Smoothness and stability in the Alt–Phillips problem

Abstract We study the one-phase Alt–Phillips free boundary problem, focusing on the case of negative exponents $$\gamma \in (-2,0)$$ γ ∈ ( - 2 , 0 ) . The goal of this paper is twofold. On the one hand, we prove smoothness of $$C^{1,\alpha }$$ C 1 , α -regular free boundaries by reducing the problem to a class of degenerate quasilinear PDEs, for which we establish Schauder estimates. Such a method provides a unified proof of the smoothness for general exponents. On the other hand, by exploiting the higher regularity of solutions, we derive a new stability condition for the Alt–Phillips problem in the negative exponent regime, ruling out the existence of nontrivial axially symmetric stable cones in low dimensions. Finally, we provide a variational criterion for the stability of cones in the Alt–Phillips problem, which recovers the one for minimal surfaces in the singular limit as $$\gamma \rightarrow -2$$ γ → - 2 .