Abstract
We prove that an a priori BMO gradient estimate for the two phase singular perturbation problem implies Lipschitz regularity for the limits. This problem arises in the mathematical theory of combustion where the reaction-diffusion is modelled by the $p$-Laplacian. A key tool in our approach is the weak energy identity. Our method proves a natural and intrinsic characterization of the free boundary points and can be applied to more general classes of solutions.
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