Variational formulation of elliptic mixed boundary value problem

Question

I have some troubles with variational formulations of some pies. For instance, let's consider $\chi\in H^1(\Omega)$ as the solution to the elliptic mixed boundary value problem $$ \begin{cases} \nabla\cdot(\mu \nabla\chi)=\nabla\cdot(\mu\chi)\;\,\text{in}\,\Omega\\ \mu\nabla\chi\cdot n=\mu v\cdot n\;\,\text{on}\,\Gamma_2\\ \chi=0\;\,\text{on}\,\Gamma_1 \end{cases} $$ where $v\in (L^2(\Omega))^3$, $\mu$ is a symmetric matrix, bounded and uniformly positive definite in $\Omega$. The domain is Lipschitz such that $\partial\Omega=\Gamma_1\cup\Gamma2$ and $n$ is the outward unitary normal on $\Gamma_2$.

In particular I don't know how to deal with the second equation

Answer 1

I am not quite sure how to understand the term $\mu\chi$ (what is its format?), but anyway.

I would multiply by $\chi$ the first equation and integrate, obtaining $$ \int_\Omega \nabla\cdot(\mu\nabla \chi)\chi\ dx -\int_\Omega \nabla\cdot(\mu\chi)\chi\ dx=0 $$ and apply Gauß-Green, obtaining $$ \int_{\Gamma_1} (\mu\nabla \chi\cdot n)\chi\ dx +\int_{\Gamma_2} (\mu\nabla \chi\cdot n)\chi\ dx-\int_{\Omega} \mu|\nabla \chi|^2\ dx -\int_\Omega \nabla\cdot(\mu\chi)\chi\ dx=0 $$ and in view of your boundary conditions $$ \int_{\Gamma_2} (\mu v\cdot n)\chi\ dx-\int_{\Omega} \mu|\nabla \chi|^2\ dx -\int_\Omega \nabla\cdot(\mu\chi)\chi\ dx=0 $$ Up to the above caveat, this looks like a perfectly well-defined equation for $H^1(\Omega)$-functions.