inverse problems with rough data · 2 complex geometrical optics solutions 11 ... the views of the...

Inverse problems with rough data

by

Boaz Eliezer Haberman

A dissertation submitted in partial satisfaction of the

requirements for the degree of

Doctor of Philosophy

in

Mathematics

in the

Graduate Division

of the

University of California, Berkeley

Committee in charge:

Professor Daniel I. Tataru, ChairProfessor Michael M. Christ

Professor Chung-Pei Ma

Spring 2015

1

Abstract


by


Doctor of Philosophy in Mathematics

University of California, Berkeley

Professor Daniel I. Tataru, Chair

In this dissertation, we will discuss some inverse problems associated to elliptic equationswith rough coefficients.

The first results in this thesis are based on the papers [HT13, Hab14], and relate toCalderón’s problem for the conductivity equation

div(γ∇u) = 0.

Given a domain Ω and a positive real-valued function γ defined on Ω, we may define theDirichlet-to-Neumann map Λγ, which maps the Dirichlet data of solutions to the conductivityequation to the corresponding Neumann data. Calderón’s problem is to determine γ fromΛγ. Uniqueness in Calderón’s problem for smooth conductivities in dimensions three andhigher was established by Sylvester and Uhlmann. Later, this result was extended by variousauthors to less regular conductivities which have at least one and a half derivatives. InChapter 4, we will show that Λγ uniquely determines a C

1 conductivity in dimensions threeand higher. In Chapter 6, we will improve this result in dimensions n = 3, 4 to γ ∈ W 1,n(Ω).We also have a new result for conductivities in W s,p(Ω) in higher dimensions for certain (s, p)such that s > 1 but W s,p(Ω) 6⊂ W 1,∞(Ω).

In Chapter 7, we will prove a new result on recovering an unbounded magnetic potentialA from the Dirichlet-to-Neumann map ΛA,q for the Schrödinger equation

(D + A)2u+ qu = 0.

Previous results on this problem assumed that the potential A is bounded. We will showthat, in three dimensions, the integrability assumption on A can be considerably relaxed inexchange for slightly more regularity. Namely, we will show that the Cauchy data uniquelydetermine curlA if q ∈ W−1,3(Ω) and A is small in W s,3(Ω), where s > 0. We will alsoshow that q is determined by the Dirichlet-to-Neumann map if A is small in W s,3(Ω) andq ∈ L∞(Ω).

i

To my mother, Elinor Haberman; may her memory be a blessing.

ii

Contents

Contents ii

1 Calderón’s problem 11.1 The Dirichlet-to-Neumann map . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Recovering a potential and inverse scattering . . . . . . . . . . . . . . . . . . 21.3 Boundary determination and the anisotropic problem . . . . . . . . . . . . . 51.4 Calderón’s problem in the plane . . . . . . . . . . . . . . . . . . . . . . . . . 81.5 An equivalent Schrödinger equation . . . . . . . . . . . . . . . . . . . . . . . 81.6 Rough coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Complex geometrical optics solutions 112.1 An integral identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 The Sylvester-Uhlmann argument . . . . . . . . . . . . . . . . . . . . . . . . 162.3 The operator ∆ζ and the spaces Ẋ

bζ . . . . . . . . . . . . . . . . . . . . . . . 17

3 L2 estimates 193.1 The operator ∆ζ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 Dyadic projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.3 Littlewood-Paley theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.4 Localization estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4 Calderón’s problem for Lipschitz conductivities 294.1 Existence of CGO solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.2 An averaged estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.3 Proof of uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5 Unique continuation and Carleman estimates 355.1 Unique continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.2 The Carleman method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.3 L2 Carleman estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.4 Lp Carleman estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.5 Unbounded gradient terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

iii

6 Conductivities with unbounded gradient 476.1 Scaling considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476.2 Estimates at fixed modulation . . . . . . . . . . . . . . . . . . . . . . . . . . 486.3 Bilinear estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496.4 Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

7 Recovering a magnetic potential 597.1 Cauchy data for Schrödinger operators . . . . . . . . . . . . . . . . . . . . . 597.2 An integral identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607.3 A transport equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627.4 The ∂ equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637.5 A lemma of Eskin and Ralston . . . . . . . . . . . . . . . . . . . . . . . . . . 647.6 Phase estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657.7 Solvability for LA,q,ζ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 677.8 Recovering the magnetic potential . . . . . . . . . . . . . . . . . . . . . . . . 707.9 Recovering the scalar potential . . . . . . . . . . . . . . . . . . . . . . . . . 747.10 Open questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

Bibliography 77

iv

Acknowledgments

I would like to express my gratitude to my friends and to my family for their supportand encouragement throughout my time in graduate school. To my sisters: Shoshanah,Chasiah, Sarah, Milcah and Devorah, for their moral support and frequent visits. To Sarahin particular, for setting me on the path to being a mathematician. To my father, whointroduced me to analysis, and whose wise counsel and much-needed encouragement wereinstrumental in getting this thesis finished in a timely manner.

For the warmth and hospitality of Berkeley’s Jewish community, without which it wouldnot have been possible for me to study here.

To the lunch group, including Baoping Liu, Boris Ettinger, Ben Harrop-Griffiths, BenDodson, Tobias Schottdorf, Paul Smith, Andy Lawrie, Sung-Jin Oh, Jason Murphy, Cris-tian Gavrus, Grace Liu and Mihaela Ifrim, for stimulating conversations, mathematical andotherwise, and for giving me something to look forward to every day.

To Daniel Appel, Zvi Rosen, Russell Brown, Alberto Ruiz, Carlos Kenig and Mikko Salofor many helpful conversations.

To Gunther Uhlmann, for introducing me to the field of inverse problems and for manyhelpful conversations.

To the mathematics faculty at Berkeley, including Craig Evans, Maciej Zworski, JohnLott and Denis Auroux, each of whose lectures exposed me to different ways of thinkingabout analysis. Special thanks to Michael Christ, for serving on my thesis committee andteaching me to think about harmonic analysis from a classical point of view.

Most of all, I thank my academic advisor, Daniel Tataru, whose patient guidance, relent-less optimism and sincere commitment made this thesis possible.

This dissertation is based upon work supported by the National Science under GrantsNo. DGE 1106400 and DMS 0354539. Any opinion, findings, and conclusions or recommen-dations expressed in this dissertation are those of the author and do not necessarily reflectthe views of the National Science Foundation.

1

Chapter 1

Calderón’s problem

1.1 The Dirichlet-to-Neumann map

Let Ω ⊂ Rn be some bounded domain with Lipschitz boundary, and let γ ∈ L∞(Ω) be areal-valued function defined on Ω satisfying

γ(x) > c > 0,

which we interpret as the conductivity of an object at a given point. An electrical potentialu in this situation satisfies the conductivity equation

Lγu = 0,

whereLγu := div(γ∇u).

Our assumptions on γ imply that Lγ is uniformly elliptic. By the theory of elliptic equations,there exists, for any smooth f , a unique solution uf to the Dirichlet problem

Lγuf = 0 in Ω

uf |∂Ω = f,and hence we may formally define the Dirichlet-to-Neumann map Λγ by

Λγf := γ∂uf∂ν

∣∣∣∣∣∂Ω

,

where ∂/∂ν is the outward unit normal derivative at the boundary.Suppose v is a smooth function such that v|∂Ω = g. By the divergence theorem and the

fact that Lγuf = 0, we haveˆΩ

γ∇uf · ∇v dx =ˆ

Ω

div(γ∇uf · v) dx

=

ˆ∂Ω

γ∂νuf · g dS

= 〈Λγf, g〉L2(∂Ω), .

(1.1)

CHAPTER 1. CALDERÓN’S PROBLEM 2

By elliptic regularity theory, we have the a priori estimate

‖uf‖H1(Ω) . ‖f‖H1/2(∂Ω).

The trace map Tru = u|∂Ω extends to a continuous surjection from H1(Ω) to H1/2(∂Ω), andthus we can identify H1/2(∂Ω) with the quotient

H1/2(∂Ω) = H1(Ω)/H10 (Ω),

where H10 (Ω) = ker Tr. Its dual is the space H−1/2(∂Ω). From (1.1) we have

|〈Λγf, g〉L2(∂Ω)| . ‖uf‖H1(Ω)‖v‖H1(Ω) . ‖f‖H1/2(∂Ω)‖v‖H1(Ω)

for any v such that Tr v = f , and thus

|〈Λγf, g〉| . ‖f‖H1/2(∂Ω)‖g‖H1/2(∂Ω),

so that Λγ extends to a continuous map

Λγ : H1/2(∂Ω)→ H−1/2(∂Ω).

The main topic of this thesis will be Calderón’s problem: [Cal80] Can we recover γ from Λγ?

1.2 Recovering a potential and inverse scattering

Consider now a Schrödinger Hamiltonian of the form

LA,q = (D + A)2 + q,

defined on a domain Ω. Here A is a magnetic vector potential, q is a scalar electric potential,and D = −i∇. Since LA,q can have nontrivial solutions vanishing on ∂Ω, the Dirichlet-to-Neumann map is a multivalued relation. We define it by

ΛA,q = {(u|∂Ω , ν · (∇+ iA)u|∂Ω) : LA,qu = 0 in Ω}.

This problem has a gauge invariance: If φ is smooth function vanishing at ∂Ω, then

ΛA+∇φ,q = ΛA,q.

This means that we can only hope to recover A up to a gradient term. Thus a naturalproblem is to show that

ΛA1,q1 = ΛA2,q2 ⇒ curlA1 = curlA2 and q1 = q2.


One reason why this problem is interesting is because it is closely related to inverse scatter-ing at fixed energy. In quantum mechanics, a wavefunction u satisfies the time-dependentSchrödinger equation

i∂tu = LA,qu. (1.2)

For a solution with initial data u0, we write

u = e−itLA,qu0.

Under suitable conditions on A and q (namely, that their influence is localized to a compactregion in space), the solution will eventually escape the influence of the potential and scatterto a solution to the free Schrödinger equation with no potential, i.e.

limt→∞‖e−itLA,qu0 − e−it(−∆)u+0 ‖ = 0.

Since the Schrödinger equations is time reversible, we can similarly define u−0 by

limt→−∞

‖e−itLA,qu0 − e−it(−∆)u−0 ‖ = 0.

Here the interpretation is that every particle which interacts with the potential was once veryfar away from its region of influence, and at that time it was governed by the free evolution.Now, the map u0 7→ u−0 is unitary and thus invertible, so we may define the scattering matrixcorresponding to the system by

S(u−0 ) = u+0 .

The scattering matrix encodes the observations of the system that can be made from adistance. An observation is made by sending a free wave into the system from far away.The free wave interacts with the system for some time, but eventually leaves the influenceof the system and scatters to a different free wave. The problem of inverse scattering is todetermine curlA and q from the scattering matrix S.

Although we have expressed the scattering matrix in terms of the time-independentSchrödinger equation, there is a stationary formulation of this problem as well. Consider thestationary Schrödinger equation

LA,qu = Eu. (1.3)

Here the energy E is a nonnegative real number. If A and q are zero, then solutions to thisequation are plane waves eik·x, where k ∈ Rn satisfies |k|2 = E. According to stationaryscattering theory, solutions to Schrödinger equation (1.3) for the perturbed Laplacian havethe asymptotic form

u(x) = eik·x + |x|−(n−1)/2ei|k||x|Ça

Çx

|x|, k

å+O(|x|−1)

å.

The function a : Sn−1 × Rn → R is known as the scattering amplitude, and completelydetermines the scattering matrix S.


