The Alt-Phillips functional for negative powers

We develop the free boundary regularity for nonnegative minimizers of the Alt-Phillips functional for negative power potentials $\int_Ω\left(\frac 1 2 |\nabla u|^2 + u^γ χ_{\{u>0\}}\right) \, dx, \quad \quad γ\in (-2,0),$ and establish a $Γ$-convergence result of the rescaled energies to the perimeter functional as $γ\to -2$.

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