# Dirichlet-to-Neumann map on $C^{k,1}$ domains

I am interested in the mapping properties of the Dirichlet-to-Neumann map (also called the Poincare-Steklov operator) for $C^{k,1}$ domains, between Sobolev spaces on the boundary. What I know is in the case $k=0$, i.e., for Lipschitz domains, this map is bounded between $H^{s+1/2}\to H^{s-1/2}$ for $s\in[-\frac12,\frac12]$, and in the cases $k\geq1$, the DtN map is bounded between $H^{s+1/2}\to H^{s-1/2}$ for $s\in[-k,k]$. These results are for example in McLean's book on strongly elliptic systems, and as you can see there seem to be room for improvement for the $k\geq1$ case. Namely one would expect the boundedness to occur for $s\in[-k-\frac12,k+\frac12]$. If it is true can you please provide a reference or give a sketch of an argument?

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