Faddeev [Fad56] and Berezanskii [Ber58] showed that a potential q can be recovered fromthe corresponding scattering amplitude at high energy. This problem is formally overde-termined, however, as the domain of a is (2n − 1)-dimensional, while the domain of q isn-dimensional.

The problem of inverse scattering at fixed energy E is to determine curlA, q from thescattering amplitudes aE(θ, ω) = a(θ,

√Eω) where ω ∈ Sn−1. Faddeev [Fad66] noticed

that the Schrödinger equation LA,qu = Eu has many nonphysical exponentially increasingsolutions which are asymptotically equal to ex·ζ , for ζ ∈ Cn. In particular, we can take |ζ|to be as large as we please, even though E is fixed. These solutions constructed by Faddeevare the same as the complex geometrical optics solutions introduced later by Sylvester andUhlmann in connection with Calderón’s problem. The problems are closely related; in fact,for compactly supported potentials the Dirichlet-to-Neumann map ΛA,q can be recoveredfrom the scattering amplitude aE at fixed energy and vice-versa.

In [Fad76], Faddeev introduced a nonphysical scattering amplitude connected with theexponentially increasing solutions, and showed that this amplitude can be recovered from thephysical scattering amplitude aE(θ, ω), where E is fixed. A similar method was discoveredindependently by Newton [New74,New82]. Faddeev’s ideas were developed further and maderigorous in the work of [NA84,BC85,NK87]. We summarize some uniqueness results for theinverse scattering problem for exponentially decreasing potentials. Since the problem forcompactly supported potentials is equivalent to Calderón’s problem, we also include resultsfor the Dirichlet-to-Neumann map. We indicate in this case that the potential is supportedon a compact set Ω by the notation X(Ω):


n q ASylvester-Uhlmann [SU86] 2 W 2,2(Ω), small 0Novikov [Nov86] 2 e−γ〈x〉C∞, small 0Novikov [Nov92] 2 e−γ〈x〉L∞, small 0Sun [Sun93b] 2 W 1,∞(Ω), small W 3,∞(Ω), smallBukhgeim [Buk08] 2 C1(Ω) 0Bl̊asten [Bl̊a11] 2 W 0+,2+(Ω) 0Imanuvilov-Yamamoto [IY12] 2 L2+(Ω) 0

Guillarmou-Salo-Tzou [GST11] 2 e−γ|x|2L∞ ∩ C1,0+ 0

Lai [Lai11] 2 L∞(Ω) L∞(Ω)

Khenkin-Novikov [NK87] ≥ 3 e−γ〈x〉L∞, small 0Nachman-Sylvester-Uhlmann [NSU88] ≥ 3 L∞(Ω) 0Novikov [Nov88] ≥ 3 L∞(Ω) 0Novikov [Nov94] 3 e−γ〈x〉L∞ 0Sun [Sun93a] ≥ 3 L∞(Ω) W 2,∞(Ω), smallNakamura-Sun-Uhlmann [NSU95] ≥ 3 L∞(Ω) C∞(Ω)Eskin-Ralston [ER95] ≥ 3 e−γ〈x〉C∞ e−γ|x|C∞Salo [Sal04,Sal06] ≥ 3 L∞(Ω) W 1,∞(Ω)Krupchyk-Uhlmann [KU14] ≥ 3 L∞(Ω) L∞(Ω)Jerison-Kenig [Cha90] ≥ 3 Ln/2+(Ω) 0Lavine-Nachman [Nac92,DSFKS13] ≥ 3 Ln/2(Ω) 0Päivärinta-Salo-Uhlmann [PSU10] ≥ 3 e−γ〈x〉L∞ e−γ〈x〉W 1,∞

From the scaling of the problem, the natural assumptions are to take A ∈ Ln and q ∈ W−1,n.So far there are no results in any dimension for A ∈ Lp with p 0)and qi ∈ W−1,3(Ω) for i = 1, 2. There exists some � > 0, depending on Ω and s, such that if‖Ai‖W s,3(Ω) ≤ � for i = 1, 2 then

LA1,q1 = LA2,q2

implies that curlA1 = curlA2. If q1, q2 ∈ L∞ then we also have q1 = q2.

1.3 Boundary determination and the anisotropic

problem

From a physical point of view, one would expect that the value of the conductivity atthe boundary would be most accessible to boundary measurements. In [KV84], Kohn andVogelius showed that for smooth conductivities, the map γ 7→ Λγ determines the values ofγ and all of its normal derivatives on ∂Ω. This was improved by Alessandrini to


Theorem 1.3.1 ([Ale90]). Suppose Ω is a Lipschitz domain and γ1, γ2 are Lipschitz in Ω.If γ1 − γ2 is Ck is a neighborhood of ∂Ω, then

Λγ1 = Λγ2 ⇒ ∂ηγ1|∂Ω = ∂ηγ2|∂Ω

for any multiindex η with |η| ≤ k.

Brown [Bro13] proved that the boundary values can be recovered whenever this makessense. In particular, his result applies when γ is continuous or γ ∈ W 1,1(Ω).

Theorem 1.3.2 ([Bro13]). Let Ω be a Lipschitz domain, and suppose that γ is continu-ous at almost every boundary point, in the sense that it is equal almost everywhere to itsnontangential limit at the boundary. Then Λγ determines γ|∂Ω almost everywhere.

It is possible to determine γ at the boundary even if γ is replaced by a positive-definitematrix {γij}. If we define

Lγ =∑i,j

∂i(γij∂ju),

then the Dirichlet-to-Neumann map is defined by

Λγ(u|∂Ω) =∑i,j

γijνi∂ju

where u satisfies Lγu = 0. The Dirichlet-to-Neumann map is invariant under diffeomor-phisms: If φ : Ω→ Ω is a diffeomorphism such that φ|∂Ω = id, then

Λφ∗γ = Λγ,

where

φ∗γ(x) =

Ç1

detDφDφ · γ ·DφT

å(φ−1(x))

The Calderón problem for an anistropic conductivity is to recover γ from Λγ up to thisdiffeomorphism invariance.

In this context, it is natural to consider the following geometric problem, which is equiv-alent to Calderón’s problem in dimensions three and higher.

Let (M, g) be a compact Riemannian manifold with boundary, and define Λg by

Λg(u|∂M) = ρ∂νu|∂M ,

where ∂ν is again the outward unit normal, ρ is the volume density associated to g, and usolves the Laplace-Beltrami equation

∆gu = 0.

We can reformulate this problem as follows:


Conjecture. Let (M1, g1) and (M2, g2) be complete connected Riemannian manifolds withboundary of dimension n ≥ 3. Suppose that ∂M1 is diffeomorphic to ∂M2, and that theDirichlet-to-Neumann maps agree with respect to this identification. Then there exists adiffeomorphism φ : M1 →M2 such that φ∗g2 = g1.

This conjecture is open in dimensions three and higher, except for some results under veryrestrictive assumtions on the geometry (see for example [DSFKSU09]). However, Lee andUhlmann [LU89] showed that Λg does determine the metric and all of its normal derivativesat the boundary. Their argument is based on the following idea: Choose a neighborhoodU of X = ∂M which admits boundary normal coordinates. That is, we assume there is adiffeomorphism

φ : [0, �]×X → U

with φ(0, ·) = id. We can foliate U into hypersurfaces

Σt = {φ(t, x) : x ∈M}

and we assume that the curves t 7→ φ(t, x) are normal to the Σt, so that the metric takesthe form

g = dt2 + gt

where t denotes projection onto the first coordinate and gt is a metric on X. For t ≥ 0, theopen submanifold

Mt = M \ {φ(s, x) : s < t}

is bounded by Σt and determines a Dirichlet-to-Neumann map Λg(t). An easy formal cal-culation using the definition of the Dirichlet-to-Neumann map gives the operator Riccatiequation

∂tΛg(t) + Λg(t)ρ−1Λg(t) + ρ∆gt = 0.

Lee and Uhlmann showed that t 7→ Λg(t) is a smooth function from the interval [0, �] to thespace of elliptic pseudodifferential operators of order 1 on X. Using the symbol calculus forpseudodifferential operators, we can deduce that

σ(Λg(t))2(x, ξ) = ρ2gt(x)(ξ, ξ)

where σ(Λg(t)) is the principal symbol of Λg(t). From this it follows that gt can be recoveredfrom Λg(t) by looking at the principal symbol. Similarly, all of the derivatives ∂

kt gt can

be formally recovered from the asymptotic expansion of the symbol of Λg(t) by equatingterms of equal homogeneity in the Riccati equation. Conversely, the terms in the asymptoticexpansion of the symbol of Λg(t) are completely determined by the metric and its normalderivatives, so that any information about the interior values of the metric must lie in thesmoothing part of symbol.


1.4 Calderón’s problem in the plane

As a linear operator from H1/2(∂Ω) to H−1/2(∂Ω), the Dirichlet-to-Neumann map can beexpressed in terms of a Schwartz kernel, i.e.

Λγf(x) =

ˆ∂Ω

K(x, y)f(y) dS(y).

Since ∂Ω is an (n − 1)-dimensional space, the domain of K : ∂Ω × ∂Ω → R has dimension2n− 2. On the other hand, the conductivity γ which we are trying to recover is defined onan n-dimensional domain. Thus the problem of recovering a conductivity from the Dirichlet-to-Neumann map is formally determined in two dimensions, and formally overdetermined inthree dimensions and higher.

This heuristic suggests that in order to solve Calderón’s problem in two dimensions,we must use all of the information contained in the Dirichlet-to-Neumann map. In higherdimensions, on the other hand, there is more wiggle room, and we can get away with ignoringmuch of this information. This freedom will be crucial to proving uniqueness in higherdimensions, especially as the conductivity becomes less regular.

Sylvester and Uhlmann proved uniqueness in the isotropic problem for conductivitiesclose to the identity in [SU86]. Sylvester [Syl90] established uniqueness in the anisotropicproblem for conductivities close to the identity, by using a version of isothermal coordinatesto reduce it to the isotropic problem.

Nachman [Nac96] proved the first uniqueness result in two dimensions with no smallnessassumption. We summarize some subsequent results in the following table:

Isotropic AnistropicNachman [Nac96] γ ∈ W 2,1 Nachman [Nac96] γij ∈ C3Brown-Uhlmann [BU97] γ ∈ W 1,2+ Sun-Uhlmann[SU03] γij ∈ W 1,2+Astala-Päivärinta[AP06] γ ∈ L∞ Astala-Lassa-Päivärinta [ALP05] γij ∈ L∞

In [ALP11], the assumption that γij is bounded was relaxed slightly to a BMO-typebound, and some counterexamples were given to show that their assumptions are essentiallysharp. Thus uniqueness and nonuniqueness for Calderón’s problem in the plane is fairlywell-understood.

1.5 An equivalent Schrödinger equation

Using the Leibniz rule, we may write the equation Lγu = 0 as a perturbation of the Laplaceequation, i.e.

−∆u−∇ log γ · ∇u = 0.

We wish to eliminate the first-order term ∇ log γ · ∇u and replace it with a zero-orderpotential term qu. We can do this, using a Liouville transformation, because the coefficient


∇ log γ is a gradient. Indeed, for any φ we have

e−φ ◦ (−∆) ◦ eφ = −∆− 2∇φ · ∇ − |∇φ|2 −∆φ.

