On the Two-phase Membrane Problem with Coefficents below the Lipschitz Threshold

We study the regularity of the two-phase membrane problem, with coefficients below the Lipschitz threshold. For the Lipschitz coefficient case one can apply a monotonicity formula to prove the C-regularity of the solution and that the free boundary is, near the so-called branching points, the union of two C-graphs. In our case, the same monotonicity formula does not apply in the same way. In the absence of a monotonicity formula, we use a specific scaling argument combined with the classification of certain global solutions to obtain C-estimates. Then we exploit some stability properties with respect to the coefficients to prove that the free boundary is the union of two Reifenberg vanishing sets near so-called branching points.