Assume $\Omega$ is an open, bounded subset of $\mathbb R^3$ with smooth boundary $\partial \Omega= \Gamma$. For $u \in H^{1/2}(\Gamma)$, let $U \in H^1(\Omega)$ denote the weak solution of the Dirichlet problem:

$\begin{cases} \Delta U=0 & \text{in } \Omega\\ U=u & \text{on } \Gamma \\ \end{cases}$

and let $Nu:=\frac{\partial U}{\partial \nu}$ be the Neumann data on $\Gamma$. We have $N: H^{1/2}(\Gamma) \to H^{-1/2}(\Gamma).$

My question is, what is known about the square root of $N$ and its properties (e.g. its boundedness between which Sobolev spaces, positivity, etc), and does anyone have a reference?

Motivation: I have a solution of some PDE $u \in L^2(0,T;H^2(\Gamma)) \cap H^1(0,T;L^2(\Gamma))$ (maybe smoother than this too) and I need to essentially write $$\int_\Gamma u_tN(u)$$ in terms of a nice non-negative quantity, in the same way that I would write for the Laplacian (with good boundary conditions) $$\int_\Omega u_t(-\Delta u) = \frac 12\frac{d}{dt}\int_\Omega |\nabla u|^2.$$ I need this to derive estimates.