Setting φ = 12

log φ, we obtain

− div(γ∇u) = −∆u−∇ log γ · ∇u= −γ−1/2∆(γ1/2u) + (1

4|∇ log γ|2 + 1

2∆ log γ)u

−γ1/2 div(γ∇u) = −∆(γ1/2u) + q(γ1/2u), (1.4)

whereq = 1

4|∇ log γ|2 + 1

2∆ log γ,

and we find that the function v = γ1/2u satisfies the time-independent Schrödinger equation

(−∆ + q)v = 0, (1.5)

By setting u = 1, we may observe that

q = γ−1/2∆γ1/2.

Since the conductivity equation is equivalent to a Schrödinger equation and Λγ determinesγ and its derivatives at the boundary, it is easy to see that knowledge of the Dirichlet-to-Neumann map Λγ is equivalent to knowledge of the corresponding Dirichlet-to-Neumannmap Λq for the Schrödinger equation.

Suppose that q can be recovered from Λq. Then for two conductivities γ1, γ2 we have

Λγ1 = Λγ2 ⇒ Λq1 = Λq2 ⇒ q1 = q2,

where qi = γ−1/2i ∆γ

1/2i . A priori, this is not enough to show that γ1 = γ2. In particular,

multiplying each γi by a constant will not change the qi. Thus we need to combine thedetermination of qi in the interior with the determination of γi at the boundary discussedabove. Indeed, the function log γ1 − log γ2 satisfies the divergence-form elliptic equation

div((γ1γ2)1/2∇(log γ1 − log γ2)) = 0.

Since the assumption Λγ1 = Λγ2 implies that log γ1 − log γ2 = 0 on ∂Ω, this implies thatγ1 = γ2 in Ω. By this argument, Sylvester and Uhlmann [SU87] reduced the problem to thatof recovering a potential and showed that, for smooth conductivities, Λγ1 = Λγ1 implies thatγ1 = γ2.


1.6 Rough coefficients

In the above discussion, we have implicitly assumed that γ is sufficiently smooth, so that qis well-defined and the boundary determination results apply. One reason to consider theinverse problem for the Schrödinger equation with potentials in Sobolev spaces of negativeregularity index is that when γ has less than two derivaties, the potential q will lie in sucha space. This line of inquiry was initiated by Brown [Bro96], who showed uniqueness inCalderón’s problem for conductivities in C3/2+. We summarize some uniqueness results forCalderón’s problem in the following table:

Boundary Interior n γKohn-Vogelius [KV84] Sylvester-Uhlmann [SU87] ≥ 3 C∞Alessandrini [Ale90] Nachman-Sylvester-Uhlmann [NSU88] ≥ 3 C1,1Nachman [Nac96] Jerison-Kenig [Cha90] ≥ 3 W 2,n/2+

Brown [Bro96] ≥ 3 C3/2+Päivärinta-Panchenko-Uhlmann [PPU03] ≥ 3 W 3/2,∞Brown-Torres [Bro03] ≥ 3 W 3/2,2n+Haberman-Tataru [HT13] ≥ 3 C1, W 1,∞ smallCaro-Rogers [CR14] ≥ 3 W 1,∞Nguyen-Spirn [NS14] 3 W 3/2+,2

Brown [Bro13] Haberman [Hab14] 3, 4 W 1,n

5,6 W 1+θ(12− 2n

),nθ

In this thesis, we will first prove the following:

Theorem 2. [HT13] Let Ω ⊂ Rn, n ≥ 3 be a bounded domain with Lipschitz boundary.For i = 1, 2, let γi ∈ W 1,∞(Ω) be real valued functions, and assume there is some c suchthat 0 < c < γi < c

−1. Then there exists a constant �d,Ω such that if each γi satisfies either‖∇ log γi‖L∞(Ω) ≤ �d,Ω or γi ∈ C1(Ω) then Λγ1 = Λγ2 implies γ1 = γ2.

The smallness condition for Lipschitz conductivities can be removed [CR14], but we willnot address this issue. We will also prove a stronger theorem where the regularity assumptionin three and four dimensions corresponds to the L∞ scaling of the problem:

Theorem 3. [Hab14] Suppose 3 ≤ n ≤ 6, and let Ω ⊂ Rn be a bounded domain withLipschitz boundary. Let γ ∈ W s,p(Ω) be real-valued functions, where

(s, p) =

(1, n) n = 3, 4(1 + (1− θ)(12− 2

n), n

1−θ ) n = 5, 6.

and θ ∈ [0, 1). Assume in addition that there is some c such that 0 < c < γi < c−1. ThenΛγi = Λγ2 implies that γ1 = γ2.

11

Chapter 2

Complex geometrical optics solutions

2.1 An integral identity

A weak definition of the DN map Calderón’s approach for recovering the values of γin the interior was to use a weak formulation of the problem. By (1.1), we may express themap Λγ by

〈Λγ(u|∂Ω), v|∂Ω〉 =ˆ

Ω

γ∇u · ∇v dx,

where u is a solution to div(γ∇u) = 0 in Ω, and v is arbitrary. If γ1 and γ2 are twoconductivities such that Λγ1 = Λγ2 , then this implies thatˆ

Ω

(γ1 − γ2)∇u1 · ∇u2 dx = 0 (2.1)

when div(γ1∇u1) = div(γ2∇u2) = 0. Thus if the set

V = span{∇u1 · ∇u2 : div(γ1∇u1) = div(γ2∇u2) = 0}

is dense in some sense, then we can conclude that γ1 = γ2.Calderón observed that V is dense in L2(Ω) (for example) when γ1 = γ2 = 1. If ζ ∈ Cn

satisfies ζ · ζ = 0, thendiv(∇eζ·x) = ∆eζ·x = ζ · ζeζ·x = 0.

Choose k ∈ Rn and ζ1, ζ2 ∈ Cn such that

ζi · ζi = 0ζ1 + ζ2 = ik

ζ1 · ζ2 6= 0.

For example, when k 6= 0 we can take

ζ1 = ik + Z

ζ2 = ik − Z

CHAPTER 2. COMPLEX GEOMETRICAL OPTICS SOLUTIONS 12

where Z ∈ Rn satisfies |Z| = |k| and Z · k = 0. Then we have

∇eζ1·x · ∇eζ2·x = (ζ1 · ζ2)eik·x,

and thus V contains the complex exponentials eik·x for any k ∈ Rn. This together with (2.1)is enough to conclude that γ1 = γ2, which is also a direct consequence of the assumptionγ1 = γ2 = 1. By using a slightly more sophisticated argument, Calderón showed that thelinearization of the map γ 7→ Λγ about γ = 1 is injective, and his paper gave an importantclue as to how boundary measurements could yield interior identifiability.

Reduction to the Schrödinger equation It is useful to reformulate the identity (2.1)in terms of solutions to the Schrödinger equations

(−∆ + qi)ui = 0 (2.2)

whereqi = γ

−1/2i ∆γ

1/2i =

14|∇ log γi|2 + 12∆ log γi. (2.3)

If Λγ1 = Λγ2 , then we will show thatˆΩ

(q1 − q2)u1u2 dx = 0 (2.4)

for any u1, u2 satisfying (2.2). Thus we can reduce the problem to showing that the set

W = span{u1u2 : (−∆ + qi)ui = 0 for i = 1, 2}

is dense in a suitable sense.

Some technical lemmas The cost of replacing the conductivity equation with a Schrödingerequation is that we must take two derivatives of the conductivity. This means that if theconductivity is not twice-differentiable, the potential will lie in a space of negative regularityindex. In order to make sense of this, we will need some technical results, which we takefrom [Bro03].

We define, for f ∈ W 1,n(Rn) compactly supported and u, v ∈ H1loc,

〈∆f · u, v〉L2(Rn) =ˆ∇f · (∇uv + u∇v) dx. (2.5)

First we show that (2.4) makes sense for γi ∈ W 1,n(Rn).

Lemma 2.1.1. Suppose n ≥ 3. Suppose that γ is a positive function on Rn such thatlog γ ∈ L∞(Rn), ∇ log γ ∈ Ln(Rn) and γ = 1 outside of a large ball B. Then

|〈qu, v〉| .γ ‖u‖H1(B)‖v‖H1(B),

where q = γ−1/2∆γ1/2 and 〈qu, v〉 is defined using (2.5).


Proof. Using the above definition and Hölder’s inequality, we have

|〈qu, v〉| . ‖∇ log γ‖2Ln‖uv‖Ln/(n−2) + ‖∇ log γ‖Ln‖u∇v +∇uv‖Ln/(n−1).γ ‖u‖L2n/(n−2)‖v‖L2n/(n−2) + ‖u‖L2n/(n−2)‖∇v‖L2 + ‖∇u‖L2‖v‖L2n/(n−2)

If n ≥ 3, we have the Sobolev embedding H1 ⊂ L2n/(n−2), and we are done, since q issupported in B.

Next, we establish (2.4) for conductivities which agree outside Ω:

Lemma 2.1.2. Suppose n ≥ 3. Let γ1, γ2 ∈ W 1,n(Rn) be functions such that 0 < c ≤ γi ≤ c−1for some c and γi = 1 outside a large ball. If γ1 = γ2 outside Ω and Λγ1 = Λγ2, then for

qi = γ−1/2i ∆γ

1/2i , we have

〈q1 − q2, v1v2〉 = 0

when each vi is a weak solution in H1loc(Rn) to ∆vi − qivi = 0.

Proof. By (1.4), the functions ui = γ−1/2i vi satisfy the conductivity equation

div(γi∇ui) = 0.

We claim that

〈Λγi u1|∂Ω , u2|∂Ω〉 =ˆ

Ω

[∇v1∇v2 −∇γ1/2i · ∇(γ−1/2i v1v2)] dx. (2.6)

We assume i = 1, and will use the fact that γ1 = γ2 outside Ω. This will imply that

(γ−1/21 − γ

−1/22 )v2

∣∣∣∂Ω

= 0. (2.7)

Assuming that this statement is justified, we can apply the formula (1.1) to γ−1/21 v1 and

γ−1/21 v2, to obtain

〈Λγ1 u1|∂Ω , u2|∂Ω〉 =ˆ

Ω

γ1∇(γ−1/21 v1) · ∇(γ−1/21 v2) dx

=

ˆΩ

[∇v1 · ∇v2 + γ1∇γ−1/21 · (∇γ−1/21 v1v2 + γ

−1/21 v2∇v1 + γ

−1/21 v1∇v2)] dx

=

ˆΩ

[∇v1 · ∇v2 −∇γ1/21 · ∇(γ−1/21 v1v2)] dx.

Since Λγ1 = Λγ2 , we have

0 = 〈(Λγ1 − Λγ2) u1|∂Ω , u2|∂Ω〉

= −ˆ

Ω

[∇γ1/21 · ∇(γ−1/21 v1v2)−∇γ

1/22 · ∇(γ

−1/22 v1v2)] dx


On the other hand, since γ1 = γ2 outside Ω, we have

0 =

ˆRn\Ω

[∇γ1/21 · ∇(γ−1/21 v1v2)−∇γ

1/22 · ∇(γ

−1/22 v1v2)] dx

Adding these two equalities we obtain the desired identity

0 = 〈q1 − q2, v1v2〉.

In order to complete the proof of Lemma 2.1.2, we must justify the assertion (2.7). Wedo so with the following

Lemma 2.1.3. Suppose Ω is a Lipschitz domain in Rn, where n ≥ 2, and β ∈ L∞ ∩W 1,n.If β|∂Ω = 0, then u 7→ βu maps H1(Ω) to H10 (Ω).

