[PPT] - Some issues on Electrical Impedance Tomography with complex PowerPoint Presentation

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Some issues on Electrical Impedance Tomography with complex coefficient

Elisa Francini (Universit` a di Firenze)

in collaboration with Elena Beretta and Sergio Vessella

E. Francini (Universit`

a di Firenze) EIT 1 / 42

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Outline

1 Electrical Impedance Tomography with complex coefficient 2 Reconstruction from the DN map: a stability result. 3 Size estimate from a single boundary measurement.

E. Francini (Universit`

a di Firenze) EIT 2 / 42

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Electrical Impedance Tomography

Electrical Impedance Tomography:

Determine electrical conductivity and permittivity in the interior of a body by measurements of electric currents and voltages at the boundary of the body. Medical diagnostics: detection of pulmonary emboli, breast tumours, screening of organs in transplant surgery Nondestructive testing of materials: detection of corrosion or cracks Geophysical prospection: detection of underground mineral deposits

E. Francini (Universit`

a di Firenze) EIT 3 / 42

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Mathematical model

γ : Ω ⊂ Rn → C complex admittivity function γ = σ + iωǫ electrical conductivity frequency electrical permittivity σ ≥ λ−1 > 0 (dissipation of energy), |γ| ≤ λ The potential u ∈ H1(Ω, C) is a weak solution to equation div (γ∇u) = 0 in Ω. Boundary data: u|∂Ω and γ ∂u

∂ν |∂Ω.

E. Francini (Universit`

a di Firenze) EIT 4 / 42

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Mathematical model

γ : Ω ⊂ Rn → C complex admittivity function γ = σ + iωǫ electrical conductivity frequency electrical permittivity σ ≥ λ−1 > 0 (dissipation of energy), |γ| ≤ λ The potential u ∈ H1(Ω, C) is a weak solution to equation div (γ∇u) = 0 in Ω. Boundary data: u|∂Ω and γ ∂u

∂ν |∂Ω.

E. Francini (Universit`

a di Firenze) EIT 4 / 42

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Mathematical model

γ : Ω ⊂ Rn → C complex admittivity function γ = σ + iωǫ electrical conductivity frequency electrical permittivity σ ≥ λ−1 > 0 (dissipation of energy), |γ| ≤ λ The potential u ∈ H1(Ω, C) is a weak solution to equation div (γ∇u) = 0 in Ω. Boundary data: u|∂Ω and γ ∂u

∂ν |∂Ω.

E. Francini (Universit`

a di Firenze) EIT 4 / 42

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Part I Reconstruction from infinite boundary measurements

E. Francini (Universit`

a di Firenze) EIT 5 / 42

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DN map

Given f ∈ H1/2(∂Ω) there exists a unique weak solution u ∈ H1(Ω) of the problem div (γ∇u) = in Ω u = f

n

∂Ω. Consider the Dirichlet to Neumann map Λγ : H1/2(∂Ω) → H−1/2(∂Ω) given by Λγf = γ ∂u ∂ν |∂Ω , where ν is the exterior unit normal vector to ∂Ω.

EIT Inverse Problem:

Determine γ from the knowledge of the Dirichlet-to Neumann map Λγ

E. Francini (Universit`

a di Firenze) EIT 6 / 42

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DN map

Given f ∈ H1/2(∂Ω) there exists a unique weak solution u ∈ H1(Ω) of the problem div (γ∇u) = in Ω u = f

n

∂Ω. Consider the Dirichlet to Neumann map Λγ : H1/2(∂Ω) → H−1/2(∂Ω) given by Λγf = γ ∂u ∂ν |∂Ω , where ν is the exterior unit normal vector to ∂Ω.

EIT Inverse Problem:

Determine γ from the knowledge of the Dirichlet-to Neumann map Λγ

E. Francini (Universit`

a di Firenze) EIT 6 / 42

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DN map

Given f ∈ H1/2(∂Ω) there exists a unique weak solution u ∈ H1(Ω) of the problem div (γ∇u) = in Ω u = f

n

∂Ω. Consider the Dirichlet to Neumann map Λγ : H1/2(∂Ω) → H−1/2(∂Ω) given by Λγf = γ ∂u ∂ν |∂Ω , where ν is the exterior unit normal vector to ∂Ω.

