By Luis Caffarelli, Sandro Salsa

Unfastened or relocating boundary difficulties look in lots of parts of research, geometry, and utilized arithmetic. a customary instance is the evolving interphase among a high-quality and liquid part: if we all know the preliminary configuration good sufficient, we should always be capable of reconstruct its evolution, specifically, the evolution of the interphase. during this booklet, the authors current a sequence of rules, tools, and methods for treating the main easy problems with any such challenge. particularly, they describe the very primary instruments of geometry and actual research that make this attainable: houses of harmonic and caloric measures in Lipschitz domain names, a relation among parallel surfaces and elliptic equations, monotonicity formulation and pressure, and so forth. The instruments and ideas offered the following will function a foundation for the examine of extra complicated phenomena and difficulties. This e-book comes in handy for supplementary examining or should be a very good self reliant examine textual content. it really is compatible for graduate scholars and researchers attracted to partial differential equations.

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**Example text**

It is Harnack’s inequality, applied to the ﬁrst derivatives of w, that supplies what is needed. In this context Harnack’s inequality says: if the distance between the surfaces S and St(en +γτ ) is of order t at one point, then it is of order t in a neighborhood of that point. 1. THE MAIN THEOREM. HEURISTIC CONSIDERATIONS AND STRATEGY 53 if γ = 1/λ, one of the two surfaces St(en ±γτ ) separates from S by a distance of order t, in at least half of the points. If we suppose that the “good” surface is St(en +γτ ) , then, by Harnack’s inequality, St(en +γτ ) stays ct-away from S in B1/2 .

In fact, the family vt (x) = sup u Bt (x) with t constant, can only detect a uniform enlargement of the monotonicity cone, and, as such, one cannot exploit the interior gain. For this purpose we ask the question: what are the conditions on a variable radius t(x) so that for any harmonic function u, vt will be always subharmonic. Here is the fundamental lemma. 7. 12) for C = C(n) large enough. Let u be continuous, deﬁned in a domain Ω so large that the function w(x) = sup u(x + ϕ(x)ν) |σ|=1 is well deﬁned in B1 .

9): B ⎧ Δwt = 0 in Ω+ (vϕt ) ∩ Ω ≡ Ωt ⎪ ⎪ ⎪ ⎨w ≡ 0 in Ω ¯ Ωt t ⎪ wt = 0 on ∂B9/10 ⎪ ⎪ ⎩ wt = u(x0 ) on ∂B1/8 (x0 ) Then for a small constant c, h and any ε > 0 small enough, Vt = vεϕt + cεwt (0 ≤ t ≤ 1) is a family of subsolution. Proof. 7. We have to check that Vt has the correct asymptotic behavior. Notice that F (Vt ) = F (vεϕt ). 17) with β¯ α ¯ ≥G 1 − ε|∇ϕt | 1 + ε|∇ϕt | 4. LIPSCHITZ FREE BOUNDARIES ARE C 1,γ 70 Since |∇ϕt | ≡ 0 outside B7/8 the right inequality is satisﬁed by vεϕt and hence by Vt since wt is positive.