Proof. We replace β with a function supported away from ∂Ω. For each �, let η� be a cutofffunction such that η� = 0 when d(x, ∂Ω) < � and η� = 1 when d(x, ∂Ω) > 2�. Since ∂Ω isLipschitz, we can choose η� such that |η�|+ �|∇η�| . 1 uniformly in �. In particular, we have|∇η�| . d(x, ∂Ω)−1.

Now set β� = η�β. First, we claim that β� → β in W 1,n(Ω) as � → 0. This follows fromHardy’s inequality and the dominated convergence theorem, since

‖∇η�β‖n . ‖d(x, ∂Ω)−1β‖n . ‖∇β‖n.

Now we claim that β�u→ βu in H1(Ω) for any u ∈ H1(Ω). First, we have β�u→ βu inL2 and β�∇u→ β∇u in L2(Ω) by the dominated convergence theorem, since ‖β�‖∞ . ‖β‖∞.It remains to show that ∇β�u → ∇βu in L2(Ω). This follows from Hölder’s inequality andSobolev embedding:

‖∇(β� − β)u‖2 . ‖β − β�‖W 1,n‖u‖2n/(n−2) . ‖β − β�‖W 1,n‖u‖H1 → 0.

Since β�u is obviously in H10 (Ω) for each �, this shows that βu ∈ H10 (Ω).

Boundary identification According to the boundary identification Theorems 1.3.1 and 1.3.2,the assumption Λγ1 = Λγ2 is enough to ensure that γ1 and γ2 admit extensions which agreeoutside of Ω. In what follows we will justify this assertion. The details depend on ourassumptions on the γi.

C1 conductivities: We use Whitney’s extension theorem

Theorem 2.1.4 ([Whi34]). Let A be a closed subset of Rn, and suppose that f isuniformly Ck on A. Then f admits a Ck extension to Rn.


For a general closed set A, we must specify in what sense the function is Ck; however,for our applications this is obvious.

Suppose now that γi ∈ C1(Ω). If Λγ1 = Λγ2 , then Theorem 1.3.1 implies that γ1 = γ2and ∂γ1 = ∂γ2 on ∂Ω. By Whitney’s extension theorem, we may extend the functionsγi to functions γ̃i ∈ C10(Rn). Furthermore, we can arrange that the γ̃i are boundedabove and below and are equal to 1 outside a compact set.

Now, since γ̃1− γ̃2 = ∂γ̃1−∂γ̃2 = 0 on ∂Ω, the function χRn\Ω(γ̃1− γ̃2) is also C1. Thuswe may replace the extension γ̃2 by γ̃2 + χRn\Ω(γ̃1 − γ̃2), which agrees with g̃1 outsideof Ω.

Lipschitz conductivities: In this case it is easy to construct Lipschitz extensions γ̃i (forexample, by reflection across the boundary). Theorem 1.3.1 implies that γ1 = γ2 on ∂Ω.Again we can arrange that they agree outside of Ω by using the fact that a Lipschitzfunction on Rn \ Ω vanishing on ∂Ω extends by zero to a Lipschitz function on Rn.

Conductivities in Ws,p: If 1 < p < ∞, a theorem of Calderón [Cal61] guarantees exten-sions of the γi which are bounded in W

s,p(Rn). By Theorem 1.3.2, we have γ1 = γ2 on∂Ω as long as the γi are in W

1,1(Ω). To construct extensions which agree outside of Ω,we use a theorem on extensions by zero, which follows from [Mar87, Theorem 1].

Theorem 2.1.5. Let Ω be a Lipschitz domain. Suppose that 1 < p 1/p ands− 1/p ≤ 1. If f ∈ W s,p(Ω) and f |∂Ω = 0, then the extension of f by zero is boundedin W s,p(Rn).

By the above discussion and Lemma 2.1.2, we have

Lemma 2.1.6. Let Ω ⊂ Rn (n ≥ 3) be a bounded Lipschitz domain. Suppose that 0 < c < γi < c−1and one of the following holds for i = 1, 2

• γi ∈ C1(Ω)

• γi ∈ Lip(Ω)

• γi ∈ W s,p, where 1 < p


2.2 The Sylvester-Uhlmann argument

We now construct complex geometrical optics (CGO) solutions to the Schrödinger equations(−∆+qi)vi = 0 in order to exploit the integral identity 〈q1−q2, v1v2〉 = 0. Since the functionex·ζ is harmonic when ζ · ζ = 0, we are led to look for solutions of the form

vi = ex·ζi(1 + ψi), (2.8)

where ζi ∈ Cn are such that ζi · ζi = 0. Writing

0 = (Re ζ + i Im ζ)2 = |Re ζ|2 − |Im ζ|2 + 2iRe ζ · Im ζ,

we see that this condition implies that |Re ζ| = |Im ζ| and Re ζ · Im ζ = 0.We will show that q1 = q2 by showing that (q̂1 − q̂2)(k) = 〈q1 − q2, eik·x〉 = 0 for each

k ∈ Rn. To this end, we would like the ζi to satisfy ζ1 + ζ2 = ik, so that

ζ1 = η1 + i(12k + η2)

ζ2 = −η1 + i(12k − η2),

where η1, η2 ∈ Rn. In order that (12k + η2) · η1 = (12k − η2) · η1 = 0, we must have

k · η1 = η2 · η1 = 0. The condition |12k + η2| = |12k − η2| implies that k · η2 = 0. Finally, we

have|η1|2 = |12k + η2| =

14|k|2 + |η2|2.

It is here that we may exploit the overdeterminacy of the problem. The vectors k, η1 andη2 are orthogonal, and |η1| ≥ |k|. In R2, we cannot have three orthogonal nonzero vectors,so if k 6= 0 this forces η2 = 0. Note also that η1 is then completely determined by k up toreflection through the origin. The situation in Rn for n ≥ 3 is completely different. Once kis chosen, we have n− 1 degrees of freedom in choosing η2, and an additional n− 3 degreesof freedom in choosing η1. In particular, for any k ∈ Rn, we can choose ζi such that |ζi| isas large as we please.

We shall construct CGO solutions (2.8) such that

‖ψi‖ → 0 as |ζi| → ∞ (2.9)

in an appropriate norm. Once we have done this, we have

0 = 〈q1 − q2, v1v2〉= 〈q1 − q2, e(ζ1+ζ2)·x(1 + ψ1)(1 + ψ2)〉= 〈q1 − q2, eik·x(1 + ψ1)(1 + ψ2)〉

(2.10)

Taking the limit as |ζi| → ∞ we have

0 = 〈q1 − q2, eik·x〉 = (q̂1 − q̂2)(k),

and we have shown that q1 = q2. To conclude that γ1 = γ2, we prove the following:


Lemma 2.2.1. Let γi, qi be as in Lemma 2.1.2, and suppose that q1 = q2 in the sense ofdistributions. Then γ1 = γ2.

Proof. First, have qi ∈ H−1(Rn) for each i. This is a consequence of Lemma 2.1.1. In fact,choosing φ ∈ C∞0 (Rn) such that φqi = qi, we have

|〈qi, u〉| = |〈qi, φu〉| . ‖φ‖H1‖u‖H1 . ‖u‖H1

uniformly in u ∈ H1.It follows that we may test q1 − q2 against the function g1g2(log g1 − log g2) ∈ H1(Rn),

where gi = γ1/2i . This gives

0 =

ˆ[∇g1 · ∇(g2(log g1 − log g2))−∇g2 · ∇(g1(log g1 − log g2))] dx

=

ˆ(g2∇g1 − g1∇g2) · ∇(log g1 − log g2) dx

=

ˆg1g2|∇(log g1 − log g2)|2 dx

which implies that g1 = g2.

2.3 The operator ∆ζ and the spaces Ẋbζ

It remains to construct CGO solutions ex·ζ(1 + ψ) with ‖ψ‖ → 0 as |ζ| → ∞. The functionv = ex·ζ(1 + ψ) solves the Schrödinger equation (−∆ + q)v = 0 if

0 = e−x·ζ(−∆ + q)ex·ζ(1 + ψ)= −∆ζ(1 + ψ) + qψ + q

where∆ζ = e

−x·ζ ◦∆ ◦ ex·ζ = ∆ + 2ζ · ∇+ ζ · ζ.If ζ · ζ = 0, then ∆ζ1 = 0 and our equation for ψ is

(∆ζ − q)ψ = q. (2.11)

The operator ∆ζ is a Fourier multiplier with symbol

pζ(ξ) = −|ξ|2 + 2iξ · ζ.

We define an inverse ∆−1ζ by ÷∆−1ζ f(ξ) = p−1ζ f̂(ξ),and we write (2.11) as

(I −∆−1ζ ◦ q)ψ = ∆−1ζ q,


where we interpret the q on the left hand side as the map u 7→ qu. We will construct ψ in aBanach space X, by showing that

‖∆−1ζ ◦ q‖X→X ≤ 1/2, (2.12)

In order to obtain a useful bound for ‖ψ‖X , we will need to show that

‖∆−1ζ q‖X → 0 as |ζ| → ∞. (2.13)

Finally, in order to justify taking limits in (2.10), we will need to show that

|〈qi, eik·xψ1ψ2〉| . ‖ψ1‖X‖ψ2‖X (2.14)|〈qi, eik·xψj〉| . ‖ψj‖X . (2.15)

In Sylvester and Uhlmann’s original proof, the space X was taken to be a weighted Sobolevspace. In fact, one can show that something like (2.12) holds for X = H1τ (Ω), where

‖u‖H1τ (Ω) = τ‖u‖L2(Ω) + ‖∇u‖L2(Om).

Using this type of argument, is was shown in [PPU03,KU14] that complex geometrical opticssolutions can be constructed when the conductivity γ is merely Lipschitz. However, solutionswhich are constructed in H1τ will not satisfy (2.13), because this would require

‖∇∆−1ζ q‖L2 → 0.

Since q contains the term ∆ log γ, to show (2.13) for Lipschitz γ would require

‖∇2∆−1ζ (log γ)‖L2 → 0.

Because ∆ζ is not elliptic, this will not hold in general. Using some averaging argumentswhich we will discuss in Chapter 4, it was shown by [NS14] that (2.13) holds for a large setof ζ when γ ∈ C1,� for � > 0. However, their argument fails for γ ∈ C1.

Following the philosophy of Bourgain [Bou93], the author and Tataru [HT13] constructedsolutions in Banach spaces which are adapted to the operator ∆ζ (as in [Tat96]). The spaceẊbζ is defined by the norm

‖u‖Ẋbζ

= ‖∆bζu‖L2 ,

where ∆bζ is the Fourier multiplier operator with symbol pζ(ξ)b. Then (2.12) is equivalent to

‖q‖Ẋbζ→Ẋb−1

ζ≤ 1/2 (2.16)

On the other hand, since the spaces Ẋbζ and Ẋ−bζ are dual to each other, an estimate (2.14)

is equivalent to‖qi‖Ẋb

ζ1→Ẋ−b

ζ2

. 1. (2.17)

These two conditions will essentially coincide if we choose b = 1/2. Similarly, we will needto show that ‖∆−1ζ qi‖Ẋ1/2

ζ

= ‖qi‖Ẋ−1/2ζi

→ 0, which corresponds to the condition (2.15). Thiswill require a new argument of [HT13] based on averaging over choices of ζ.

19

Chapter 3

L2 estimates

3.1 The operator ∆ζ

In what follows, it will be convenient to take

ζ = τ(e1 − iη),

where |e1| = 1 and |η| ≤ 1. The conjugated Laplacian ∆ζ = e−x·ζ∆ex·ζ is a Fourier multiplierwith symbol

pζ(ξ) = −(ξ − iζ)2 = −(ξ − τη)2 + 2iτe1 · (ξ − τη) + τ 2.

