# Dirichlet-to-Neumann map on Lipschitz domains

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 $$[0,1]$$?

• $\Lambda_g=\Lambda$? Aug 1, 2022 at 12:50

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.
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 $$v$$.