EIT Inverse Problem:

Determine γ from the knowledge of the Dirichlet-to Neumann map Λγ

E. Francini (Universit`

a di Firenze) EIT 6 / 42

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Known results

Uniqueness n ≥ 3 If γ is sufficiently smooth the results obtained by Sylvester and Uhlmann (1987, Ann. of Math.) for the conductivity case can be extended to the complex case. (Λγ1 = Λγ2 implies γ1 = γ2) n = 2 F.(2000, Inverse Problems) for small frequencies ω, Bukhgeim for arbitrary frequencies and smooth coefficients (2008, Inv. Ill Posed Problems). n ≥ 3, γ ∈ L∞(Ω) completely open n = 2 Astala and P¨ aiv¨ arinta for real conductivities σ ∈ L∞(Ω) ( 2006,

Ann. of Math.)
E. Francini (Universit`

a di Firenze) EIT 7 / 42

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Known results

Stability n ≥ 3 Alessandrini (1988, Appl. Anal.) has proved for the conductivity case that if σjW 2,∞(Ω) ≤ E for j = 1, 2, then σ1 − σ2L∞(Ω) ≤ ω(Λσ1 − Λσ2) where ω(t) = C| log t|−η. n = 2 Barcelo, Faraco and Ruiz (2007, Jour. Math. Pures Appl.) have proved for the conductivity case an analogous logarithmic estimate under the a priori assumption σC α(Ω) ≤ E. Clop, Faraco, Ruiz (2009, IPI) for σj ∈ W α,p, α > 0, j = 1, 2, and 1 < p < ∞ then σ1 − σ2L2(Ω) ≤ ω(Λσ1 − Λσ2) where ω(t) = C| log t|−η.

E. Francini (Universit`

a di Firenze) EIT 8 / 42

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A priori assumptions

Mandache (2001, Inverse Problems) has proved that logarithmic stability is the best possible stability also using as a-priori assumption σC k(Ω) ≤ E, ∀k = 0, 1, 2....

E. Francini (Universit`

a di Firenze) EIT 9 / 42

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Strategy:

Look for a-priori assumptions on γ that are physically relevant give rise to a better type of stability

Assume:

γ is piecewise constant on known domains. In the real case: Alessandrini and Vessella (2005, Adv. in Appl. Math.)

E. Francini (Universit`

a di Firenze) EIT 10 / 42

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Strategy:

Look for a-priori assumptions on γ that are physically relevant give rise to a better type of stability

Assume:

γ is piecewise constant on known domains. In the real case: Alessandrini and Vessella (2005, Adv. in Appl. Math.)

E. Francini (Universit`

a di Firenze) EIT 10 / 42

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Strategy:

Look for a-priori assumptions on γ that are physically relevant give rise to a better type of stability

Assume:

γ is piecewise constant on known domains. In the real case: Alessandrini and Vessella (2005, Adv. in Appl. Math.)

E. Francini (Universit`

a di Firenze) EIT 10 / 42

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Main assumptions

(H1) Ω ⊂ Rn, bounded, |Ω| ≤ A, Lipschitz boundary (H2) γ(x) = N

j=1 γj✶Dj(x)

unknown complex numbers known open sets Di ∩ Dj = ∅, i = j, Lipschitz boundaries and ∪N

j=1Dj = Ω

(H3) ∂D1 ∩ ∂Ω contains an open flat portion Σ1. For every j ∈ {2, . . . , N} there exists a chain of subdomains connecting D1 to Dj so that consecutive domains share a flat portion of boundary

E. Francini (Universit`

a di Firenze) EIT 11 / 42

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A possible configuration Σ1

Dj D1

E. Francini (Universit`

a di Firenze) EIT 12 / 42

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Chain Σ1

Dj D1

E. Francini (Universit`

a di Firenze) EIT 13 / 42

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Main result

Theorem

Let Ω satisfy assumption (H1). Let γ(k), k = 1, 2 be two complex piecewise constant functions of the form γ(k)(x) =

