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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 $[0,1]$?

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  • $\begingroup$ $\Lambda_g=\Lambda$? $\endgroup$
    – username
    Aug 1, 2022 at 12:50

1 Answer 1

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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. 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.


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 $v$.

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