The Parabolic Two-Phase Membrane Problem: Regularity in Higher Dimensions

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For the parabolic obstacle-problem-like equation Δu - ∂ₜ u = λ+ χ{ᵤ>₀} - λ- χ{ᵤ<₀}, where λ+ and λ- are positive Lipschitz functions, we prove in arbitrary finite dimension that the free boundary ∂{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 imrn to the parabolic case. The result is optimal in the sense that the graphs are in general not better than Lipschitz, as shown by a counter-example.