We prove that an a priori bounded mean oscillation (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 modeled by the $p$-Laplacian. A key tool in our approach is the weak energy identity. Our method provides a natural and intrinsic characterization of the free boundary points and can be applied to more general classes of solutions.

中文翻译：

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两相奇异摄动问题的规律

我们证明了两相奇异扰动问题的先验有界平均振荡 (BMO) 梯度估计意味着限制的 Lipschitz 规律性。这个问题出现在燃烧的数学理论中，其中反应扩散由$磷$- 拉普拉斯。我们方法中的一个关键工具是弱能量恒等式。我们的方法提供了自由边界点的自然和内在特征，可以应用于更一般的解决方案类别。