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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!

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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

1 Answer 1

I don't have time to check if this includes what you are looking for, but check out:

Numerical Solution of Partial Differential Equations: Finite Difference Methods (Oxford Applied Mathematics & Computing Science Series) by G. D. Smith (Author)

HTH ~A

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