Geometry of Four Dimensions (volume I) by A. R. Forsyth

By A. R. Forsyth

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Let us rephrase the previous argument in more geometric terms. First of all, the Lipschitz continuity of w can be expressed in the following way. Let Γ(θ, en ) be the cone with axis en and opening θ given by λ = cotan θ For any vector τ , denote by Sτ the surface obtained translating S by τ . Then S is Lipschitz with constant λ if for any τ ∈ Γ(θ, en ), “Sτ stays above S”. In other words, if |τ | = 1, τ, en = 0 and γ ≤ 1/λ, the family of surfaces St(en +γτ ) , t > 0, stays above S. If we choose γ = 1/λ it may happen that St(en +γτ ) becomes tangent to S at some point.

To prove that in a neighborhood of a “flat” point the free boundary is a Lipschitz graph. 2. Lipschitz free boundaries are C 1,γ . We start with step 2, in the next chapter. 1. The main theorem. p. in the sequel): to find a function u such that, in the cylinder C1 = B1 (0) × (−1, 1), B1 (0) ⊂ Rn−1 , Δu = 0 in Ω+ (u) = {u > 0} and Ω− (u) = {u ≤ 0}0 − + u+ ν = G(uν ) on F (u) = ∂Ω (u) . 1) We assume that F (u) is given by the graph of a Lipschitz function xn = f (x ), x ∈ B1 (0), with Lipschitz constant L and f (0) = 0.

Let’s see what are the minimal assumptions to achieve this kind of results. A parallel with viscosity solutions of elliptic equations Lv = Tr(A(x)D2 v) = 0 is in order. A linear growth and the non degeneracy for u+ correspond to having a Harnack inequality (controlled growth) for v and this is true, for instance, if L is strictly elliptic (A(x) ≥ λI) and A is bounded measurable. 43 44 3. THE REGULARITY OF THE FREE BOUNDARY For our free boundary problem, the parallel requirement is that 0 < c < u+ ν ≤ C in the viscosity sense.

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