A new discrete monotonicity formula with application to a two-phase free boundary problem in dimension two

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We continue the analysis of the two-phase free boundary problems initiated in DK, where we studied the linear growth of minimizers in a Bernoulli type free boundary problem at the non-flat points and the related regularity of free boundary. There, we also defined the functional ϕₚ(r,u,x₀)=1r⁴∫Bᵣ₍ₓ₀₎|∇ u+(x)|ᵖ|x-x₀|ⁿ⁻²dx∫Bᵣ₍ₓ₀₎|∇ u-(x)|ᵖ|x-x₀|ⁿ⁻²dx where x₀ is a free boundary point, i.e. x₀∈∂{u>0} and u is a minimizer of the functional J(u):=∫Ω|∇ u|ᵖ +λ+ᵖχ{ᵤ>₀} +λ-ᵖχ{ᵤ≤ ₀}, for some bounded smooth domain Ω⊂ Rⁿ and positive constants λ± with Λ:=λ+ᵖ-λᵖ->0. Here we show the discrete monotonicity of ϕₚ(r,u,x₀) in two spatial dimensions at non-flat points, when p is sufficiently close to 2, and then establish the linear growth. A new feature of our approach is the anisotropic scaling argument discussed in Section 4.