# 2nd Order Elliptic PDEs with functional BCs

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!

-
What is your background? – timur Sep 5 '12 at 2:21
PhD candidate in theoretical and computational mechanics. – Luis Costa Sep 5 '12 at 2:23
Do you know how to treat (linear) Robin boundary conditions? – timur Sep 5 '12 at 2:50
I do. Though the arguments of $g$ are defined on the entire domain. – Luis Costa Sep 5 '12 at 2:55
For example, $g$ can be equal to some constant, $c = \int_\Omega u(\vec{x}) d\Omega$ – Luis Costa Sep 5 '12 at 3:04