Calderón's problem for quasilinear conductivities · Karthik Iyer University of Washington 4th...

Calderon’s problem for quasilinear conductivities

Karthik Iyer

University of Washington

4th April 2016

Karthik Iyer General Exam 1/21

Let σ(x) be a symmetric uniformly elliptic matrix function inL∞(Ω).

Consider the BVP

∇ · (σ∇u) = 0 on Ω with u = f on ∂Ω

Define the Dirichlet to Neumann map Λσ

Λσ : u∣∣∂Ω→ σ∇u · ν

∣∣∂Ω

The inverse conductivity problem is to get information about σfrom Λσ


Different aspects of the inverse problem:

Uniqueness

Stability

Reconstruction

Numerical Implementation

Partial Data


We can extend the definition of Dirichlet-to-Neumann (DN)

map Λσ : H12 (∂Ω)→ H−

12 (∂Ω)

〈Λσ(f ), g〉 =∫

Ω σ(x)∇u·∇vg dx where u is the unique solutionto conductivity equation with boundary data f and vg

∣∣∂Ω

= g

Does Λσ1 = Λσ2 =⇒ σ1 = σ2 on Ω ?

Yes if σl are scalar valued. No if σl are matrix valued for l = 1, 2


Non-uniqueness in the anisotropic case

Let y = F (x) be smooth diffeomorphism F : Ω → Ω withF (x) = x on ∂Ω. Then∫

Ω

∑σij∂u

∂xi

∂u

∂xjdx =

∫Ω

∑σij

∂u

∂yk

∂yk∂xi

∂u

∂yl

∂yl∂xj

det(∂x

∂y) dy

More compactly,∫Ω

(σ(x)∇xu · ∇xu) dx =

∫Ω

(F∗σ(y)∇yu · ∇yu) dy

where F∗σ(y) = 1det DF (x)DF (x)Tσ(x)(DF (x)) with DFij =

∂yi∂xj

Λσ = ΛF∗σ


Anisotropic uniqueness results for n = 2

Let Λσ1 = Λσ2 . Then

σ2 = Φ∗σ1, σi ∈ C 2,α(Ω), Sylvester 1993 (with a smallnessassumption)

σ2 = Φ∗σ1, σi ∈W 1,p(Ω), p > 2 Sun and Uhlmann 2003

σ2 = Φ∗σ1, σi ∈ L∞(Ω) Astala, Paivarinta, Lassas 2006


Sun and Uhlmann anisotropic 2003 result

Theorem 1

Let Ω ⊂ R2 be a bounded domain with C 1 boundary. Let σ1 andσ2 be two anisotropic conductivities in W 1,p(Ω) for p > 2 withσ1 − σ2 ∈W 1,p

0 (Ω) and Λσ1 = Λσ2 . Then ∃ a diffeomorphismΦ : Ω→ Ω in W 2,p class with Φ

∣∣∂Ω

= I such that σ2 = Φ∗σ1.


Conjecture

Conjecture 1

Let Ω ⊂ R2 be a bounded domain with C 1 boundary. Let σ1(x , t)and σ2(x , t) be two anisotropic conductivities. Assume thatσ1 − σ2 ∈W 1,p

0 (Ω) for each t with p > 2 and Λσ1 = Λσ2 . Then ∃a diffeomorphism Φ : Ω→ Ω in W 2,p class with Φ

∣∣∂Ω

= I suchthat σ2 = Φ∗σ1.


Assumptions

σ(x , t) is a symmetric matrix function defined on Ω×R whereΩ is a bounded domain in R2 with C 1 boundary

[Ellipticity] ∃λ > 0 so that 1λ I σ(x , t) λI for any (x , t) ∈

Ω× R

[Smoothness in t] σ(x , t) is C 1 w.r.t t such that ∂σij (x ,t)∂t is in

L∞(Ω) for any t with an L > 0 so that ||∂σij (x ,t)∂t ||L∞(Ω) ≤ L

for any t.

[Smoothness in x ] For each fixed t, σ(x , t) ∈ W 1,p(Ω) withp > 2


Known results for quasi-linear case

Let Λσ1 = Λσ2 . Then

σ2 = σ1, σi = γi I , γi ∈ C 1,α(Ω× R), Sun 1991

σ2 = Φ∗σ1, σi ∈ C 2,α(Ω× R), Sun and Uhlmann 1997


Conjecture makes sense as Φ preserves class.

Strategy: Define the quasi-linear inverse problem, reduce to thecase of linear inverse problem, use results from linear inverseproblems, make reductions.

Technique adapted from Sun and Uhlmann, 1997 who get unique-ness for anisotropic σ ∈ C 2,α(Ω× R)


DN map

Consider

∇ · (σ(x , u)∇u) = 0 in Ω with u∣∣∂Ω

= f

where f ∈ H1/2(∂Ω). ∃! solution u ∈ H1(Ω).

