I have a non-linear form of the Poisson equation (with a diffusion coefficient that is a function of the derivatives of the dependent variable) that I'm trying to solve numerically (using FEM which requires the pde to be posed in its weak form).
$$ \nabla \cdot (\eta \nabla v) = G(x) $$
where $\eta = \sqrt{1/2 (\nabla v : \nabla v)}$ and $v = v(y,z)$. Multiplying by a test function $\theta$ and integrating wrt the domain, $\Omega$, gives the weak form
$$ \oint_\Gamma \theta \cdot (\eta\nabla v \cdot \mathbf{n})d\Gamma - \int_\Omega \left( \eta\nabla v \cdot \nabla \theta + G \right) d\Omega = 0$$
However, none of my boundary conditions correspond to the Dirichelet type as I have $\partial_n v = v$ on three of the boundaries in a rectangular domain and $\partial_{n} v = 0$ on the other ($n$ is the normal vector to surface). Is this problem actually solvable using FEM, or at all? I have the solution to a simpler problem which might be a reasonable guess at the solution here.
The former boundary condition corresponds to a Robin-type boundary condition and the latter corresponds to a Neumann-type boundary condition. Just split the boundary integral into the two sub-boundaries and apply the boundary conditions. The boundary term goes to zero on the Neumann part. On the Robin part you get an additional contribution to the nonlinear form. You end up with a nonlinear weak formulation which you can easily solve using a fixed-point or Newton's scheme.