Let $\Omega$ be a compact, riemannian manifold with non-empty smooth boundary $\partial \Omega = \Gamma$. For a smooth function $u \in C^\infty(\Gamma)$ we define the harmonic extension $\hat{u}$ as the solution of the PDE $\begin{align} \Delta \hat u = 0 \text{ on $\Omega$} \\ \hat u = u \text { on $\Gamma$ } \end{align} $.

Now we define the Dirichlet-to-Neumann Operator $N$ by

$N \colon C^\infty (\Gamma) \to C^\infty (\Gamma), \ u \mapsto \frac{\partial \hat u}{\partial \nu}$ where $\nu$ is the outward unit conormal on the boundary $\Gamma$.

In Partial Differential Equations 2, Taylor shows that $N$ is an elliptic, symmetric Pseudodifferential Operator of order 1.

Is it possible to conclude from this information, that $N$ is essentially self-adjoint?

  • 2
    $\begingroup$ The special case of $\Omega$ being an $n-$dimensional ball is very nicely explained in Lax: Functional Analysis, Section 36.2. $\endgroup$ – András Bátkai Jan 25 '16 at 18:39

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