Let $D: C^{\infty}(\partial K) \rightarrow C^{\infty}(\partial K)$ be the Steklov operator (or sometimes so-called Dirichlet-to-Neumann operator) defined via the relation: $\psi \mapsto u|_{\partial K}$.
Here $u$ solves the following boundary value problem:
$$ \Delta u = 0$$ $$ \frac{\partial u}{\partial \vec{n}}|_{\partial K} = \psi|_{\partial K}$$
Therefore, we can calculate the spectrum of the Steklov operator for a disk and thus obtain some general propetries of the spectrum via Riemann Mapping Theorem up to some conformal equivalence.
Let also $u := u_{\psi}$, $w := v_{\varphi}$ be smooth functions that solve the boundary value problem in the preceding setting. Then we define so-called second Steklov operator $S_{2}$ as $$\int_{\partial K}{(S_{2} \psi) \varphi dx } = \int_{K} {\langle \nabla^{2} u, \nabla^{2} w \rangle dx}$$
By differentiating the Bochner-Lichnerowicz-Weitzenbock formula, we get that $S_{2}$ can be expressed as $$S_{2}(f) = - \nabla_{\partial K}{Df} - D(\nabla_{\partial K}{f}) - H_{\partial K}{f} + D \nabla_{\partial K} \cdot II_{\partial K} \nabla_{\partial K} D(f)$$ Here $II_{\partial K}$ stands for the second fundamental form, $H_{\partial K}$ is the mean curvature, D is the Dirichlet-to-Neumann operator on $K$.
My questions are:
(1) Are there any implicitly stated relationships between the Dirichlet-to-Neumann operator and 2nd Steklov operator apart from those that can be directly derived from the definition? What can we say about some propetries that the spectrums of these two operators share?
(2) Is it possible to treat the 2nd Steklov operator as a "connecting" operator for some mixed boundary value problem, perhaps, such as Dirichlet-to-Neumann is related to the Diriclet problem and, correspondingly, Neumann problem?
Any piece of advice or references provided are much appreciated!