A parabolic two-phase obstacle-like equation

For the parabolic obstacle-problem-like equationΔu−∂tu=λ+χ{u>0}−λ−χ{u<0},Δu−∂tu=λ+χ{u>0}−λ−χ{u<0}, where λ+λ+ and λ−λ− are positive Lipschitz functions, we prove in arbitrary finite dimension that the free boundary ∂{u>0}∪∂{u<0}∂{u>0}∪∂{u<0} is in a neighborhood of each “branch point” the union of two Lipschitz graphs that are continuously differentiable with respect to the space variables. The result extends the elliptic paper [Henrik Shahgholian, Nina Uraltseva, Georg S. Weiss, The two-phase membrane problem—regularity in higher dimensions, Int. Math. Res. Not. (8) (2007)] to the parabolic case. There are substantial difficulties in the parabolic case due to the fact that the time derivative of the solution is in general not a continuous function. Our result is optimal in the sense that the graphs are in general not better than Lipschitz, as shown by a counter-example.