By Luis Caffarelli, Sandro Salsa

Loose or relocating boundary difficulties seem in lots of components of research, geometry, and utilized arithmetic. a customary instance is the evolving interphase among a fantastic and liquid section: if we all know the preliminary configuration good adequate, we should always have the capacity to reconstruct its evolution, particularly, the evolution of the interphase. during this ebook, the authors current a sequence of rules, equipment, and methods for treating the main simple problems with this kind of challenge. specifically, they describe the very primary instruments of geometry and actual research that make this attainable: homes of harmonic and caloric measures in Lipschitz domain names, a relation among parallel surfaces and elliptic equations, monotonicity formulation and stress, and so on. The instruments and ideas offered the following will function a foundation for the examine of extra advanced phenomena and difficulties. This ebook comes in handy for supplementary examining or could be an excellent autonomous research textual content. it truly is appropriate for graduate scholars and researchers attracted to partial differential equations.

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**Example text**

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.