Solvability of Poisson equation with Cauchy boundary condition I am interested in any explanation/comment/reference you could provide me regarding the solvability of the Poisson problem with Cauchy boundary data
$$
\begin{cases}
-\Delta u  = f \ &\textrm{in}\ \Omega \\ 
\hspace{0.3cm} \partial_\nu u = g &\textrm{on}\ \partial\Omega \\
\hspace{0.65cm} u = h \ &\textrm{on}\ \partial\Omega \\
\end{cases}
$$
for $f \in L^2(\Omega)$ and $g,h \in H^{\frac{1}{2}}(\partial\Omega)$ (or more regular data if needed).
It is obvious that the problem does not always have a solution. For example, there is always a solution
$$u = \text{const.} \quad \text{for} \quad f \equiv 0,\ g =0,\ h=\text{const.} $$
But there is no solution $u$ when
$$ f \equiv 0,\ g \neq 0,\ h = \text{const.}$$
Because if there was one, it would violate the strong maximum principle (harmonic functions attain their min/max on the boundary $\partial\Omega$).
Furthermore, the weak form of the problem does not really make sense if we set $h = 0$ and take test functions $\phi \in H^1_0(\Omega)$. We then have
$$
\int_\Omega \nabla u \nabla \phi - \int_{\partial\Omega} \partial_\nu u \phi = \int_\Omega f\phi
$$
and due to $\phi$ being in $H^1_0(\Omega)$, $ \int_{\partial\Omega} \partial_\nu u~ \phi$ vanishes. Thus how do we impose the boundary condition?
I suspect that a kind of energy method is needed, to treat the problem as an optimization problem with relaxed (weak) satisfaction of the b.c.. But again, as I pointed before, there may not be a solution.
Thus, I would like to learn if there are conditions for the solvability of this problem.
Thanks in advance!
 A: There is a theory for coercive operators on well behaved domains $\partial \Omega = \partial \Omega_D \cup \partial \Omega_N$, where $|\Omega_D|>0$ that
\begin{equation}
\begin{aligned}
-\Delta u = f  & \quad \mbox{in} \quad \Omega\\ 
u(x) = g & \quad \mbox{on} \quad \partial \Omega_D\\
\partial_\nu u(x) = h & \quad \mbox{on} \quad \partial \Omega_N 
\end{aligned}
\end{equation}
has an unique weak solution. See https://www2.karlin.mff.cuni.cz/~pokorny/LectureNotes/moderni_teorie_color.pdf
(Czech, pg 69-72) or https://www.amazon.de/-/en/Lawrence-C-Evans/dp/0821849743
Now assume $\Omega_D = \partial \Omega$ and find that unique weak solution. Now assume that in the sense of traces (you can have a look for traces in Evans, chapter 5.5) there exists $\partial_\nu u(x) = \hat{h}$. Your problem will be solvable if and only if $h = \hat{h}$. In other words you are imposing too many conditions.
That being said however it is still of interest for which $f$'s the solution vanish completely at the boundary.
