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.