N

j=1

γ(k)

j

✶Dj(x), where γ(k) satisfy for k = 1, 2 assumption (H2) and Dj, j = 1, . . . , N satisfy assumption (H3). Then there exists a positive constant C such that γ(1) − γ(2)L∞(Ω) ≤ CΛγ(1) − Λγ(2)L(H1/2(∂Ω),H−1/2(∂Ω)). The constant C depends on the Lipschitz constant of the boundaries, on the volume of Ω, on n, N and λ. [Beretta, F., Comm. PDE, 2011]

E. Francini (Universit`

a di Firenze) EIT 14 / 42

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Main idea of the proof (uniqueness)

Assume Λγ(1) = Λγ(2) div

γ(1)∇u1
= 0

div

γ(2)∇u2
= 0

in Ω ⇓ 0 =< (Λγ(1) − Λγ(2))u1, u2 >=

Ω
γ(1) − γ(2)

∇u1∇u2 dx We want to use this identity for ”singular” solutions

E. Francini (Universit`

a di Firenze) EIT 15 / 42

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Main idea of the proof (uniqueness)

Assume Λγ(1) = Λγ(2) div

γ(1)∇u1
= 0

div

γ(2)∇u2
= 0

in Ω ⇓ 0 =< (Λγ(1) − Λγ(2))u1, u2 >=

Ω
γ(1) − γ(2)

∇u1∇u2 dx We want to use this identity for ”singular” solutions

E. Francini (Universit`

a di Firenze) EIT 15 / 42

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Main idea of the proof (uniqueness)

Assume Λγ(1) = Λγ(2) div

γ(1)∇u1
= 0

div

γ(2)∇u2
= 0

in Ω ⇓ 0 =< (Λγ(1) − Λγ(2))u1, u2 >=

Ω
γ(1) − γ(2)

∇u1∇u2 dx We want to use this identity for ”singular” solutions

E. Francini (Universit`

a di Firenze) EIT 15 / 42

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Main idea of the proof (uniqueness)

Assume Λγ(1) = Λγ(2) div

γ(1)∇u1
= 0

div

γ(2)∇u2
= 0

in Ω ⇓ 0 =< (Λγ(1) − Λγ(2))u1, u2 >=

Ω
γ(1) − γ(2)

∇u1∇u2 dx We want to use this identity for ”singular” solutions

E. Francini (Universit`

a di Firenze) EIT 15 / 42

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Main idea of the proof (uniqueness)

y z DM Σ1 D1 D0 γ(1)−γ(2)L∞(DM) = γ(1)−γ(2)L∞(Ω) Extend γ(k) = 1 in D0, an open cylin- der connected to Σ1. Set Gk(x, y) the Green’s function re- lated to γ(k) and Ω ∪ D0 For y, z ∈ D0

D1
γ(1)

1

− γ(2)

1

∇G1(x, y)∇G2(x, z) dx

+

Ω\D1
γ(1)(x) − γ(2)(x)
∇G1(x, y)∇G2(x, z) dx = 0.

by letting y and z tend to Σ1 ⇒ γ(1)

1

= γ(2)

1 .

E. Francini (Universit`

a di Firenze) EIT 16 / 42

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Main idea of the proof (uniqueness)

y z DM Σ1 D1 D0 γ(1)−γ(2)L∞(DM) = γ(1)−γ(2)L∞(Ω) Extend γ(k) = 1 in D0, an open cylin- der connected to Σ1. Set Gk(x, y) the Green’s function re- lated to γ(k) and Ω ∪ D0 For y, z ∈ D0

D1
γ(1)

1

− γ(2)

1

∇G1(x, y)∇G2(x, z) dx

+

Ω\D1
γ(1)(x) − γ(2)(x)
∇G1(x, y)∇G2(x, z) dx = 0.

by letting y and z tend to Σ1 ⇒ γ(1)

1

= γ(2)

1 .

