I wish to clear up a slight confusion that I have. This isn't very complicated but I'm just a bit stumped.
we know the the Robin Boundary Condition states:
$$ au+b\frac{\partial u}{\partial n} = g\hspace{15pt}on\hspace{15pt}\partial\Omega$$
My question is, assuming a=b=1, what's the difference between
$$u=g \biggr\rvert_{\frac{\partial u}{\partial n}=0},$$or $$\frac{\partial u}{\partial n}=g \biggr\rvert_{u=0},$$or $$ u+\frac{\partial u}{\partial n} = g$$ ?
In my case, a=b=1, and my Neumann and Dirchilet Boundaries are
$$u=1, \hspace{15pt} \frac{\partial u}{\partial n} =0$$
Which yields
$$ u+\frac{\partial u}{\partial n} = 1,\hspace{15pt} \frac{\partial u}{\partial n}=0$$
However, this can just as easily be
$$ u+\frac{\partial u}{\partial n} = 1,\hspace{15pt} u=0,\frac{\partial u}{\partial n}=1$$ or even
$$ u+\frac{\partial u}{\partial n} = 1,\hspace{15pt} u=0.5,\frac{\partial u}{\partial n}=0.5$$
Clearly, each of these possibilities are mathematically/physically distinct. How is this reconciled?