The two-phase Alt-Phillips problem for quasilinear operators

Abstract: We establish interior regularity, optimal growth, and nondegeneracy estimates for sign-changing minimizers of the $p-$singular or $p-$degenerate quasilinear Alt--Phillips functional throughout the full range of $1&lt;p&lt;+\infty$ and of the nonlinearity power $0<γ<p$. In addition, we obtain local finite perimeter and density estimates, from which we deduce the local $(N-1)$-rectifiability of the reduced and two-phase free boundaries and the local finiteness of their $(N-1)$-dimensional Hausdorff measure.