E. Francini (Universit`

a di Firenze) EIT 16 / 42

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Main idea of the proof (uniqueness)

y z DM Σ1 D1 D0 γ(1)−γ(2)L∞(DM) = γ(1)−γ(2)L∞(Ω) Extend γ(k) = 1 in D0, an open cylin- der connected to Σ1. Set Gk(x, y) the Green’s function re- lated to γ(k) and Ω ∪ D0 For y, z ∈ D0

D1
γ(1)

1

− γ(2)

1

∇G1(x, y)∇G2(x, z) dx

+

Ω\D1
γ(1)(x) − γ(2)(x)
∇G1(x, y)∇G2(x, z) dx = 0.

by letting y and z tend to Σ1 ⇒ γ(1)

1

= γ(2)

1 .

E. Francini (Universit`

a di Firenze) EIT 16 / 42

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Main idea of the proof (uniqueness)

Ω\D1
γ(1)(x) − γ(2)(x)
∇G1(x, y)∇G2(x, z) dx = 0,

∀y, z ∈ D0. Can we let y, z into D1? Yes because S1(y, z) :=

Ω\D1
γ(1)(x) − γ(2)(x)
∇G1(x, y)∇G2(x, z) dx

Σ1 DM D1 D0 solves in D0 ∪ D1 divy(γ(1)(y)∇yS1(y, z))) = 0 and divz(γ(2)(z)∇zS1(y, z))) = 0 By letting y and z tend to ∂D2 ⇒ γ(1)

2

= γ(2)

2 .

E. Francini (Universit`

a di Firenze) EIT 17 / 42

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Main idea of the proof (uniqueness)

Ω\D1
γ(1)(x) − γ(2)(x)
∇G1(x, y)∇G2(x, z) dx = 0,

∀y, z ∈ D0. Can we let y, z into D1? Yes because S1(y, z) :=

Ω\D1
γ(1)(x) − γ(2)(x)
∇G1(x, y)∇G2(x, z) dx

Σ1 DM D1 D0 solves in D0 ∪ D1 divy(γ(1)(y)∇yS1(y, z))) = 0 and divz(γ(2)(z)∇zS1(y, z))) = 0 By letting y and z tend to ∂D2 ⇒ γ(1)

2

= γ(2)

2 .

E. Francini (Universit`

a di Firenze) EIT 17 / 42

SLIDE 30

Main idea of the proof (uniqueness)

Ω\D1
γ(1)(x) − γ(2)(x)
∇G1(x, y)∇G2(x, z) dx = 0,

∀y, z ∈ D0. Can we let y, z into D1? Yes because S1(y, z) :=

Ω\D1
γ(1)(x) − γ(2)(x)
∇G1(x, y)∇G2(x, z) dx

Σ1 DM D1 D0 solves in D0 ∪ D1 divy(γ(1)(y)∇yS1(y, z))) = 0 and divz(γ(2)(z)∇zS1(y, z))) = 0 By letting y and z tend to ∂D2 ⇒ γ(1)

2

= γ(2)

2 .

E. Francini (Universit`

a di Firenze) EIT 17 / 42

SLIDE 31

Main idea of the proof (uniqueness)

Ω\D1
γ(1)(x) − γ(2)(x)
∇G1(x, y)∇G2(x, z) dx = 0,

∀y, z ∈ D0. Can we let y, z into D1? Yes because S1(y, z) :=

Ω\D1
γ(1)(x) − γ(2)(x)
∇G1(x, y)∇G2(x, z) dx

Σ1 DM D1 D0 solves in D0 ∪ D1 divy(γ(1)(y)∇yS1(y, z))) = 0 and divz(γ(2)(z)∇zS1(y, z))) = 0 By letting y and z tend to ∂D2 ⇒ γ(1)

2

= γ(2)

2 .

E. Francini (Universit`

a di Firenze) EIT 17 / 42

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Main technical ingredients

Construction of singular solutions and study of their asymptotic behavior near the discontinuity interfaces. Quantitative estimates of unique continuation.

E. Francini (Universit`

a di Firenze) EIT 18 / 42

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Main technical ingredients

Construction of singular solutions and study of their asymptotic behavior near the discontinuity interfaces. Quantitative estimates of unique continuation.