We want to prove estimates that are uniform in τ , which we will take to be large. When|ξ| � τ , we have

|pζ | ∼ |ξ|2, (3.1)

so that ∆ζ is elliptic in this region. Thus the interesting behavior of ∆ζ occurs at frequenciesthat are less than or comparable to τ . In this case multiplication by τ is a worse operatorthan differentiation, so we will consider the term 2iξ · ζ to be of second order, and thus partof the principal symbol of pζ .

The characteristic set Σζ = {ξ : pζ(ξ) = 0} is the intersection of the plane perpendicularto e1 and a sphere centered at τη:

Σζ = {ξ : ξ · e1 = 0, |ξ − τη| = τ}.

We will refer to the distance d(ξ,Σζ) from this set as the modulation. We have

|pζ |2 = [(τ + |ξ − τη|)(τ − |ξ − τη|)]2 + 4τ 2((ξ − τη) · e1)2,

so symbol pζ is elliptic at high modulation and vanishes simply on Σζ :

|pζ | ∼

τd(ξ,Σζ) when d(ξ,Σζ) ≤ τ/8τ 2 + |ξ|2 when d(ξ,Σζ) ≥ τ/8 (3.2)

CHAPTER 3. L2 ESTIMATES 20

3.2 Dyadic projections

In order to study the operator ∆ζ , it will be useful to decompose Fourier space into dyadicregions where the modulation d(ξ,Σζ) does not vary too much. We will use the Greek lettersλ, µ, ν to represent dyadic integers of the form 2k, where k ∈ Z.

Fix a dyadic partition of unity, that is, a smooth nonnegative function ṁ : R>0+→ R≥0supported in [1/4, 1] such that ∑

λ∈2Zmλ(ρ) = 1 for ρ > 0,

wheremλ(ρ) = m(ρ/λ).

For example, we may take m = ψ(x) − ψ(2x), where ψ is smooth on [0, 1] and ψ = 1 on[0, 1/2].

For any dyadic integer λ, we define

m≤λ =∑µ≤λ

mµ.

By construction, m≤λ is smooth, bounded, and supported in [0, λ]. Similarly, we define m≥λ,mλ and so on.

Now let Σ ⊂ Rn be a closed set. We define EΣλ to be the set of ξ with modulationcomparable to λ:

EΣλ := {ξ : d(ξ,Σ) ∈ (λ/4, λ]}.Then the function mΣλ defined by

mΣλ (ξ) = mλ(d(ξ,Σ))

is supported in EΣλ .Let QΣλ , Q

Σ≤λ be the Fourier multipliers with symbols m

Σλ ,m

Σ≤λ. We will wish to distinguish

the cases λ ≤ τ/8 and λ & τ/8, so we define projections onto low and high modulation by

QΣl =∑λ≤τ/8

QΣλ

QΣh =∑λ>τ/8

QΣλ ,

where the λ vary over dyadic integers.Now take Σ = Σζ . By (3.2), we have

‖Qhu‖H1τ ∼ ‖u‖Ẋ1/2ζ

, (3.3)

where‖v‖H1τ = τ‖u‖L2 + ‖∇u‖L2 ,


so the Ẋ1/2ζ norm is only interesting at low modulation.

Since the functions mλ and mµ have disjoint support unless λ ∼ µ, we have the approxi-mate reproducing formula

1 = (∑λ

mλ)(∑µ

mµ)

=∑λ∼µ

mλmµ

In particular, by Cauchy-Schwartz we have

1 .∑λ

(mλ)2.

Thus we can write the Ẋbζ norms in the dyadic form

‖u‖2Ẋbζ∼∑λ∈Z

λ2b‖Qλu‖2L2 (3.4)

3.3 Littlewood-Paley theory

If we take Σ to be the origin, then we have

m{0}λ (ξ) = m(|ξ/λ|).

By abuse of notation we will usemλ to denote the functionmλ(|ξ|). We define the Littlewood-Paley projection Pλ to be the Fourier multiplier operator with symbol mλ. These projectionshave a particularly nice structure, because they are obtained by dilating a single functionm. In particular, by the dilation invariance of the Fourier transform, we have

Pλf = φλ ∗ f,

where φλ = λnφ(λx) and φ̂ = m. The Littlewood-Paley projections satisfy

• Almost orthogonality: The operators Pλ are self-adjoint and satisfy PλPµ = 0 unlessλ ∼ µ. In particular,

‖u‖2L2 ∼∑λ

‖Pλu‖2L2 .

• Lp boundedness: For any interval J , let PJ =∑λ∈J Pλ. Then for p ∈ [1,∞] we have

‖PJf‖Lp . ‖f‖Lp .

• Finite band property: For p ∈ [1,∞] we have

‖Pλf‖Lp ∼ λ−1‖∇Pλf‖Lp .

In two dimensions we also have

‖Pλf‖Lp ∼ λ−1‖∂Pλf‖Lp ,

where ∂ = ∂1 + i∂2.


• Bernstein’s inequality: For q ≥ p ≥ 1 we have

‖φλ ∗ f‖Lq . λn(1/p−1/q)‖f‖Lp ,

which holds not only for the Littlewood-Paley projections but more generally for anyφ such that ‖φ‖L1 ∼ 1.

• Square function estimate: For p ∈ (1,∞),

‖f‖Lp ∼∥∥∥∥∥∥(∑

λ

|Pλf(x)|2)1/2∥∥∥∥∥∥

Lp

.

3.4 Localization estimates

In this section we will show that functions in Ẋ1/2ζ are locally in L

2. More precisely, we willestablish that

‖u‖L2(B) .B τ−1/2‖∆1/2ζ u‖L2 , (3.5)

where B is a ball. This will allow us to show that

q : Ẋ1/2ζ → Ẋ

−1/2ζ ,

where q acts on X1/2ζ by multiplication. For example, if q is bounded and supported in some

ball B, then Hölder’s inequality and (3.5) give

|〈qu, v〉| .B ‖q‖L∞‖u‖L2B‖v‖L2B. τ−1‖q‖L∞‖u‖Ẋ1/2

ζ

‖v‖Ẋ

1/2ζ

.

By duality, this shows that‖q‖

Ẋ1/2ζ→Ẋ−1/2

ζ

. τ−1‖q‖L∞ .

The presence of the factor τ−1 suggests that there is some room for improvement in thisestimate. Since τ behaves like a derivative, we will be able to show that, essentially,

‖q‖Ẋ

1/2ζ→Ẋ−1/2

ζ

. ‖q‖W−1,∞ .

The idea behind (3.5) is the uncertainty principle. Essentially, the main obstacle toproving an elliptic estimate is the presence of the characteristic set, but once we localize inspace the singularity at the characteristic set is mitigated.

To see why this should be true, note that by (3.2) we have |pζ |1/2 & τ 1/2d(ξ,Σζ)1/2.Localization in space should blur the Fourier variable so that distances less than 1 are notdistinguished. This means that after localization, we have |pζ |1/2 & τ 1/2, which is exactlywhat we need to establish (3.5).


More precisely, localization to the unit ball allows us to replace u with φu, where φ = φ(x)for some fixed Schwartz function φ that is equal to one on the unit ball. The Fourier transformof φu is φ̂ ∗ û. We can interpret convolution with φ̂ as averaging on the unit scale.

Let v and w be two weights on Rd. Defining Tf = φ ∗ f , where φ is a rapidly decreasingfunction, we would like to find sufficient conditions for ‖T‖L2v→L2w to be bounded. This isequivalent to bounding ‖Sf‖L2→L2 , where Sf = w(ξ)1/2φ ∗ (v(ξ)−1/2f).

We prove the following lemma:

Lemma 3.4.1. Let v and w be nonnegative weights defined on Rd. If φ is a fixed rapidlydecreasing function, then

‖φ ∗ f‖L2w ≤ Cφ min{

supξ

√ˆJ(ξ, η)dη, sup

η

√ˆJ(ξ, η)dξ

}‖f‖L2v ,

where

J(ξ, η) = |φ(ξ − η)|w(ξ)v(η)

andCφ = Cd‖〈ξ〉Nφ‖L∞

for some large N depending on d.

Proof. We can write

‖Sf‖2L2 =ˆ Çˆ

φ(ξ − η)v(η)−1/2f(η)dηåÇˆ

φ(ξ − ζ)v(ζ)−1/2f(ζ)dζåw(ξ)dξ.

Applying the inequality ab ≤ 12(a2 + b2) we have

‖Sf‖2L2 ≤˚|φ(ξ − η)φ(ξ − ζ)|v(η)−1|f(η)|2w(ξ) dη dζ dξ.

Integrating first in ζ, we find that

‖Sf‖2L2 .¨|φ(ξ − η)|w(ξ)

v(η)|f(η)|2 dη dξ,

Equivalently, we may bound the adjoint S∗. To describe the adjoint, we compute

〈Sf, g〉 =ˆ Çˆ

φ(ξ − η)v(η)−1/2f(η)dηåw(ξ)1/2g(ξ)dξ

=

ˆf(η)

Çˆφ(ξ − η)g(ξ)w(ξ)

1/2

v(η)1/2dξ

ådη

= 〈f, S∗g〉,


so that

S∗g(η) =

ˆφ(ξ − η)g(ξ)w(ξ)

1/2

v(η)1/2dξ.

This means that bounding T from Lv to Lw is the same as bounding T∗g = φ(−ξ) ∗ g from

L2w−1 to L2v−1 . To do this it suffices to show that

ˆ|φ(ξ − η)|w(ξ)

v(η)dη =

ˆJ(ξ, η) dη

is uniformly bounded.

We use the preceding lemma to show that the homogeneous spaces Ẋ±1/2ζ are equivalent,

after localization, to the inhomogeneous spaces X±1/2ζ defined by

‖u‖Xbζ

= ‖(τ + |pζ(ξ)|)bû(ξ)‖L2 .

More precisely, we have

Lemma 3.4.2. Suppose that φ is a smooth function such that

Cφ = Cd∑

k+l≤N(n)‖〈x〉k∂lφ‖L∞


flatter as τ → ∞. For our purposes it will suffice to treat the case at hand. By translationand rotation, we can replace Σζ by

Σ = {ξ : |ξ| = τ, ξ1 = 0}.

We split the integral as´Rd−Σ1 +

´Σ1

, where

Σ1 = {ξ : d(ξ,Σ) ≤ 1}.

For the first term, we haveˆRd−Σ1

d(ξ,Σ)−1〈ξ − η〉−M dξ .ˆ〈ξ − η〉−Mdξ . 1.

Next we treat the integral over Σ1. Write ξ = (ξ1, ξ′), and pass to polar coordinates in ξ′.

Then for ξ = ξ1e1 + rω and η = η1e1 + tν, we have

〈ξ − η〉−M . 〈rω − tν〉−M . 〈rω − rν〉−M

(by the elementary inequality |rω − tν| ≥ 12

√r2 + t2|ω − ν|.) Also, note that

d(ξ,Σζ) & |r − τ |+ |ξ1|,

so that ˆΣζ,1

.ˆSn−2

ˆ 1−1

ˆ τ+1τ−1〈rω − rν〉−M(|r − τ |+ |ξ1|)−1rn−2drdξ1dω.

Since (|r − τ |+ |η1|)−1 is integrable with respect to drdη1,ˆΣζ,1

. (τ + 1)n−2ˆSn−2〈(τ − 1)ω − (τ − 1)ν〉−Mdω.

The quantity τn−2´Sn−2〈τω − τν〉−Mdω is uniformly bounded in τ , so

´Σζ,1

. 1.

