C^∞ regularity in semilinear free boundary problems

We study the higher regularity of solutions and free boundaries in the Alt-Phillips problem Δ u=uγ-1, with γ∈(0,1). Our main results imply that, once free boundaries are C1,α, then they are C^∞. In addition u/d2/2-γ and u2-γ/2 are C^∞ too. In order to achieve this, we need to establish fine regularity estimates for solutions of linear equations with boundary-singular Hardy potentials -Δ v = κ v/d² in Ω, where d is the distance to the boundary and κ≤1/4. Interestingly, we need to include even the critical constant κ=1/4, which corresponds to γ=2/3.