E. Francini (Universit`

a di Firenze) EIT 18 / 42

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Singular solutions

Difficulty:

Lack of a result of existence of the Green’s function in Ω for n ≥ 3 n ≥ 3 Hofmann and Kim (2007, Manuscripta Math. ) existence of Green’s function for weak solutions satisfying interior H¨

lder

continuity estimates . n = 2 Dong and Kim (2009, Trans. Amer. Math. Soc. ) existence of Green’s function

E. Francini (Universit`

a di Firenze) EIT 19 / 42

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Singular solutions

Difficulty:

Lack of a result of existence of the Green’s function in Ω for n ≥ 3 n ≥ 3 Hofmann and Kim (2007, Manuscripta Math. ) existence of Green’s function for weak solutions satisfying interior H¨

lder

continuity estimates . n = 2 Dong and Kim (2009, Trans. Amer. Math. Soc. ) existence of Green’s function

E. Francini (Universit`

a di Firenze) EIT 19 / 42

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Walkway

Define a walkway K ⊂ ∪M−1

j=0 Dj that crosses only flat interfaces

DM D1 D0

E. Francini (Universit`

a di Firenze) EIT 20 / 42

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Tools

Fundamental solutions at the interfaces

Let γ, δ ∈ C. Then, if Γ(x, y) is the fondamental solution to the Laplace equation Γγ,δ(x, y) =     

1 γ Γ(x, y) + sΓ(x, y∗)

if xn > 0, yn > 0,

1

γ + s

Γ(x, y)

if xn · yn < 0,

1 δΓ(x, y) + tΓ(x, y∗)

if xn < 0, yn < 0, where y∗ = (y1, . . . , yn−1, −yn), s =

γ−δ γ(γ+δ) and t = δ−γ γ(γ+δ), is a

fundamental solution for the differential operator div

δχRn

− + γχRn +

∇·
in

Rn.

Li,Nirenberg, Comm. Pure Appl. Math., 2003

H¨

lder gradient estimates up to the interfaces for solutions to elliptic

systems.

E. Francini (Universit`

a di Firenze) EIT 21 / 42

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Singular solutions

Theorem

Let γ satisfy assumptions (H1)-(H3) in Ω0 = Ω ∪ D0. For y ∈ K there exists a unique function G(·, y), continuous in Ω0 \ {y}

Ω0

γ∇G(·, y) · ∇φ = φ(y), ∀φ ∈ C ∞

0 (Ω0),

G(·, y)H1(Ω0\Br(y)) ≤ Cr1−n/2, ∀r < d(y, ∂Ω)/2, G(x, y) = G(y, x) for every x, y ∈ K. Moreover for x ∈ Dk, x0 ∈ Σk and yr = x0 + rν, ∇xG(x, yr) = ∇xΓk(x, yr) + O(1)

E. Francini (Universit`

a di Firenze) EIT 22 / 42

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Estimates of unique continuation

Theorem

Let v ∈ H1(K) be a solution to div (γ∇v) = 0 in K, such that vL∞(K0) ≤ ǫ0, |v(x)| ≤ (ǫ0 + E0)dist(x, ΣM)1− n

2

for every x ∈ K. Then |v(˜ xr)| ≤ C

ǫ0

E0 + ǫ0 τ (M)N1δM

1 τr

(E0 + ǫ0)r(1− n

2)(1−τr),

where ˜ xr = PM − 2rν (PM), PM ∈ ΣM, r ∈ (0, r0) and τ, τr, δ1 ∈ (0, 1), N1 = A

rn

0 and C depends on r0, L, A, n, λ, N.

E. Francini (Universit`

a di Firenze) EIT 23 / 42

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DM ΣM D1 D0 ˜ xr

E. Francini (Universit`

a di Firenze) EIT 24 / 42

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Some remarks

We expect that Lipschitz continuous dependence of the admittivities on the data still holds replacing the flatness condition on the interfaces with C 1,α regularity.

