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Let $\Omega\subset \mathbb R^n$ be an open domain and $\tau>0$. Consider the following boundary value problem

$-\Delta v=f $ in $\Omega$, $\partial_\nu v+\tau v=g$ on $\partial\Omega$.

If $\Omega$ is a bounded with sufficiently smooth boundary it is known that we have "maximal elliptic $L_p$-regularity", i.e. for $f\in H^k_p(\Omega)$ and $g\in W^{k+1-1/p}_p(\partial\Omega)$ there is a unique solution $v\in H^{k+2}_p(\Omega)$, where $k\in \mathbb N$ and $p\in(1,\infty)$, see e.g. Triebel, Interpolation theory, Function spaces, Differential operators.

Is anybody aware of a corresponding result for the case when $\Omega$ is unbounded, e.g. a half space or an infinite layer?

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For the case $p=2$, I would have a look at this paper by Wolfgang Arend and Mahamadi Warma, and its follow-up papers: Potential Analysis 19: 341–363, 2003.

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