We study the local behavior of the nodal sets of the solutions to elliptic quasilinear equations with nonlinear conductivity part, $\hbox{div}(A_s (x, u)u) = \hbox{div} { \mathbf f (x)}$, where $A_s (x, u)$ has " broken " derivatives of order $s \geq 0$, such as $A_s (x, u) = a(x) + b(x)(u +)^s$ , with $(u +)^0$ being understood as the characteristic function on $ \{u > 0\}$. The vector $\mathbf f (x)$ is assumed to be $C^\alpha$ in case $s = 0$, and $C^{ 1,\alpha}$ (or higher) in case $s > 0$. Using geometric methods, we prove almost complete results (in analogy with standard PDEs) concerning the behavior of the nodal sets. More exactly, we show that the nodal sets, where solutions have (linear) nondegeneracy, are locally smooth graphs. Degenerate points are shown to have structures that follow the lines of arguments as that of the nodal sets for harmonic functions, and general PDEs.

In this paper we consider a quasilinear elliptic PDE, $\hbox{div} (A(x,u) \nabla u) =0$, 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.