Applications of Elliptic Carleman Inequalities to Cauchy and by Mourad Choulli PDF

By Mourad Choulli

ISBN-10: 331933641X

ISBN-13: 9783319336411

ISBN-10: 3319336428

ISBN-13: 9783319336428

This e-book provides a unified method of learning the steadiness of either elliptic Cauchy difficulties and chosen inverse difficulties. in accordance with effortless Carleman inequalities, it establishes three-ball inequalities, that are the foremost to deriving logarithmic balance estimates for elliptic Cauchy difficulties and also are priceless in proving balance estimates for definite elliptic inverse difficulties.

The booklet provides 3 inverse difficulties, the 1st of which is composed in choosing the outside impedance of a disadvantage from the some distance box development. the second one challenge investigates the detection of corrosion via electrical size, whereas the 3rd matters the choice of an attenuation coefficient from inner info, that is influenced via an issue encountered in biomedical imaging.

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Extra info for Applications of Elliptic Carleman Inequalities to Cauchy and Inverse Problems

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51) entails u L ∞ (Γ ) ≤ C4 β M + e| ln μ|/ u H 1 (ω) . This proves (i). The proof of (ii) is similar to that of (i). 26, (i) and (ii) can be substituted by (i ) for any u ∈ H0 = H 1 (Ω) ∩ C 0,α (Ω) and 0 < < 1, u L ∞ (Γ ) ≤C β u H0 + Lu (ii ) for any u ∈ C 1,α (Ω) and 0 < < 1, L 2 (Ω) + ec/ u H 1 (ω) . 28 2 Uniqueness of Continuation and Cauchy Problems u W 1,∞ (Γ ) ≤C β u C 1,α (Ω) + Lu L 2 (Ω) + ec/ u H 1 (ω) . 28 Let ω and ω be two non empty open subsets of Ω. There exist C > 0 and γ > 0, that can depend on ω and ω, so that, for any u ∈ H 1 (Ω) satisfying Lu = 0, u H 1 (ω) ≤C γ u H 1 (Ω) + −1 u H 1 (ω) , > 0.

1/2 L 2 (B(x, r2 )∩Ω) u 1/2 H 2 (Ω) . 5 Cr 2 u ≤ u L 2 (B(x, r2 )∩Ω) 1−γ H 2 (Ω) u L 2 (B(x,r )∩Γ ) + ∇u γ L 2 (B(x,r )∩Γ )n . 24). As the trace mapping w ∈ H 2 → (w, ∇w) ∈ L 2 (Γ )n+1 is bounded and u L 2 (B(x,r )∩Γ ) + ∇u L 2 (B(x,r )∩Γ )n ≤ u L 2 (Γ ) + ∇u L 2 (Γ )n , we get u L 2 (B(x,r )∩Γ ) + ∇u L 2 (B(x,r )∩Γ )n ≤K u H 2 (Ω) . Here K is a constant independent on r . 24) yields Cr 2 u ≤ u L 2 (B(x, r2 )∩Ω) 1−γ/2 H 2 (Ω) u L 2 (B(x,r )∩Γ ) + ∇u L 2 (B(x,r )∩Γ )n γ/2 . 25). In the rest of this section we quantify the local behavior at the boundary of a solution of an obstacle problem.

In this proof, for sake of simplicity we use the following notations : B(x, r1 ) = B(x0 , ρ + r1 ) B(x, r1 , r2 ) = B(x, r2 )\B(x, r1 ), B = B(x, d), where d = diam(Γ ). Fix r > 0 so that B(x, r ) ∩ Γ ⊂ Γ0 . Define ψ(x) = ln (ρ + d)2 /|x − x0 |2 . Then |∇ψ(x)| = 2 2 ≥ = m > 0, x ∈ B ∩ Ω. |x − x0 | ρ+d Let χ ∈ C0∞ (B(x, r )), χ = 1 on B(x, 3r4 ) and |∂ α χ| ≤ K r −|α| , |α| ≤ 2, where K is a constant independent on r . Let u ∈ H 2 (Ω) satisfying Lu = 0 in Ω. 14 to v = χu in the domain B ∩ Ω in order to get, for λ ≥ λ0 fixed and τ ≥ τ0 , C B(x, r2 )∩Ω e2τ ϕ u 2 d x ≤ B(x,r )∩Ω + 1 r2 e2τ ϕ (Qu)2 d x B(x,r )∩Γ e2τ ϕ (u 2 + |∇u|2 )dσ.

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Applications of Elliptic Carleman Inequalities to Cauchy and Inverse Problems by Mourad Choulli

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