Download e-book for kindle: A geometric approach to free boundary problems by Luis Caffarelli, Sandro Salsa

By Luis Caffarelli, Sandro Salsa

ISBN-10: 0821837842

ISBN-13: 9780821837849

Loose or relocating boundary difficulties seem in lots of parts of research, geometry, and utilized arithmetic. a regular instance is the evolving interphase among an exceptional and liquid section: if we all know the preliminary configuration good sufficient, we must always be capable to reconstruct its evolution, particularly, the evolution of the interphase. during this publication, the authors current a chain of principles, tools, and strategies for treating the main uncomplicated problems with any such challenge. specifically, they describe the very basic 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 tension, and so on. The instruments and ideas offered right here will function a foundation for the examine of extra complicated phenomena and difficulties. This ebook comes in handy for supplementary analyzing or may be a superb autonomous examine textual content. it's appropriate for graduate scholars and researchers attracted to partial differential equations.

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3. The enlarged cone. ¯ e) put any σ ∈ Γ(θ, E(σ) = π − α(σ, ν) . 2 Moreover, for a small positive μ put ρ(σ) = |σ|μ sin(E(σ)) , Sμ = Bρ(σ) (σ) . ¯ σ∈Γ(θ,e) Then, there exist θ¯ and λ = λ(μ, θ0 ) < 1 such that ¯ e¯) ⊂ Sμ Γ(θ, e) ⊂ Γ(θ, and π π ¯ −θ ≤λ −θ . 2 2 Proof. Put δ = π 2 − θ and let σ1 , σ2 (unit vectors) be the two generatrices of Γ(θ, e) belonging to span{ν, e}. Suppose that σ1 is the nearest to ν of the two. 5) α(σ1 , ν) ≤ − 2θ , α(σ2 , ν) ≤ . 2 2 These two directions give the maximum and the minimum gain in the opening of the cone Γ(θ, e), respectively.

Indeed, if x ∈ F (vϕ ), dist(x, F (u)) = ϕ(x) so that dist(x, π) ≤ ϕ(x). Both surfaces are smooth with unit normal vector at x1 parallel to ν¯ = ν + ∇ϕ(x1 ) . 4. 8 Therefore, if a(τ1 , τ2 ) denotes the angle between the vectors τ1 , τ2 , a(¯ ν , en ) ≤ a(ν, en ) + a(ν, ν¯ ) ≤ artan λ + arsin |∇ϕ(x1 )| ≤ artan λ + c0 |∇ϕ(x1 )| Now, since |∇ϕ| < 1 tan(artan λ + c0 |∇ϕ(x1 )|) ≤ If |∇ϕ| ≤ 1 c1 (1+λ) , λ + c1 |∇ϕ(x1 )| 1 − λc1 |∇ϕ(x1 )| then tan a(¯ ν , en ) ≤ λ + c1 |∇ϕ(x1 )| that is F (vϕ ) is Lipschitz with Lipschitz constant λ ≤ λ + c1 sup |∇ϕ| .

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. More precisely if at x0 ∈ F (u) there is a touching ball B ⊂ Ω+ (u), no matter how small, then u has a linear behavior u(x) = α x − x0 , ν + − β x − x0 , ν − + o(|x − x0 |) with, for instance α = G(β) . Strict ellipticity corresponds to the strict monotonicity of G, with G(0) > 0. 7 part 3). In chapter 6 we shall construct solutions of our free boundary problem, precisely satisfying the properties a), b).

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