In order to prove (3.6) and the adjoint estimate (3.7), we need to show that

ˆ|φ(ξ − η)| |p(η)|+ τ

|p(ξ)|dξ . 1, (3.9)

where p = pζ . We estimate φ(ξ − η) by 〈ξ − η〉−M with M large and split this into twointegrals

´|ξ|>100τ and

´|ξ|≤100τ .

When |ξ| ≥ 100τ , we have |p(ξ)| ∼ |ξ|2, soˆ|ξ|>100τ

.ˆ|ξ|>100τ

〈ξ − η〉−M |p(η)|+ τ|ξ|2

dξ.

When |η| > 8τ , we have |p(η)| . |η|2 . |ξ|2 + |ξ−η|2, so it is easy to see that the integralis bounded uniformly in τ .


On the other hand, when |η| < 4τ , we use that |p(η)| . τ 2, and |ξ−η| & τ in the domainof integration, and so the integral is bounded above by

τ−Nˆ|ξ|>100s

τ 2

|ξ|2〈ξ − η〉−Mdξ . 1

We now turn to the second integral, where we haveˆ|ξ|≤100τ

.ˆ|ξ|≤100τ

|φ(ξ − η)| |p(η)|+ ττd(ξ,Σζ)

dξ.

When |η| > 200τ , we have |ξ − η| ≥ τ , so the integral is bounded by

τ−Nˆ|ξ|2 + |ξ − η|2 + τ

τd(ξ,Σζ)〈ξ − η〉−Mdξ . 1,

by (3.8). On the other hand, when |η| ≤ 200τ , we have |p(η)| ∼ τd(η,Σζ), and by thetriangle inequality,

|p(η)|+ τ|p(ξ)|

.|ξ − η|+ d(ξ,Σζ) + 1

d(ξ,Σζ). 1 + d(ξ,Σζ)

−1〈ξ − η〉,

and our integral is bounded byˆ〈ξ − η〉−Mdξ +

ˆd(ξ,Σζ)

−1〈ξ − η〉−Mdξ.

The first integral is obviously finite, and the second integral is finite by (3.8).By Plancherel’s theorem and Lemma 3.4.1, the estimate (3.9) implies (3.6).

This lemma implies that we can replace the spaces Ẋ1/2ζ with X

1/2ζ in (2.16) and (2.17):

Corollary 3.4.3. Suppose q is supported in a compact set B, and suppose ζi · ζi = 0. Then

‖q‖Ẋ

1/2ζ1→Ẋ−1/2

ζ2

.B ‖q‖X1/2ζ1→X−1/2

ζ2

.

Proof. Let φ be a Schwartz function that it equal to one on B. Then

|〈qu, v〉| = |〈qφu, φv〉|. ‖q‖

X1/2ζ1→X−1/2

ζ2

‖φu‖X

1/2ζ1

‖φv‖X

1/2ζ2

. ‖q‖X

1/2ζ1→X−1/2

ζ2

‖u‖Ẋ

1/2ζ1

‖v‖Ẋ

1/2ζ2

Definition 3.4.1. We say that u ∈ Ẋ1/2ζ if u is a tempered function on Rn whose Fouriertransform is a function satisfying p

1/2ζ û ∈ L2(Rn).


Corollary 3.4.4. The space Ẋ1/2ζ with the norm described above is a Banach space contained

in H1loc and‖φu‖H1 .φ τ 1/2‖u‖Ẋ1/2

ζ

. (3.10)

for φ as in Lemma 3.4.2.

Proof. If u ∈ Ẋ1/2ζ , then by (3.7), the definition of the X1/2ζ norm and (3.2)

‖φu‖L2 . τ−1/2‖φu‖X1/2ζ

. τ−1/2‖u‖Ẋ

1/2ζ

.(3.11)

To estimate ‖∇(φu)‖H1 , note that E≤τ/8 ⊂ B(0, 3τ). Thus (3.5) and (3.3) imply that

‖∇(φQlu)‖L2 . τ−1/2‖Qlu‖Ẋ1/2ζ

+ τ−1/2‖∇Qlu‖Ẋ1/2ζ

. τ 1/2‖Qlu‖Ẋ1/2ζ

‖∇(φQhu)‖L2 . ‖Qhu‖H1. ‖u‖Ẋ1/2

Thus the estimate (3.10) holds. In particular, if ‖u‖Ẋ

1/2ζ

= 0, then u = 0.

Now suppose that un is a Cauchy sequence in Ẋ1/2ζ . We need to show that it has a

limit in Ẋ1/2ζ . By (3.11), the sequence un is Cauchy in L

2(〈x〉−N dx) for N sufficiently large.Thus there is some u ∈ L2(〈x〉−N dx) such that un → u in L2(〈x〉−N dx). In particular thisconvergence holds in the sense of tempered distributions. On the other hand, the sequenceûn is a Cauchy sequence in L

2(|pζ | dξ), and converges to some g ∈ L2(|pζ | dξ).We are done if we can show that g = û. Indeed, for any Schwartz function φ, we have

φun → φu in L2, which implies that φ̂ ∗ ûn → φ̂ ∗ û in L2. At the same time, the proof ofLemma 3.4.2 implies that φ̂ ∗ ûn → φ̂ ∗ g in the L2((|pζ |+ τ) dx) norm and thus in L2. Thusφ̂ ∗ g = φ̂ ∗ û for any test function φ, which implies that g = û.

In view of (2.16) and (2.17), we now show that

Lemma 3.4.5. Let f ∈ L∞(Rn). For ζ1, ζ2 ∈ Cn satisfying Re ζi = τ and |Im ζi| ≤ τ , wehave

‖f‖X

1/2ζ1→X−1/2

ζ2

. τ−1‖f‖∞ (3.12)

andsupj‖∂jf‖X1/2

ζ1→X−1/2

ζ2

. ‖f‖∞. (3.13)


Proof. By duality, it will suffice to show that

‖fuv‖L1 . τ−1‖f‖L∞‖u‖X1/2ζ1

‖v‖X

1/2ζ2

(3.14)

‖f∂i(uv)‖L1 . ‖f‖L∞‖u‖X1/2ζ1

‖v‖X

1/2ζ2

(3.15)

The first estimate (3.14) follows from Hölder’s inequality and the trivial estimate

‖w‖L2 . τ−1/2‖w‖X1/2ζ

. (3.16)

For the second estimate (3.15), we write u = Qhu + Qlu = uh + ul, and similarly for v. Bythe product rule, we have

∂i(uv) = ∂iuhv + u∂ivh + ∂iulv + u∂ivl.

By (3.2), we have

‖∂iwh‖L2 . ‖wh‖X1/2ζ

.

Together with (3.16), this suffices to control ‖∂iuhv + u∂ivh‖L1 .For the last two terms, we use the fact that ûl and v̂l are supported at frequencies less

than or comparable to τ , so the derivative operator is bounded by τ , i.e.

‖∂iwl‖L2 . τ‖wl‖L2 .

Combined with (3.16), this suffices to control ‖∂iulv + u∂ivl‖L1 .

If f is continuous, then we can show a bit more:

Lemma 3.4.6. Suppose that f in Lemma 3.4.5 is taken to be uniformly continuous. Then

‖∂if‖X1/2ζ1→X−1/2

ζ2

= oτ→∞(1).

Proof. Let Pλ denote the Littlewood-Paley projections. Write f = f≤τ1/2 + f>τ1/2 , wheref≤τ1/2 = P≤τ1/2f . By Lemma 3.4.5 and the finite band property,

‖∂if‖X1/2ζ1→X−1/2

ζ2

. ‖∂if≤τ1/2‖X1/2ζ1→X−1/2

ζ2

+ ‖∂if>τ1/2‖X1/2ζ1→X−1/2

ζ2

. τ−1‖∂if≤τ1/2‖L∞ + ‖f>τ1/2‖L∞

. τ−1/2‖f‖L∞ + ‖f>τ1/2‖L∞

Since f is uniformly continuous, ‖f>τ1/2‖L∞ → 0 as τ →∞. Taking limits, we conclude that‖m∂if‖ = oτ→∞(1).

29

Chapter 4

Calderón’s problem for Lipschitzconductivities

In this chapter we will prove



4.1 Existence of CGO solutions

Using the results of the previous section, we can construct solutions to the CGO equa-tion (2.11).

Lemma 4.1.1. Let n ≥ 3. Suppose that γ ∈ W 1,∞ satisfies 0 < c < γ < c−1 and γ = 1outside a fixed compact set B. Let q = γ−1/2∆γ1/2. If ‖∇ log γi‖L∞ is sufficiently small orγi ∈ C1, then for |Re ζ| sufficiently large, the equation

(∆ζ − q)ψ = h (4.1)

has a solution ψ in Ẋ1/2ζ for any h ∈ Ẋ

−1/2ζ . Furthermore, ψ satisfies the estimate

‖ψ‖Ẋ

1/2ζ

. ‖h‖Ẋ−1/2ζ

.

Proof. We write (4.1) as(I −∆−1ζ ◦ q)ψ = ∆−1ζ h.

We show that I −∆−1ζ q has a bounded inverse on Ẋ1/2ζ . For this it suffices to show that

‖∆−1ζ ◦ q‖Ẋ1/2ζ→Ẋ1/2

ζ

≤ 1/2.

CHAPTER 4. CALDERÓN’S PROBLEM FOR LIPSCHITZ CONDUCTIVITIES 30

Since ∆−1ζ is an isometry from Ẋ1/2 to Ẋ−1/2, this is the same as showing that

‖q‖Ẋ

1/2ζ→Ẋ−1/2

ζ

≤ 1/2.

By Corollary 3.4.3, it suffices to show that ‖q‖X

1/2ζ→X−1/2

ζ

is small. We have

q = 14|∇ log γ|2 + 1

2div(∇ log γ).

By Lemma (3.4.5), we have

‖q‖X

1/2ζ→X−1/2

ζ

. τ−1‖∇ log γ‖2L∞ + ‖∇ log γ‖L∞ .

We can make this small by taking τ large and ‖∇ log γ‖L∞ small. If γ ∈ C1, then

‖q‖X

1/2ζ→X−1/2

ζ

. τ−1‖∇ log γ‖2L∞ + oτ→∞(1) = oτ→∞(1)

by Lemma 3.4.6.

Recently, Caro and Rogers [CR14] constructed local solutions (which satisfy (−∆ζ+q)ψ = qin the set B) for γ ∈ W 1,∞ with no smallness condition, by conjugating the operator ∆ζ + qby a convex weight of the form e−M(x·e1)

2. Since this adds some technicalities, we will not

pursue this direction.

4.2 An averaged estimate

To obtain control of our solutions to the equation (∆ζ − mq)φ = q in Ẋ1/2ζ , it remains toestimate ‖q‖

Ẋ−1/2ζ

. The worst part of q looks like ∆ log g = ∇ · (∇ log g), and so we are ledto bound expressions of the form

‖∇f‖2Ẋ−1/2ζ

:=∑i

‖∂if‖2Ẋ−1/2ζ

,

where f is some continuous function with compact support. At high modulation, p(ξ)−1/2 ∼ (|ξ|+τ)−1,so ‖Qh∇f‖X−1/2 ≤ ‖f‖2. At low modulation, however, p(ξ) could be small. If we use thefact that q is localized to replace the Ẋ

−1/2ζ norm with the X

−1/2ζ norm, then we would have

the straightforward estimate

‖Ql∇f‖X−1/2ζ

. τ−1/2‖Ql∇f‖L2 .

