Hi!


I was wondering about the definition of the conormal derivative of a function
$u$ which is given on a domain $\Omega$. It is known that if $-\Delta u = f$, considered
as functionals on $H^1\_0(\Omega)$, this does not provide enough information to
define a conormal derivative of $u$.
However, a lot of textbooks, for example "Strongly elliptic systems..." by W. McLean,
state that if $-\Delta u = f$ as functionals on $H^1(\Omega)$, then the conormal derivative of $u$ can be defined (just by enforcing Green's formula). I'm not very convinced of this, because saying that $-\Delta u = f$ as functionals on $H^1(\Omega)$ means that i need to have a conormal derivative of $u$ defined already! Or, i can put it like that: either,


* the definition of a weak solution $u$ and its conormal derivative has to be done simultaneously. However, i don't see how to do that.
* Or, the definition of the conormal derivative depends on both, $u$ and $f$. But does this make any sense?


Maybe someone has thought about this and can share his ideas with me.



The conormal derivative of a function
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](http://dx.doi.org/10.1016/j.jde.2011.06.017).


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 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).