E. Francini (Universit`

a di Firenze) EIT 25 / 42

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Some remarks

Local Dirichlet to Neumann map γ1 − γ2L∞(Ω) ≤ CΛΣ1

1 − ΛΣ1 2 L(H1/2

co ,H−1/2 co

).

H1/2

co

=

φ ∈ H1/2(∂Ω) : suppφ ⊂ Σ1 ⊂ ∂Ω
E. Francini (Universit`

a di Firenze) EIT 26 / 42

SLIDE 43

An important application

V de Hoop, Qiu, Scherzer, Inverse problems, 2012

H¨

lder stability implies local convergence of Landweber iteration.

The constant in H¨

lder stability determines the ray of convergence of

iterations. Lipschitz continuity implies linear convergence.

E. Francini (Universit`

a di Firenze) EIT 27 / 42

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Part II Size estimates

E. Francini (Universit`

a di Firenze) EIT 28 / 42

SLIDE 45

The problem

Ω D γ = γ0✶Ω\D + γ1✶D

Size estimate

Determine |D| from FINITELY many boundary measurements. Motivation: Necrosis in liver tissue

E. Francini (Universit`

a di Firenze) EIT 29 / 42

SLIDE 46

The solutions

Let u0 ∈ H1(Ω) be the background solution div(γ0∇u0) = 0 in Ω, γ0 ∂u0

∂ν

= h on ∂Ω, γ0 = σ0 + iǫ0, σ0 ≥ c0 > 0 Let u1 ∈ H1(Ω) solution to div(γ∇u1) = 0 in Ω, γ ∂u1

∂ν

= h su ∂Ω. γ = γ0✶Ω\D + γ1✶D γ1 = σ1 + iǫ1, σ1 ≥ c0 > 0

Goal

We want to bound |D| in terms of δW :=

∂Ω h(u1 − u0)
E. Francini (Universit`

a di Firenze) EIT 30 / 42

SLIDE 47

The solutions

Let u0 ∈ H1(Ω) be the background solution div(γ0∇u0) = 0 in Ω, γ0 ∂u0

∂ν

= h on ∂Ω, γ0 = σ0 + iǫ0, σ0 ≥ c0 > 0 Let u1 ∈ H1(Ω) solution to div(γ∇u1) = 0 in Ω, γ ∂u1

∂ν

= h su ∂Ω. γ = γ0✶Ω\D + γ1✶D γ1 = σ1 + iǫ1, σ1 ≥ c0 > 0

Goal

We want to bound |D| in terms of δW :=

∂Ω h(u1 − u0)
E. Francini (Universit`

a di Firenze) EIT 30 / 42

SLIDE 48

The solutions

Let u0 ∈ H1(Ω) be the background solution div(γ0∇u0) = 0 in Ω, γ0 ∂u0

∂ν

= h on ∂Ω, γ0 = σ0 + iǫ0, σ0 ≥ c0 > 0 Let u1 ∈ H1(Ω) solution to div(γ∇u1) = 0 in Ω, γ ∂u1

∂ν

= h su ∂Ω. γ = γ0✶Ω\D + γ1✶D γ1 = σ1 + iǫ1, σ1 ≥ c0 > 0

Goal

We want to bound |D| in terms of δW :=

∂Ω h(u1 − u0)
E. Francini (Universit`

a di Firenze) EIT 30 / 42

SLIDE 49

The conductivity case

Ω D γ0 = σ0 γ1 = σ1 dist(D, ∂Ω) ≥ d0 > 0 Kang, Seo and Sheen, Alessandrini and Rosset, Capdeboscq and Vogelius (low volume fraction) Kang, Kim and Milton ( Different approach for the two dimensional

case. Two boundary measurement. No need of background. Explicit

constants)

E. Francini (Universit`

a di Firenze) EIT 31 / 42

SLIDE 50

Strategy (Alessandrini, Seo, Rosset)

δW :=

∂Ω h(u1 − u0)

Energy estimates C1δW ≤

D

|∇u0|2 ≤ C2δW Lower and upper bounds for |D| in terms of

D |∇u0|2 using

respectively regularity estimates and quantitative estimates of unique continuation K2W0|D|p ≤

