Dirichlet-to-Neumann operator

Definition

Let $\Omega \subset \R^n$, $n \ge 2$, be a bounded connected open set with smooth boundary $\partial\Omega$ and outward unit normal $\nu$. Let $H^{1/2}(\partial\Omega)$ and $H^{-1/2}(\partial\Omega)$ be the boundary Sobolev spaces of order $\tfrac12$ and $-\tfrac12$. For $f \in H^{1/2}(\partial\Omega)$, let $u_f \in H^1(\Omega)$ be the weak solution of the Dirichlet problem $$\Delta u = 0 \text{ in } \Omega, \qquad u = f \text{ on } \partial\Omega.$$ The boundary condition is taken in the sense of the trace operator. The Dirichlet-to-Neumann operator is the map $$\Lambda \colon H^{1/2}(\partial\Omega) \to H^{-1/2}(\partial\Omega), \qquad \Lambda f = \partial_\nu u_f|_{\partial\Omega}.$$ More generally, let $\gamma$ be measurable on $\Omega$ with $c \le \gamma \le c^{-1}$ for some $c > 0$. The operator $\Lambda_\gamma$ uses the equation $\nabla \cdot (\gamma \nabla u) = 0$ and the weakly defined Neumann data $\gamma \, \partial_\nu u|_{\partial\Omega}$.