  In this paper we consider a quasilinear elliptic PDE, $\text{div} (A(x,u)
\nabla u) =0$, where the underlying physical problem gives rise to a jump for
the conductivity $A(x,u)$, across a level surface for $u$. Our analysis
concerns Lipschitz regularity for the solution $u$, and the regularity of the
level surfaces, where $A(x,u)$ has a jump and the solution $u$ does not
degenerate.
  In proving Lipschitz regularity of solutions, we introduce a new and
unexpected type of ACF-monotonicity formula with two different operators, that
might be of independent interest, and surely can be applied in other related
situations. The proof of the monotonicity formula is done through careful
computations, and (as a byproduct) a slight generalization to a specific type
of variable matrix-valued conductivity is presented.
