# Is the weak form of a pde solvable without “essential” (Dirichlet) boundary conditions?

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.