Nondegeneracy for stable solutions to the one-phase free boundary problem

We prove the nondegeneracy condition for stable solutions to the one-phase free boundary problem. The proof is by a De Giorgi iteration, where we need the Sobolev inequality of Michael and Simon and, consequently, an integral estimate for the mean curvature of the free boundary. We then apply the nondegeneracy estimate to obtain local curvature bounds for stable free boundaries in dimension $n$, provided the Bernstein type theorem for stable, entire solutions in the same dimension is valid. In particular, we obtain this curvature estimate in $n=2$ dimensions.

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