A Two-phase Problem with a Lower-dimensional Free Boundary

For a bounded domain D ⊂ Rn, we study minimizers of the energy functional ∫ D |∇u| dx+ ∫ D∩(Rn−1×{0}) λχ{u>0} + λ χ{u<0} dHn−1, without any sign restriction on the function u. One of the main result states that the free boundaries Γ = ∂{u(·, 0) > 0} and Γ− = ∂{u(·, 0) < 0} never touch. Moreover, using Alexandrov-type reflection technique, we can show that in dimension n = 3 the free boundaries are C1 regular on a dense subset.