Define the DN map Λσ : H1/2(∂Ω)→ H−1/2(∂Ω) as 〈Λσ(f ), g〉 =∫Ω

σ(x , u)∇u · ∇vg dx

If Φ is a W 2,p diffeomorphism of Ω which is identity on theboundary, we still have Λσ = ΛΦ∗σ


First Linearization

Fix t ∈ R and f ∈ H1/2(∂Ω). Denote σt(x) as the linearanisotropic conductivity obtained from σ(x , t) by freezing t.

lims→0||1s Λσ(t + sf )− Λσt (f ) ||H−1/2(∂Ω) = 0

Λσ1 = Λσ2 =⇒ Λσt1

= Λσt2∀t ∈ R.

Obtain family of diffeomorhphisms φt ∈W 2,p(Ω) and φt∣∣∂Ω

=Id so that φt∗σ

t1 = σt2 for all t ∈ R.


Define Φ(x , t) = φt(x). Φ is well defined.

Need to show that for any t0, ∂Φ∂t

∣∣t=t0

= 0 in Ω.

First show smoothness of Φ with respect to t.


Construction and properties of Φ

For each t, construct a unique W 2,p diffeomorphism of theplane so that F t

l : Ω → Ωtl so that (F t

l )∗σtl is isotropic for

l = 1, 2.

F tl is constructed solving a homogeneous Beltrami equation

with a compactly supported dilation factor µtl ∈W 1,p(C).

Since µtl is C 1 in t, F tl will be C 1 in t

F t1 = F t

2 on R2\Ω. φt = (F t2 )−1F t

1 is a W 2,p diffeomorphismwhich is identity on the boundary.


F t1 and F t

2 are C 1 w.r.t t, φt = (F t2 )−1 F t

1 also inherits thesame smoothness w.r.t to t.

In fact it is possible to show that ∂Φ∂t

∣∣t=t0∈ W 1,p(Ω) for any

t0 ∈ R

The task is thus reduced to showing ∂Φ∂t

∣∣t=0

= 0. The sameargument works for t 6= 0.

By a transformation one may assume that Φ0 = Id so thatσ1(x , 0) = σ2(x , 0) = σ(x)


Some reductions

Fix f ∈W 2− 1p,p(∂Ω). For l = 1, 2 and each t, solve

∇ · σtl∇utl = 0 in Ω with utl∣∣∂Ω

= f

Consider the unique H10 (Ω) solution v tl to

∇ · σtl∇v tl = −∇ ·[∂σtl∂t∇utl

]t → ul(x , t) is differentiable in the H1 topology with derivativev tl .


∇ ·[(

∂σ1

∂t− ∂σ2

∂t

) ∣∣∣∣t=0

∇u]

+∇ · σ∇(v01 − v0

2 ) = 0

where u = u1(x , 0) = u2(x , 0)

Let X = ∂Φt

∂t

∣∣∣t=0

. For each t ∈ R

ut1(x) = ut2(Φt(x))

Differentiating in t at t = 0 gives

v01 − v0

2 − X · ∇u = 0 in Ω


Need for second linearization

Lemma 2

For every f in H1/2(∂Ω) ∩ L∞(∂Ω) and t ∈ R

lims→0

∣∣∣∣∣∣∣∣1s[

Λσ(t + sf )

s− Λσt (f )

]− Kσ,t(f )

∣∣∣∣∣∣∣∣H−1/2(∂Ω)

= 0

Kσ,t : H1/2(∂Ω) ∩ L∞(∂Ω)→ H−1/2(∂Ω) is defined implicitly as

〈Kσ,t(f ), g〉 =

∫Ω

∇vg · σt(x , t)v0∇(v0) dx

where vg , v0 are unique solutions to ∇ · σ(x , t)∇u = 0 with traceg and f respectively.


Second linearization implies∫Ω

∇u1 ·∂σ1

∂t

∣∣∣∣t=0

∇u22 dx =

∫Ω

∇u1 ·∂σ2

∂t

∣∣∣∣t=0

∇u22 dx

where u, u1, u2 are W 2,p solutions to the linear equation ∇ ·σ(x)∇u = 0 in Ω

Let B(x) =(∂σ1∂t −

∂σ2∂t

) ∣∣∣t=0

. Above can be re-written as∫Ω

B(x)∇u · ∇(u1u2) dx = 0


What needs to be done

Need to show ∇ · B(x)∇u = 0 for any u ∈ W 2,p solution to∇ · σ∇u = 0.

Need to show that X · ∇u = 0 for any u ∈ W 2,p solution to∇ · σ∇u = 0 implies X = 0 in Ω.

Can we improve the result even further to get the quasi-linearanalog of the optimal regularity for the conductivities i.e removethe smoothness assumption in x?

Stability issues for the quasi-linear case?


Calderón's problem for quasilinear conductivities · Karthik Iyer University of Washington 4th...

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