Two-phase semilinear free boundary problem with a degenerate phase

We study minimizers of the energy functional integral(D)[|del u|(2) + lambda(u+)(p)]dx for p is an element of ( 0, 1) without any sign restriction on the function u. The distinguished feature of the problem is the lack of nondegeneracy in the negative phase. The main result states that in dimension two the free boundaries Gamma(+) = partial derivative{u > 0} boolean AND D and Gamma(-) = partial derivative{u < 0} boolean AND D are C-1,C-alpha-regular, provided 1 - epsilon(0) < p < 1. The proof is obtained by a careful iteration of the Harnack inequality to obtain a nontrivial growth estimate in the negative phase, compensating for the apriori unknown nondegeneracy.