C ∞ regularity in semilinear free boundary problems.

We study the higher regularity of solutions and free boundaries in the Alt-Phillips problem $\Delta u=u^{\gamma -1},$ with $\gamma \in (0,1).$ Our main results imply that, once free boundaries are $C^{1,\alpha },$ then they are $C^{\infty }.$ In addition $u/d^{\frac{2}{2-\gamma }}$ and $u^{\frac{2-\gamma }{2}}$ are $C^{\infty }$ too. In order to achieve this, we need to establish fine regularity estimates for solutions of linear equations with boundary-singular Hardy potentials $-\Delta v=\kappa v/d^2$ in $\Omega ,$ where d is the distance to the boundary and $\kappa \le \frac{1}{4}.$ Interestingly, we need to include even the critical constant $\kappa =\frac{1}{4},$ which corresponds to $\gamma =\frac{2}{3}.$