I came across the following which I would like to understand but I don't know what parts of which pde books would be best to read. It does not seem like this pde is standard material in any random book I look up so I need help in knowing what to read. Ideally I would like an exposition that would explain the details such as why the domain of $\Lambda$ is chosen to be half differentiable functions among other things. And more generally I'm totally unfamiliar with pseudodifferential operators and distributions.
Let $\Omega$ be a relatively compact, simply connected open set in $\mathbb{R}^2$ with $C^2$ boundary. Let $\gamma(p)>0$ be a $C^2$ function on $\overline{\Omega}$. Let $f$ be a function defined on $\partial\Omega$. Then there is a unique function $u$, defined on $\overline\Omega$, such that $$\nabla(\gamma \nabla u)= 0$$ and $u(p)=f(p)$ for $p\in \partial\Omega$. Let $\frac{\partial u}{\partial n}(p)$ be the directional derivative of $u$ in the direction of the outward pointing unit normal $n$ at the point $p\in \partial\Omega$. Then the Dirichlet-to-Neumann map, $\Lambda$, is defined by the formula $$\Lambda f(p)=\gamma(p)\frac{\partial u}{\partial n}(p)$$ The domain of $\Lambda$ may be taken to be $H^{\frac{1}{2}}(\partial\Omega)$ and the image is in $H^{-\frac{1}{2}}(\partial\Omega)$. $\Lambda$ is a pseudodifferential operator of order 1 and as such has a kernel, $K(x,y)$, defined as a distribution on $\partial\Omega\times \partial\Omega$. The kernel gives a representation of $\Lambda$ by the formula $$\Lambda f(x)=\int_{\partial\Omega} K(x,y)f(y)dy$$ where $x$ and $y$ are arc length coordinates on $\partial\Omega$.