D

|∇u0|2 ≤ K1W0|D|

E. Francini (Universit`

a di Firenze) EIT 32 / 42

SLIDE 51

Strategy (Alessandrini, Seo, Rosset)

δW :=

∂Ω h(u1 − u0)

Energy estimates C1δW ≤

D

|∇u0|2 ≤ C2δW Lower and upper bounds for |D| in terms of

D |∇u0|2 using

respectively regularity estimates and quantitative estimates of unique continuation K2W0|D|p ≤

D

|∇u0|2 ≤ K1W0|D|

E. Francini (Universit`

a di Firenze) EIT 32 / 42

SLIDE 52

Strategy (Alessandrini, Seo, Rosset)

δW :=

∂Ω h(u1 − u0)

Energy estimates C1δW ≤

D

|∇u0|2 ≤ C2δW Lower and upper bounds for |D| in terms of

D |∇u0|2 using

respectively regularity estimates and quantitative estimates of unique continuation K2W0|D|p ≤

D

|∇u0|2 ≤ K1W0|D|

E. Francini (Universit`

a di Firenze) EIT 32 / 42

SLIDE 53

General assumptions

(H1) Assumptions on Ω: Bounded with Lipschitz boundary. (H2) Assumptions on D: Lebesgue measurable subset of Ω; (H3) Assumptions on the coefficients: we assume the reference medium and the inclusion have admittivities γ0 = σ0 + iǫ0 and γ1 = σ1 + iǫ1 satisfying σj ≥ c0, |γj| ≤ c−1 in Ω, for j = 0, 1. (H4) Assumptions on the boundary data: h ∈ H−1/2(∂Ω), h = 0 such that

∂Ω

h = 0

E. Francini (Universit`

a di Firenze) EIT 33 / 42

SLIDE 54

Assumptions for energy estimates

(H3a) γ0 and γ1 are different constants with positive real part. (H3b) ǫ0(x) ≡ 0 in Ω, and there exists a positive constant µ0 such that |ǫ1(x)| ≥ µ0

r

σ1(x) − σ0(x) ≥ µ0 in Ω.

E. Francini (Universit`

a di Firenze) EIT 34 / 42

SLIDE 55

Energy estimates

Proposition

Assume (H3a) then 1 (1 + |γ1−γ0|

c0

)|γ1 − γ0| |δW | ≤

D

|∇u0|2 ≤ 1 c0 + 2 |γ1 − γ0|

|δW |

Proposition

Assume γ0 and γ1 satisfy assumption (H3b) then F1|δW | ≤

D

|∇u0|2 ≤ F2|δW |, where F1 = c3 2(2 + c2

0)

and F2 = 2

1

µ0c2 + 1 µ0 + 1 c0

.
E. Francini (Universit`

a di Firenze) EIT 35 / 42

SLIDE 56

The one dimensional case

Let Ω = (−1, 1) and let D = (a, b) ⊂ Ω. Consider

(γ0u′

0)′

= 0 in (−1, 1) (γ0u′

0)(−1)

= (γ0u′

0)(1) = K ∈ C,

u0(−1) + u0(1) = 0

(γu′

1)′

= 0 in (−1, 1) (γu′

1)(−1)

= (γu′

1)(1) = K ∈ C,

u1(−1) + u1(1) = 0

E. Francini (Universit`

a di Firenze) EIT 36 / 42

SLIDE 57

The one dimensional case

δW = |K|2 2 b

a

1 γ1 − 1 γ0

dx

If ℜ 1 γ1 − 1 γ0

=

σ1 σ2

1 + ǫ2 1

− σ0 σ2

0 + ǫ2

< (>)0 in (−1, 1)

r

ℑ 1 γ1 − 1 γ0

=

ǫ1 σ2

1 + ǫ2 1

− ǫ0 σ2

0 + ǫ2

< (>)0 in (−1, 1) we can estimate b − a from δW . ǫ0 = 0 −σ1(σ1 − σ0) − ǫ2

1

σ0(σ2

1 + ǫ2 1)

< (>)0, ǫ1 σ2

1 + ǫ2 1

< (> 0)