In general, this will not be bounded unless f ∈ H1/2, which would require the conductivityγ to be in H3/2. This is why the previous work on Calderón’s problem for low regularity(i.e. [Bro96,PPU03,Bro03]) was restricted to conductivities with 3/2 derivatives.


To overcome this problem, we will exploit the fact that we have at least two degrees offreedom in choosing ζ. If we average over these parameters, we can obtain a better estimatethat does not involve a factor of τ 1/2.

Intuitively, this is possible because the bad case is when f̂ concentrates near the char-acteristic set Σζ , which is a codimension 2 sphere. If we fix a point ξ 6= 0 and vary theparameter ζ, then ξ will be far away from most of the spheres Σζ .

To make this precise, let us define the sense in which we are varying ζ. Given U in theorthogonal group O(n) and τ > 0, we define

ζ(τ, U) = τU(e1 − ie2),

where e1 and e2 are the standard basis vectors in Rn. Let µ denote Haar measure on O(n),normalized so that if σ is the usual spherical measure on Sn−1 and f : Sn−1 → R is integrable,then for any θ ∈ Sn−1 we have

ˆO(n)

f(U · θ) dµ(U) =ˆSn−1

f(ω) dσ(ω). (4.2)

The following key lemma was first proved in a slightly different form in [HT13]. Theproof was substantially simplified in [NS14], and we present the following streamlined versionfrom [Hab14].

Lemma 4.2.1. If f ∈ Ḣ−1, then

M−1ˆ 2MM

ˆO(n)

‖f‖2Ẋ−1/2ζ(τ,U)

dµ(U) dτ . ‖P≥100Mf‖2Ḣ−1 +M−1‖P


Writing v = (u1, u2), and integrating over the remaining variables, we bound by

1

MnMn−2

ˆB(0,2M)

(2|v1|+ |−|ξ|+ 2v2|)−1 dv ≤1

M2

ˆB(0,2M)

|v|−1 dv

∼ 1M

4.3 Proof of uniqueness

We now prove Theorem 2, which we state below for convenience. To simplify things, we firstrecord the following observation:

Lemma 4.3.1. Suppose ζ, ζ̃ ∈ Cn satisfy ζ · ζ = ζ̃ · ζ̃ = 0. Then

‖u‖Xbζ. (1 + |ζ − ζ̃|)|b|‖u‖Xb

ζ̃.

Proof. We have

|pζ | ≤ |pζ̃ |+ 2|(ζ − ζ̃) · ξ|≤ |pζ̃ |+ 2|ζ − ζ̃||ξ|. (1 + |ζ − ζ̃|)(|pζ̃ |+ τ)

by (3.2).



Proof. Fix r > 0 and three orthonormal vectors {e1, e2, e3}, and define

ζ1(τ, U) = τU(e1 − ie2)ζ2(τ, U) = −ζ1(τ, U)ζ̃1(τ, U) := τUe1 + i(rUe3 −

√τ 2 − r2Ue2)

ζ̃2(τ, U) := −τUe1 + i(rUe3 +√τ 2 − r2Ue2)

In what follows, all of inequalities will implicitly depend on r. For example, we have|ζi − ζ̃i| . 1. In particular, by Lemma 4.3.1, the spaces Xbζi and X

bζ̃i

have equivalent norms.

Using Lemma 4.1.1, there exist solutions ψi to the equations the equations (∆ζ̃i(τ,U)−mqi)ψi = qi,satisfying

‖ψi‖Ẋ1/2ζ̃i(τ,U)

. ‖q‖Ẋ−1/2ζ̃i(τ,U)

.


Note that by (3.10), such a solution lies in H1loc(Rn). This implies that the correspondingsolution vi = e

x·ζ̃i(τ,U)(1 +ψi) to the Schrödinger equation (∆− qi)vi lies in H1loc(Rn) as well.Let k = 2rUe3. By Lemma 2.1.6,

0 = 〈q1 − q2, eik·x(1 + ψ1)(1 + ψ2)〉= 〈q1 − q2, eik·x〉+ 〈q1 − q2, eik·xψ1ψ2〉+ 〈q1 − q2, eik·x(ψ1 + ψ2)〉

We need to show that the second and third terms are small. Let φ be a Schwartz functionthat is equal to one on the support of q. Then

|〈q1, eik·xψ1ψ2〉| = |〈q1e−ik·xψ2, ψ1〉|. ‖eik·xφψ2‖X1/2

ζ1(τ,U)

‖φψ1‖X1/2ζ1(τ,U)

= ‖e−ik·xφψ2‖X1/2ζ2(τ,U)

‖φψ1‖X1/2ζ1(τ,U)

. ‖ψ2‖Ẋ1/2ζ̃2(τ,U)

‖ψ1‖Ẋ1/2ζ̃1(τ,U)

. ‖q2‖X−1/2ζ2(τ,U)

‖q1‖X−1/2ζ1(τ,U)

since the seminorms of e−ik·xφ are bounded with a bound depending only on r. We canbound the q2 term in the same way. On the other hand, we have

|〈qi, eikψ1〉| . ‖qi‖X−1/2ζ̃1(τ,U)

‖ψ1‖Ẋ1/2ζ̃1(τ,U)

. ‖qi‖X−1/2ζ1(τ,U)

‖q1‖X−1/2ζ1(τ,U)

by duality of Ẋ1/2ζ1(τ,U)

and Ẋ−1/2ζ1(τ,U)

. The terms with ψ2 are the same. In summary, we obtain

|(q̂1 − q̂2)(2rUe3)| .∑

1≤i,j,k,l≤2‖qi‖X−1/2

ζj(τ,U)

‖qk‖X−1/2ζl(τ,U)

. (4.3)

Recalling that qi =12∆ log γi +

14|∇ log γi|2, we can apply (3.6) and Lemma 4.2.1 to obtain

M−1ˆ 2MM

ˆO(n)

‖qi‖2X−1/2ζj(τ,U)

dµ(U) dτ . ‖P≥100M∆ log γi‖2Ḣ−1 +M−1‖P


as M →∞. The right hand side does not depend on M , so this implies thatˆSn−1|(q̂1 − q̂2)(2rω)| dω = 0.

Since r is arbitrary, this implies that q̂1 = q̂2, so that q1 = q2. By Lemma 2.2.1, we concludethat γ1 = γ2.

35

Chapter 5

Unique continuation and Carlemanestimates

5.1 Unique continuation

Suppose that f is a real-analytic function on a connected open set Ω ⊂ Rn. If f vanishes toinfinite order at some point x0, then it is easy to see that it vanishes identically in Ω. Sincesolutions to the Laplace equation ∆u = 0 are real-analytic, the Laplace operator satisfiesthe strong unique continuation property (SUCP):

If ∆u = 0 in Ω and u vanishes to infinite order at a point x0 ∈ Ω, then u vanishesidentically in Ω.

If we perturb the Laplacian by adding smooth lower order terms, then solutions are no longeranalytic. However, the property SUCP still holds. The strongest theorem of this type is dueto Koch and Tataru. A simplified version of their results is as follows

Theorem 5.1.1 ([KT01]). Let P be an elliptic differential operator of the form

Pu =∑ij

∂i(gij(x)∂ju) +W1(x) · ∇u+∇(W2(x)u) + V (x)u,

where the coefficients g,Wi, V are defined in a ball B = B(x0, R). Assume gij satisfies the

uniform ellipticity condition

c|ξ|2 ≤ gijξiξj ≤ C|ξ|2 for ξ ∈ Rn \ 0

and that gij ∈ W 1,∞(B), Wi ∈ Ln(B) and V ∈ Ln/2(B). If u ∈ H1(B) is such that Pu = 0in B and ˆ

B(0,r)

|u|2 dx ≤ cNrN

for r < R and N > 0, then u vanishes identically in B.

CHAPTER 5. UNIQUE CONTINUATION AND CARLEMAN ESTIMATES 36

These assumptions are essentially sharp, in that the result is false when n ≥ 3 if theassumption on gij is replaced with a Hölder condition of order less than 1 [Pli63,Mil73,Man98]or if the exponent in the Lp assumptions on Wi and V is lowered (as can be seen by computingthe Laplacian of the function exp[−|lnx|1+�]).

5.2 The Carleman method

The first results on unique continuation which did not rely on analyticity are due to Carle-man [Car39]. Carleman’s method is based on weighted estimates of the form

‖eτφu‖ . ‖eτφ∆u‖,

where φ is a real-valued function and τ is a large parameter. In this section we will illustratehow to use such estimates to prove unique continuation theorems.

For simplicity we will focus on the weak unique continuation property (WUCP) for lower-order perturbations of the Laplacian. We say an elliptic operator P satisfies WUCP if

Pu = 0 in Ω, and u vanishes on a nonempty open set ⇒ u vanishes identicallyin Ω.

Let us first prove the following basic theorem:

Theorem 5.2.1. Let Ω ⊂ Rn be a connected open set, with n ≥ 3. Let A be a boundedvector field on Ω, and let V be a bounded function. If

(∆ + A · ∇+ V )u = 0

on Ω, and u ∈ H1(Ω) vanishes in an open subset of Ω, then u vanishes in Ω.

The proof of this theorem will be based on the following Carleman estimate

Lemma 5.2.2. Under the assumptions of Theorem 5.2.1, there exists some δ = δ(A, V ) > 0such that

‖eτx1u‖L2 . ‖eτx1(∆ + A · ∇+ V )u‖L2 (5.1)whenever u is supported in a ball of radius δ.

Before proving this, let us show how this implies a unique continuation theorem. First wewill show that unique continuation holds across spheres. The point is that a linear weight iswell-suited to proving unique continuation across planes, and the region in between a sphereand a plane just inside the sphere is a small compact set. Given an open set U , define

U+ = {x ∈ U : |x| > 1}U0 = {x ∈ U : |x| = 1}U− = {x ∈ U : |x| < 1}.


Lemma 5.2.3. Suppose that u ∈ H1(U) and (∆ + A · ∇ + V )u = 0 in U . If u = 0 in U+then u = 0 in a neighborhood of U0.

Proof. By compactness of the sphere and rotation invariance, it suffices to prove that if Ucontains the point (1, 0, . . . , 0), then u vanishes in a neighborhood of (1, 0, . . . , 0).

In order to apply Lemma 5.2.2 to u, we need to produce an approximate solutionu ∈ H1(Rn) to (∆ + A · ∇ + V )u = 0 which is localized to a small ball. Fortunately,the Carleman estimate (5.1) is compatible with such a localization.

Let χ(x1) be a smooth cutoff function, such that χ = 1 when x1 ≥ 1 − � and χ = 0when x1 ≤ 1 − 2�. By choosing � sufficiently small, we can ensure that the support of χuis contained in U−. In particular, the function χu extends by zero to a function in H

1(Rn).By taking � sufficiently small we can arrange that the support of χu is contained in a verysmall ball, such that Lemma 5.2.2 applies.

Since χ = 1 when x1 ≥ 1− �, we have

‖eτx1u‖L2(x1≥1−�) . ‖eτx1χu‖L2(Rn). ‖eτx1(∆ + A · ∇+ V )(χu)‖L2(Rn). ‖eτx1(∆χu+ 2∇χ · ∇u+∇χ · Au)‖L2(Rn),

since (∆ + A · ∇ + V )u = 0 in the support of χ. Now we use the fact that ∇χ = 0 awayfrom {x1 ∈ [1− �, 1− 2�]}:

‖eτx1u‖L2(x1≥1−�) . eτ(1−�)‖u‖H1(U)

In particular,‖u‖Lp(x1≥1−�/2) . e−�τ/2,

with a constant independent of τ . Letting τ →∞, we find that u vanishes when x1 ≥ 1−�/2.In particular, u vanishes in a neighborhood of (1, 0, . . . , 0).

