Suppose that $E=\nabla u + \nabla \times \Psi$ is an electric field, satisfying $$E \times n=0 \textrm{ on }\partial \Omega$$ and $$ \textrm{div} ( \varepsilon E +J )=0 \textrm{ in } \Omega, $$ where $\Omega$ is a smooth domain in $\mathbb R^3$ and $J$ is a smooth vector valued function (a current).
Suppose that the (smooth) vector potential $\Psi$ is known. Then it becomes an elliptic PDE in $\nabla u$, namely, setting $f=\textrm{div} ( \varepsilon \nabla \times \Psi + J)$, and $g= - \nabla\times \Psi \times n $ ,
$$-\textrm{div} ( \varepsilon \nabla u)= f \textrm{ in } \Omega, \quad \nabla u \times n = g \textrm{ on } \partial\Omega.$$
The boundary condition puzzles me a bit, as it isn't a usual Neumann or Dirichlet one. It feels like Dirichlet up to a constant, namely I am tempted to think that in fact $$ \nabla u \times n = g \textrm{ on } \partial\Omega \Leftrightarrow u = G \textrm{ on } \partial\Omega, $$ up to possibly imposing something on an integral of $u$. Is it true, and if yes, how is $G$ defined in terms of $g$?