E. Francini (Universit`

a di Firenze) EIT 37 / 42

SLIDE 58

The one dimensional case

δW = |K|2 2 b

a

1 γ1 − 1 γ0

dx

If ℜ 1 γ1 − 1 γ0

=

σ1 σ2

1 + ǫ2 1

− σ0 σ2

0 + ǫ2

< (>)0 in (−1, 1)

r

ℑ 1 γ1 − 1 γ0

=

ǫ1 σ2

1 + ǫ2 1

− ǫ0 σ2

0 + ǫ2

< (>)0 in (−1, 1) we can estimate b − a from δW . ǫ0 = 0 −σ1(σ1 − σ0) − ǫ2

1

σ0(σ2

1 + ǫ2 1)

< (>)0, ǫ1 σ2

1 + ǫ2 1

< (> 0)

E. Francini (Universit`

a di Firenze) EIT 37 / 42

SLIDE 59

The one dimensional case

δW = |K|2 2 b

a

1 γ1 − 1 γ0

dx

If ℜ 1 γ1 − 1 γ0

=

σ1 σ2

1 + ǫ2 1

− σ0 σ2

0 + ǫ2

< (>)0 in (−1, 1)

r

ℑ 1 γ1 − 1 γ0

=

ǫ1 σ2

1 + ǫ2 1

− ǫ0 σ2

0 + ǫ2

< (>)0 in (−1, 1) we can estimate b − a from δW . ǫ0 = 0 −σ1(σ1 − σ0) − ǫ2

1

σ0(σ2

1 + ǫ2 1)

< (>)0, ǫ1 σ2

1 + ǫ2 1

< (> 0)

E. Francini (Universit`

a di Firenze) EIT 37 / 42

SLIDE 60

Estimate of

D |∇u0|2 in terms of |D| (far from the

boundary)

Proposition

Assume that σ0 is real valued and Lipschitz continuous dist(D, ∂Ω) ≥ d0. There is a constant K1 depending on the a priori constants and on d0 and two constants K2 and p > 1 depending on the a priori constants, on d0 and also on F(h) =

hH−1/2(∂Ω) hH−1(∂Ω) , such that

K2W0|D|p ≤

D

|∇u0|2 ≤ K1W0|D|.

E. Francini (Universit`

a di Firenze) EIT 38 / 42

SLIDE 61

Ingredients

Three Spheres Inequality Lipschitz Propagation of Smallness Doubling Inequality

E. Francini (Universit`

a di Firenze) EIT 39 / 42

SLIDE 62

Size estimates

Theorem

Let Ω satisfy (H1) and let D be a measurable subset of Ω such that dist(D, ∂Ω) ≥ d0. Let γ0 and γ1 be coefficients satisfying assumptions (H3a) or (H3b) and let h with zero mean value. Then, C1

δW

W0

≤ |D| ≤ C2
δW

W0

1/p

where C1 depend on the a priori constants and on d0 and the numbers p > 1 and C2 depend on the a priori constants, on d0 and, in addition, on F(h). [Beretta, Francini, Vessella, submitted]

E. Francini (Universit`

a di Firenze) EIT 40 / 42

SLIDE 63

Inclusions touching the boundary

Ω D D P

E. Francini (Universit`

a di Firenze) EIT 41 / 42

SLIDE 64

Estimate of

D |∇u0|2 in terms of |D| (touching the

boundary)

Assume ∂D ∩ ∂Ω = ∅, but there exist r1 > 0 and P ∈ ∂Ω such that D ⊂ Ω\Br1(P). h ∈ H−1/2(∂Ω) be a complex valued nontrivial current density at ∂Ω such that

∂Ω h = 0, and

supp h ⊂ Γ0 := ∂Ω ∩ Br1/2(P). There is a constant K1 depending on the a priori constants and r1 and two constants K2 and p > 1 depending on the same a priori constants, on r1 and also on F(h) =

hH−1/2(∂Ω) hH−1(∂Ω) , such that

K2W0|D|p ≤

D

|∇u0|2 ≤ K1W0|D|.

E. Francini (Universit`

a di Firenze) EIT 42 / 42