I'm interested in studying linear second order elliptic PDEs with boundary conditions that are functionals of the solution and possibly its derivative. For example,
\begin{align} \nabla^2 u(\vec{x}) &= f(\vec{x}) \\ \text{BC:} \ \ \ \ \ \ \ u(\vec{x}) &= g \left[ \vec{x},u(\vec{x}),u'(\vec{x}) \right] \end{align}
where $\vec{x} \in \Omega \subset \mathbb{R}^n$ with $n = 1,2 $ or $3$.
I'm looking for general references including existence and uniqueness theorems, analytical approaches and numerical methods. Thank you!