$C^{\mathrm{\infty }}$ regularity of the Alt-Phillips Functional for negative powers

In this paper, we study the regularity of the free boundary for minimizers of the Alt-Phillips functional with negative powers $${\mathcal{E}}_{\gamma }(u)={\int }_{\mathrm{\Omega }}\frac{1}{2}\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^2+\frac{1}{\gamma }{u}^{-\gamma }{\chi }_{\{u>0\}}dx,\gamma \in (0,2)\mathrm{.}$$ We proved that the free boundaries are $C^{\mathrm{\infty }}$ at regular points. A key technical tool is the linearized operator for the PDE satisfied by the partial derivatives of a solution to the Alt-Phillips Euler-Lagrange equation in the negative power case. For this operator we establish a comparison principle, which may have further applications to the Alt-Phillips problem with negative powers.