# 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.

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{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} 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.