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You may define a conormal derivative of $u$ in a very weak sense, just as a distribution. Or you can indeed take a normal derivative in a stronger recalling that each $u\in H^\frac{3}{2}(\Omega)$ s.t. $\Delta u\in L^2(\Omega)$ has a weak normal derivative in $L^2(\Omega)$, see e.g. the classical book of Lions-Magenes if you allow for a smooth boundary of $\Omega$; things are more delicate if $\Omega$ is rougher, say, merely Lipschitz, but can still be dealt with, cf. e.g. this article.
Here a weak normal derivative is defined as follows: If $u\in H^1(\Omega)$, then $g\in L^2(\partial \Omega)$ is called its weak normal derivative if the Gau{\ss}-Green Gauss-Green formula $$\int_\Omega \nabla u \nabla \phi +\int_\Omega \Delta u \phi = \int_{\partial \Omega} g \phi$$ holds for all $\phi \in H^1(\Omega)$. ($g$ need not exist for general $u\in H^1(\Omega)$; but if it exists, it is clearly unique).
You may define a conormal derivative of $u$ in a very weak sense, just as a distribution. Or you can indeed take a normal derivative in a stronger recalling that each $u\in H^\frac{3}{2}(\Omega)$ s.t. $\Delta u\in L^2(\Omega)$ has a weak normal derivative in $L^2(\Omega)$, see e.g. the classical book of Lions-Magenes if you allow for a smooth boundary of $\Omega$; things are more delicate if $\Omega$ is rougher, say, merely Lipschitz, but can still be dealt with, cf. e.g. this article.
Here a weak normal derivative is defined as follows: If $u\in H^1(\Omega)$, then $g\in L^2(\partial \Omega)$ is called its weak normal derivative if the Gau{\ss}-Green formula $$\int_\Omega \nabla u \nabla \phi +\int_\Omega \Delta u \phi = \int_{\partial \Omega} g \phi$$ holds for all $\phi \in H^1(\Omega)$. ($g$ need not exist for general $u\in H^1(\Omega)$; but if it exists, it is clearly unique).