# What is the result of the integration by parts of $\int_{\Omega} \nabla u \cdot \mathbf{n}\, v \, d\Omega$?

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

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?

• 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 – PierreCarre May 8 at 10:38
• 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. – Neo May 8 at 12:35
• How is your normal vector defined? The normal vector is normal to what? – SprocketsAreNotGears May 8 at 15:20
• $\mathbf{n}$ stands for the exterior unit normal vector field. – Neo May 8 at 16:24

Are you sure about your integral? The normal derivative is defined on the boundary. Green's theorem would give you (with some extra regularity): $$\int_{\Gamma} (\nabla u \cdot n) v = \int_{\Omega} \nabla u \cdot \nabla v - \int_{\Omega} \Delta u v.$$
• Unfortunately, the integral is in $\Omega$. I'd like to know if it is possible to integrate by part the following equation $\int_{\Omega} \nabla u \cdot \mathbb{n} \, v \, d\Omega$. – Neo May 8 at 11:16
• 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 – PierreCarre May 8 at 11:42
• 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. – Neo May 8 at 12:27
• @Neo The boundary of $\Gamma$ is $\Gamma$... – PierreCarre May 8 at 12:41
• No, I am looking for something in a line of $\Gamma$. My guess is something $\int_{\Gamma} = \int_{C}$. – Neo May 8 at 12:51