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Let $\Omega$ be a smooth bounded domain with $\partial\Omega = \Gamma$. I have read this.

For all $u \in H^1(\Omega)$ such that $-\Delta u = g \in L^2(\Omega)$ in distribution, we can define the normal derivative $u_\nu \in H^{-\frac 12}(\Gamma)$ such that $$\langle u_\nu, \psi \rangle_{H^{-\frac 12}(\Gamma), H^{\frac 12}(\Gamma)} = -\int_\Omega gD\psi + \int_\Omega \nabla u \nabla D\psi$$ where $D\psi \in H^1(\Omega)$ is an extension of $\psi \in H^{\frac 12}(\Gamma)$.

(eg. here) In other words, we have this notion of a weak normal derivative for $H^1$ functions (usually we need $u \in H^2$).

Question: why do we need to ask for $\Delta u \in L^2(\Omega)$? Why not just define the normal derivative like so: $$\langle u_\nu, \psi \rangle_{H^{-\frac 12}(\Gamma), H^{\frac 12}(\Gamma)} = \langle \Delta u, D\psi\rangle_{H^{1}(\Omega)^*, H^1(\Omega)} + \int_\Omega \nabla u \nabla D\psi?$$

I have not seen something like this in any text except these lecture notes. Can anyone tell me why?

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    $\begingroup$ It might be just a matter of convenience: it is easier to work with an $L^2$ function than a more general distribution. $\endgroup$ Sep 27, 2014 at 14:35
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    $\begingroup$ $H^1(\Omega)^*$ is not a space of distributions in $\Omega$. For instance, the functional on the left hand side of your equation is in $H^1(\Omega)^*$. This defeats the objective of separating the inhomogeneous term in the PDE from the boundary condition. $\endgroup$ Sep 28, 2014 at 2:26

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It is not possible to define a normal derivative for all $u \in H^1(\Omega)$ which depends continuously on $u$.

The reason is that all $C_c^\infty(\Omega)$ is dense in $H^1_0(\Omega)$, but all $u \in C_c^\infty(\Omega)$ have zero normal derivative. If we would have a normal derivative depending continuously on $u \in H^1(\Omega)$, this implies that it is zero for all $u \in H^1_0(\Omega)$ and this is not correct.

Your definition fails since $\Delta u \in H^1(\Omega)^*$ is meaningless for $u \in H^1(\Omega)$ (how do you define it?).

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    $\begingroup$ I do agree that is not possible to define the normal derivative for all $H^1$ functions. Well $\Delta u \in H^1(\Omega)^*$ can be defined $\langle \Delta u, v \rangle = \int_\Omega \nabla u \nabla v$. Also $C_c^\infty$ is not dense in $H^1$. $\endgroup$
    – riem
    Dec 13, 2014 at 17:08

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