I see, thank you. Let me try another way to formulate my question then:
Given the PDE
\nabla ^2 u^i = f^i
with BCs
[\vec{v}(\vec{u})].\vec{u} = 0 \textrm{ on } \partial \Omega
[plus another BC for u, unimportant here].
Here \vec{v} is a vector field which depends on \vec{u}: v^i(u) = A^i_j(u) \, n^j, where n is the facet normal of \partial \Omega which is independent of u.
Like I discussed here, I would like to implement the BC [\vec{v}(\vec{u})].\vec{u} = 0 \textrm{ on } \partial \Omega with Nitsche’s method, but this is tricky because it is a nonlinear BC with respect to u.
However, I would like to try this workaround: set \phi^i \equiv u^j A^i_j(u), then the BC reads \phi^i \hat{n}^i= 0 which is linear, so maybe Nitsche’s method will work now.(?)
Can I just solve the original PDE by simply reformulating it in terms of u and \phi as
\nabla ^2 u^i = f^i
u^i = \phi^i
with BCs
\phi^i \hat{n}^i= 0 \textrm{ on } \partial \Omega
[plus the other BC for u]
?