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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!

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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.

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