Classification of blow-ups for the Alt--Phillips problem in three dimensions

We prove a flatness theorem for classical stable homogeneous solutions of the $\gamma$-Alt–Phillips free boundary problem in three dimensions in the range $0<\gamma \le 2\mathrm{/}3$, where $\gamma$ is the exponent in the energy density $\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^2+u^{\gamma }$. In particular, this implies full regularity of the free boundary for minimizers of the corresponding Alt–Phillips energy in dimension $3$.