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

Abstract

In this paper, we study the regularity of the free boundary for minimizers of the Alt-Phillips functional with negative powers \[\mathcal{E}_γ(u)=\int_Ω\frac{1}{2}|\nabla u|^2+\frac{1}γu^{-γ}χ_{\{u>0\}}dx,\quadγ\in(0,2).\] We proved that the free boundaries are $C^{\infty }$ 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.

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