By Luis Caffarelli, Sandro Salsa

Loose or relocating boundary difficulties look in lots of parts of research, geometry, and utilized arithmetic. a customary instance is the evolving interphase among an exceptional and liquid part: if we all know the preliminary configuration good adequate, we must always be ready to reconstruct its evolution, specifically, the evolution of the interphase. during this ebook, the authors current a sequence of principles, equipment, and strategies for treating the main uncomplicated problems with one of these challenge. specifically, they describe the very basic 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 tension, and so forth. The instruments and ideas provided right here will function a foundation for the learn of extra complicated phenomena and difficulties. This booklet turns out to be useful for supplementary examining or may be an outstanding autonomous examine textual content. it truly is compatible for graduate scholars and researchers attracted to partial differential equations.

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16) ν¯ = α ¯ = α|ν + ∇ϕ(x1 )| , β¯ = β|ν + ∇ϕ(x1 )| . 18). 18) says that vϕ is “almost” a subsolution, due to the fact that ∇ϕ is not identically zero. We shall later perturb vϕ to make it a subsolution. 7, such that vεϕt carries the monotonicity gain from B1/8 (x0 ) to the free boundary as t goes from 0 to 1. This means that we want ϕt = 1 along, say ∂B1 , ϕt ≈ 1 + ctb on ∂B1/8 (x0 ) and ¯tb in B1/2 . 11. Let 0 < r ≤ 18 . Then, there exist positive λ = λ(r), h = h(r) and a C 2 ¯1 Br/2 ( 3 en ) such that family of functions ϕt , 0 ≤ t ≤ 1, deﬁned in B 4 (i) 1 ≤ ϕt ≤ 1 + th (ii) ϕt Δϕt ≥ C|∇ϕt |2 (iii) ϕt ≡ 1 outside B7/8 (iv) ϕt|B1/2 ≥ 1 + λth (v) |∇ϕt | ≤ Cth.

Then, if u is harmonic in {u > 0}, w is subharmonic in {w > 0}. Proof. Let us normalize the situation assuming that w(0) > 0, ϕ(0) = 1 and w(0) = supB1 u is attained at x = en . We will show that lim inf r→0 1 − w(x) dx − w(0) ≥ 0 . r2 Br Choose the system of coordinates so that ∇ϕ(0) = αe1 + βen . We estimate w(x) from below for x near the origin by w(x) ≥ u(x + ϕ(x)σ) with an appropriate choice of σ = ∗ σ∗ |σ∗ | , given by n−1 ∗ σ = σ (x) = en + (βx1 − αxn )e1 + γ xi ei i=2 where γ is to be chosen later.

Then it is natural to require the existence at x0 of a touching ball from the right (not from the left) and a proper asymptotic behavior for v near x0 that could force a contradiction. Therefore it is useful to introduce another kind of “subsolution” with these characteristics. It turns out that a natural way to construct such type of functions is to start from a subsolution or a solution (in the viscosity sense) and build parallel surfaces. Here is the simplest example. 3. COMPARISON PRINCIPLES 39 Let us examine the properties of vt .