_{\Omega} \nabla u \cdot \mathbb{n} \, v \, d\Omega$. $\endgroup$ – Neo May 8 at 11:16
  • $\begingroup$ How do you even compute $\nabla u \cdot n$ in $\Omega$? Can you provide some context or source for this question? I beleive there is something wrong woith the question. @Neo $\endgroup$ – PierreCarre May 8 at 11:42
  • $\begingroup$ By integration by parts I obtained $-\int_{\Omega} \nabla^2 u v= \int_{\Omega} \nabla u \nabla v -\int_{\Gamma} \nabla u \cdot \mathbb{n} v$. Then, I need to integrate by parts $\int_{\Gamma} \nabla u\cdot \mathbb{n} v$ again to obatin something in the boundary of $\Gamma$. But I did not find any results on that. $\endgroup$ – Neo May 8 at 12:27
  • $\begingroup$ @Neo The boundary of $\Gamma$ is $\Gamma$... $\endgroup$ – PierreCarre May 8 at 12:41
  • $\begingroup$ No, I am looking for something in a line of $\Gamma$. My guess is something $\int_{\Gamma} = \int_{C}$. $\endgroup$ – Neo May 8 at 12:51

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