Let $\Omega$ be a bounded domain with a Lipschitz boundary. Consider the Dirichlet-to-Neumann map $\Lambda:H^{\frac{1}{2}}(\partial \Omega)\to H^{-\frac{1}{2}}(\partial \Omega)$ defined via
$$ \langle \Lambda f, h\rangle = \int\_{\Omega} \nabla u\cdot \nabla v \, dx,$$
for any $f,h \in H^{\frac{1}{2}}(\partial \Omega)$ where $u\in H^{1}(\Omega)$ is the unique solution to $\Delta u=0$ with Dirichlet data $f$ on $\partial \Omega$ while $v\in H^{1}(\Omega)$ is any function with trace $h$ on $\partial \Omega$.


My question is whether it is true that $\Lambda$ is continuous from $H^1(\partial \Omega)$ to $L^2(\partial \Omega)$. Moreover, is it also continuous from $H^s \to H^{s-1}$ for all $s$ in the closed interval $[0,1]$?



Dirichlet-to-Neumann map on Lipschitz domains
Let $u$ be the solution of the Dirichlet problem for Laplacian in a Lipschitz domain with boundary data $g$. Then, for every $s\in [1/2,3/2]$,
$$
\| u \|\_{H^{s}\,(U)} \leq C \| g \|\_{H^{s-1/2}\,\,\,\,(\partial U)} .
$$
This is a classical result of [Jerison & Kenig](https://www.sciencedirect.com/science/article/pii/S0022123685710671). See the remarks below Theorem 0.5 of that paper.


I believe this estimate and a duality argument implies the result you are looking for.




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Edit: Since this answer received a downvote, let me add the details of the little duality argument. If I am making a mistake, I would welcome specific comments pointing out my error.


Let $f \in H^1(\partial \Omega)$ and $h\in L^2 (\partial \Omega)$. Let $u$ and $v$ be their respective harmonic extensions to $\Omega$. Then
\begin{align\*}
\bigl| \bigl\langle \Lambda f, h \bigr\rangle \bigr|
=
\biggl| \int\_{\Omega} \nabla u \cdot \nabla v \biggr|
& \leq 
\| \nabla u \|\_{H^{1/2}\,\,(\Omega)} \| \nabla v \|\_{H^{-1/2}\,\,\,(\Omega)} 
\\ & 
\leq
C\| u \|\_{H^{3/2}\,\,(\Omega)} \| v \|\_{H^{1/2}\,\,(\Omega)} 
\\ & 
\leq 
C\| f \|\_{H^1\,(\partial\Omega)} \| h \|\_{L^2\,(\partial\Omega)} \,\,,
\end{align\*}
where in the last line we used the Jerison-Kenig estimate at both endpoints: $s=1/2$ for $v$ and $s=3/2$ for $u$.
By duality, we obtain the estimate
$$
\| \Lambda f \|\_{L^2\,(\partial \Omega)}
\leq 
C \| f \|\_{H^1\,(\Omega)},
$$
which concludes the argument that $\Lambda$ is continuous from $H^1(\Omega)$ to $L^2(\partial \Omega)$.


I have made this argument for $s=1$, but it is easy to check it works for $s$ precisely in the range $[0,1]$.
You would apply the Jerison-Kenig estimate with $(s+1/2)$ to $u$ and $(-s+3/2)$ to $v$.


