Abstract:
We consider a family of manifolds $(X_\varepsilon)_\varepsilon$ that shrinks
to a metric graph $X_0$ as $\varepsilon \to 0$, i.e., a topological graph
where each edge is assigned a length. A simple example is given by the
(smoothed) surface of the $\varepsilon$-tubular neighbourhood of $X_0$. Let
$Y_0$ be the set of vertices of degree $1$, and $Y_\varepsilon$ the
corresponding boundary of $X_\varepsilon$. Using boundary triples based on
first order Sobolev spaces, we can define the Dirichlet-to-Neumann map
$\Lambda_\varepsilon(z)$ associated to the boundary $Y_\varepsilon$ in
$X_\varepsilon$ and a Laplace-type operator $\Delta_\varepsilon \ge 0$
for $\varepsilon \ge 0$. In particular, for suitable $\varphi$ on
$Y_\varepsilon$, we let $\Lambda_\varepsilon(z)\varphi $ be the
``normal'' derivative of the Dirichlet solution $h$ of $(\Delta_\varepsilon
-z) h = 0$ with boundary data $\varphi$. The talk is based on a
joint work with J.\ Behrndt.
