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

Abstract

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