$C^{\mathrm{\infty }}$ regularity in semilinear free boundary problems

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