Robin-Laplacian in unbounded domains - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T02:08:52Z http://mathoverflow.net/feeds/question/118619 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/118619/robin-laplacian-in-unbounded-domains Robin-Laplacian in unbounded domains Richard Gustier 2013-01-11T11:33:38Z 2013-05-03T19:22:00Z <p>Let $\Omega\subset \mathbb R^n$ be an open domain and $\tau>0$. Consider the following boundary value problem</p> <p>$-\Delta v=f$ in $\Omega$, $\partial_\nu v+\tau v=g$ on $\partial\Omega$.</p> <p>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.</p> <p>Is anybody aware of a corresponding result for the case when $\Omega$ is unbounded, e.g. a half space or an infinite layer?</p> http://mathoverflow.net/questions/118619/robin-laplacian-in-unbounded-domains/118620#118620 Answer by András Bátkai for Robin-Laplacian in unbounded domains András Bátkai 2013-01-11T12:31:01Z 2013-01-11T12:31:01Z <p>For the case $p=2$, I would have a look at <a href="http://cantor.mathematik.uni-ulm.de/m5/arendt/publications/arendt-pub/short/2003-AreWar-LplRbnBndCndArbDmn.pdf" rel="nofollow">this</a> paper by Wolfgang Arend and Mahamadi Warma, and its follow-up papers: Potential Analysis 19: 341–363, 2003.</p>