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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!

Luis Costa
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  • What is your background? – timur Sep 05 '12 at 02:21
  • PhD candidate in theoretical and computational mechanics. – Luis Costa Sep 05 '12 at 02:23
  • Do you know how to treat (linear) Robin boundary conditions? – timur Sep 05 '12 at 02:50
  • I do. Though the arguments of $g$ are defined on the entire domain. – Luis Costa Sep 05 '12 at 02:55
  • For example, $g$ can be equal to some constant, $c = \int_\Omega u(\vec{x}) d\Omega$ – Luis Costa Sep 05 '12 at 03:04
  • Then it is not really a boundary condition? – timur Sep 05 '12 at 03:10
  • The values of $u$ imposed on the boundary are a functional of the values of $u$ on, let's say, the interior of the domain. Does this not make it a boundary condition. My original problem includes a time dependence but I neglected it because I wanted to understand the time independent case. In the time dependent case consider a body loaded with BCs that instantaneously depend on the internal field and how it evolves. Maybe that clears it up? – Luis Costa Sep 05 '12 at 03:15
  • You can have side conditions like this, but typically you should still have a boundary condition. For instance, for Neumann problems you have a Neumann boundary condition, and you need an additional condition (such as the mean zero condition) to make the solution unique. – timur Sep 05 '12 at 03:15
  • Please discard my last comment. I understand it now. So is $g$ linear or nonlinear? – timur Sep 05 '12 at 03:16
  • Let's start with $g$ being linear. I want to understand how big of a rabbit hole I'm getting myself into. – Luis Costa Sep 05 '12 at 03:18
  • This is not really a PDE, because it is not local. I don't know how to call this but it makes sense. Anyways, you can solve the Dirichlet problem with fixed right hand side $f$, and varying boundary condition, say $h$. Denote this solution map sending $h$ to the solution $u$ by $u=\Phi(h)$. Then your equation can be written as $u=\Phi(g(u))$, and fixed point techniques can be tried at it. – timur Sep 05 '12 at 03:31
  • You asked about references but I don't really know anything offhand. It looks like you can somehow build it yourself, using the standard stuff as building blocks. – timur Sep 05 '12 at 03:33
  • 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 – Amzoti Sep 05 '12 at 04:07

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