Analysis and Mathematical Physics

We discuss a one-phase degenerate free boundary problem which arises from the minimization of the so-called Alt-Phillips functional $$\int_\Omega (|\nabla u|^2 + u^\gamma \chi_{\{u>0\}}) dx$$ for $\gamma \in (-2,0).$ We establish partial regularity results for the free boundary and discuss the rigidity of global minimizers when $\gamma$ approaches the endpoints of the interval of admissible values.