Read e-book online Applications of Elliptic Carleman Inequalities to Cauchy and PDF

By Mourad Choulli

ISBN-10: 331933641X

ISBN-13: 9783319336411

ISBN-10: 3319336428

ISBN-13: 9783319336428

This publication provides a unified method of learning the steadiness of either elliptic Cauchy difficulties and chosen inverse difficulties. in line with uncomplicated Carleman inequalities, it establishes three-ball inequalities, that are the major to deriving logarithmic balance estimates for elliptic Cauchy difficulties and also are necessary in proving balance estimates for definite elliptic inverse difficulties.

The ebook provides 3 inverse difficulties, the 1st of which is composed in deciding on the skin impedance of a drawback from the a ways box trend. the second one challenge investigates the detection of corrosion via electrical size, whereas the 3rd issues the decision of an attenuation coefficient from inner facts, that's encouraged by means of an issue encountered in biomedical imaging.

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

Example text

As Ω is Lipschitz, it possesses the UICP. That is there exist R > 0 and θ ∈]0, π/2[ so that, to any x ∈ Γ corresponds ξ = ξ(x) ∈ Sn−1 with the property that C(x) = {x ∈ Rn ; |x − x| < R, (x − x) · ξ > |x − x| cos θ} ⊂ Ω. 26 Let 0 < α ≤ 1. There exist ω can depend only on n, Ω, κ, K and α, so that Ω, C > 0, c > 0 and β > 0, that (i) for any u ∈ H0 = H 1 (Ω) ∩ C 0,α (Ω) satisfying Lu = 0 in Ω and 0 < < 1, u L ∞ (Γ ) ≤C β u H0 + ec/ u (ii) for any u ∈ C 1,α (Ω) satisfying Lu = 0 in Ω H 1 (ω) . 4 Stability Estimates for Cauchy Problems 25 and 0 < < 1, u W 1,∞ (Γ ) β ≤C u C 1,α (Ω) + ec/ u H 1 (ω) .

11) Also 1 div(B) − b − 2 2 ϕ = − 2 τ ϕ|∇ψ|2 − ϕΔψ + 2 2 τ ϕ|∇ψ|2 − = 2 τ ϕ|∇ψ|2 − ϕτ Δψ − ≥ 2 τ ϕm − ϕτ M − 2 2 ϕ, 2 2 ϕ ϕ ≥ 4M/m 2 . 11) implies Fξ · ξ ≥ 1 4 2 τ ϕm 2 , ≥ 6M/m 2 , τ ≥ 4/m 2 , ξ ∈ Rn , |ξ| = 1. 12) For g(w), note that |bw∇w · ν| = |∇ψ||b||w| ( |∇ψ|)−1 |b||∇w · ν| ≤ |∇ψ||b|w 2 + ( |∇ψ|)−1 |b||∇w|2 . We deduce from this inequality |g(w)| ≤ 2(M 3 3 τ 3 ϕ3 w 2 + M τ ϕ|∇w|2 ). 13) yield Ω m 4 4 τ 3 ϕ3 w 2 + m 2 2 τ ϕ|∇w|2 d x ≤4 for Ω (Lw)2 d x + 8 Γ M 3 3 τ 3 ϕ3 w 2 + M τ ϕ|∇w|2 dσ, ≥ 6M 4 /m 4 and τ ≥ 88M 6 /m 4 .

1) and Ω Δw B · ∇wd x = − =− + Ω Ω Γ ∇w · ∇(B · ∇w)d x + B ∇w · ∇wd x − Ω Γ B · ∇w∇w · νdσ ∇ 2 w B · ∇wd x (B · ∇w)(∇w · ν)dσ. 2) Here B = (∂i B j ) is the Jacobian matrix of B and ∇ 2 w = (∂i2j w) is the Hessian matrix of w. But Ω Bi ∂i2j w∂ j wd x = − Ω ∂ j w Bi ∂i2j wd x − Ω ∂i Bi (∂ j w)2 d x + Γ Bi (∂ j w)2 νi dσ. Therefore Ω ∇ 2 w B · ∇wd x = − 1 2 Ω div(B)|∇w|2 d x + 1 2 Γ |∇w|2 B · νdσ. 3) Ω 1 −B + div(B)I ∇w · ∇wd x 2 Ω 1 B · ∇w∇w · νdσ − |∇w|2 B · νdσ. 4) Making integrations by parts one more time, we obtain Ω Δwbwd x = − Ω b|∇w|2 d x − Ω w∇b · ∇wd x + bw∇w · νdσ.

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


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