Using the conformal inversion u 7→ |x|−(n−2)u(|x|−2x), we can prove that unique contin-uation holds across spheres in the opposite direction as well.

Lemma 5.2.4. Suppose 0 /∈ U . Under the assumptions of Lemma 5.2.3, if u = 0 in U−then u = 0 in a neighborhood of U0.

Proof. Let

ũ(x) = |x|−(n−2)u(|x|−2x)

A computation using polar coordinates shows that

∆ũ(x) = |x|−n∆u(|x|−2x). (5.2)


Let Ũ be the image of U under the inversion x 7→ |x|−2x. Then Ũ0 = U0 and Ũ± = U∓.Thus ũ vanishes in Ũ+. By (5.2), we have

(∆ + Ã · ∇+ Ṽ )ũ = 0.

where Ã and Ṽ are bounded. Applying Lemma 5.2.3, we find that ũ vanishes in a neighbor-hood of Ũ0. This implies, in turn, that u vanishes in a neighborhood of U0.

Finally, we conclude that WUCP holds for ∆ + A · ∇+ V :

Proof of Theorem 5.2.1. Let U be the maximal open set on which u vanishes. We will showthat ∂U ⊂ ∂Ω. This implies that Ω = U ∪ (Ω \U). Since Ω is connected and U is nonempty,we can conclude that Ω \ U = Ω \ U = ∅.

For each x ∈ U , we have x ∈ B(x, r(x)) ⊂ U with r(x) = d(x, ∂U). In particular, wemay write

U =⋃x∈U

B(x, r(x)),

so that∂U ⊂

⋃x∈U

∂B(x, r(x)).

Now let y0 ∈ ∂U be arbitrary, and choose some x0 ∈ U such that y0 ∈ ∂B(x0, r(x0)). Byconstruction, u vanishes in B(x0, r(x0)). If y0 is an interior point of Ω, then Lemma 5.2.4implies that u vanishes in a neighborhood of y0. But this means that y0 ∈ U , a contradiction.Thus y0 ∈ ∂Ω, and we have shown that ∂U ⊂ ∂Ω.

5.3 L2 Carleman estimates

In this section we will explain how to prove estimates of the form

‖eτx1u‖L2 . ‖eτx1(∆ + A · ∇+ V )u‖L2 .

To prove such an estimate, we show that the lower order terms are perturbative. This willfollow from estimates of the form

τ‖eτx1u‖L2 + ‖eτx1∇u‖L2 � ‖eτx1∆u‖L2 .

By making the change of variables v = eτx1u, we can reduce to showing that

‖v‖H1τ . ‖∆τx1v‖L2 ,

where‖v‖H1τ = τ‖v‖L2 + ‖∇v‖L2 ,

and∆τe1 = e

τx1∆e−τe1


is exactly the type of operator ∆ζ which we studied in Chapter 3. Suppose that u supportedin the unit ball B(0, 1). Then by (3.5), (3.10) and (3.7), we have

‖u‖H1τ . τ1/2‖u‖

Ẋ1/2τe1

= τ 1/2‖∆τx1u‖Ẋ−1/2τe1. τ 1/2‖∆τe1u‖X−1/2τe1. ‖∆τe1u‖L2 .

This shows thatτ‖eτx1u‖L2 + ‖eτx1∇u‖L2 . ‖eτx1∆u‖L2 . (5.3)

uniformly in τ .We will get a better estimate for the gradient term if u is supported in a smaller ball.

Indeed, if u is supported in B(x, r), then ur(x) = u(rx) is supported in the unit ball. Thus

r‖eτx1∇u(rx)‖L2 . r2‖eτx1∆u(rx)‖L2 ,

and by rescaling we have

‖eτx1/r∇u(x)‖L2 . r‖eτx1/r∆u‖L2 . (5.4)

Now suppose that A, V ∈ L∞. If r is sufficiently small and τ is sufficiently large, then (5.3)and (5.4) imply the perturbed estimate

‖eτx1u‖L2 . ‖eτx1(∆ + A · ∇u+ V )u‖L2

whenever u is supported in a ball of radius r. Thus we have proven Lemma 5.2.2 and theproof of Theorem 5.2.1 is concluded.

5.4 Lp Carleman estimates

The L2 → L2 Carleman estimates of the previous section require the coefficients to bebounded, since otherwise multiplication by the coefficient is not a bounded operator on L2.In order to get unique continuation for equations with unbounded coefficients, we will needto prove Carleman estimates in Lp spaces.

For operators of the form ∆+V , strong unique continuation under the sharp assumptionV ∈ Ln/2loc was first proven by Jerison and Kenig [JK85]. Later, Kenig, Ruiz and Sogge gave asimpler proof of the weak unique continuation property based on the following Lp Carlemanestimate for linear weights:

Theorem 5.4.1 ([KRS87]). Let n ≥ 3. The inequality

‖eτx1u‖Lp . ‖eτx1∆u‖Lp′ (5.5)

holds uniformly in τ for u ∈ W 2,p′(Rn), where 1/p + 1/p′ = 1 and 1/p′ − 1/p = 2/n (i.e.p′ = 2n/(n+ 2) and p = 2n/(n− 2).)


The numerology is such that by localizing this estimate, we can replace the Laplacian by∆ + V :

Corollary 5.4.2. Let V ∈ Ln/2(Rn) be fixed. Under the assumptions of Theorem 5.4.1,there exists δ(V ) > 0 such that

‖eτx1u‖Lp . ‖eτx1(∆ + V )u‖Lp′

whenever u is supported in a ball of radius δ.

Proof. Let � > 0, to be chosen later. By the uniform continuity of Lebesgue integrals, thereexists δ such that ‖V ‖Ln/2(B) < � for any ball B of radius δ. For u supported in such a ball,we have

‖eτx1u‖Lp(B) . ‖eτx1∆u‖Lp′ (B). ‖eτx1(∆ + V )u‖Lp′ (B) + ‖eτx1V u‖Lp′ (B). ‖eτx1(∆ + V )u‖Lp′ (B) + �‖eτx1u‖Lp(B).

By choosing � sufficiently small, we can absorb the second term into the left hand side of theequation to obtain the desired estimate.

By replacing Lemma 5.2.2 with Corollary 5.4.2, we can prove unique continuation for theoperator −∆+V with V ∈ Ln/2loc by the exact method which we used to prove Theorem 5.2.1.

As before, the estimate (5.5) is equivalent to

‖u‖Lp . ‖∆τe1u‖Lp′ ,

where ∆τe1 = eτx1∆e−τx1 .

The proof of Theorem 5.4.1 is based on an idea of Hörmander [Hör83], who observed thatsince the symbol pτ = pτe1 vanishes on a codimension-2 sphere, the Tomas-Stein restrictiontheorem gives improved Lp bounds for u when û is localized near Στ = {ξ : pτ (ξ) = 0}.

The usual statement of the Tomas-Stein theorem relates to the Fourier transform ofmeasures supported on the sphere.

Theorem 5.4.3 ([Ste93, Tom75]). Suppose p ≥ (2d + 2)/(d − 1). Let σ denote the surfacemeasure on Sd−1. Then

‖’f dσ‖Lp(Rd) . ‖f‖L2(Sd−1).Let τSd−1 denote the sphere of radius τ . Given a set E we define its λ-neighborhood

Nλ(E) := {ξ : d(ξ, E) ≤ λ}.

We use the following rescaled and localized variant of the restriction theorem:


Corollary 5.4.4. Let p be as above. Suppose that ĝ is supported in Nλ(τSd−1), where

λ ≤ τ/8. Then‖g‖Lp . λ1/2τ (d−1)/2−d/p‖ĝ‖L2(Nλ(τSd−1)

Proof. By Fourier inversion, we have

g(x) =

ˆĝ(ξ)eix·ξ dξ

=

ˆ τ+λτ−λ

ˆSd−1

ĝ(ρω)eiρ〈x,ω〉 ρd−1 dρ dσ

=

ˆ τ+λτ−λ

ρd−1(ĝ(ρω) dσ)∨(ρx) dρ

By Minkowski’s inequality, the restriction theorem, Cauchy-Schwarz, and Plancherel thisimplies that

‖g‖Lp .ˆ τ+λτ−λ‖(ĝ(ρω) dσ)∨(ρx)‖Lp(Rd) ρd−1 dρ

.ˆ τ+λτ−λ

ρ−d/p‖(ĝ(ρω) dσ)∨(x)‖Lp(Rd) ρd−1 dρ

. τ−d/pˆ τ+λτ−λ‖ĝ(ρω)‖L2(Sd−1) ρd−1 dρ

. τ−d/p(λτ d−1)1/2(ˆ τ+λ

τ−λ‖ĝ(ρω)‖2L2(Sd−1) ρd−1 dρ

)1/2= λ1/2τ (d−1)/2−d/p‖ĝ‖L2(Nλ(τSd−1)).

We now apply the restriction theorem to the dyadic regions

Eλ = {ξ : d(ξ,Σ) ∈ (λ/4, λ]}

defined in Section 3.2. Recall that Qλ is a Fourier multiplier which localizes to Eλ.

Lemma 5.4.5. Let p = 2n/(n−2), λ ≤ τ/8. Let ζ = τ(e1− iη), where |e1| = 1 and |η| ≤ 1.Then

‖Qλf‖p . (λ/τ)1/n‖∆1/2ζ f‖L2 . (5.6)

‖f‖p . ‖∆1/2ζ f‖L2 . (5.7)


Proof. Note that pζ(ξ) = pτe1(ξ−τη). Since Lp is invariant under multiplication by complexexponentials, and L2 is invariant under translation, we may assume that η = 0.

By rotation invariance, we may assume e1 = (1, 0, . . . , 0). We use the notation ξ = (ξ1, ξ′).

For (5.6), write g = Qλf . Note that

Eλ ⊂ {ξ : |ξ1| ≤ cλ, ||ξ′| − τ | ≤ cλ}.

We can write g = φλ ∗x1 g, where φλ(x1) = λφ(λx1) for some Schwartz function φ, andthe convolution is taken in the x1 variable only. By Minkowski’s inequality and Young’sinequality, we have

‖g‖p =∥∥∥∥∥ˆφλ(x1 − y1)g(y1, x′) dy1

∥∥∥∥∥p

.

∥∥∥∥∥ˆ|φλ(x1 − y1)|‖g(y1)‖Lp

x′dy1

∥∥∥∥∥Lpx1

. λ1/2−1/p‖g‖L2x1Lpx′ .

If we regard g as a function in the x′ variable, we see that its Fourier transform lies inNcλ(τS

n−2). By Corollary 5.4.4, we have ‖g(x1)‖Lpx′

. λ1/2τ (n−2)/2n‖ĝ(x1)‖L2x′

for each x1.

It follows that

‖g‖p . λ1/2λ1/2−1/pτ 1/2−1/n‖ĝ‖L2 . λ1/nτ−1/n‖∆1/2τ f‖L2 ,

since |pτ | ∼ τλ on Eλ.For (5.7), we apply (5.6) near Σζ and Sobolev embedding away from Σζ . On E≤τ/8 we

have

‖Qlf‖p .∑λ≤τ/8

‖Qλf‖p

.∑

(λ/τ)1/n‖Qλ∆1/2τ f‖L2

.Ä∑‖Qλ∆1/2τ f‖2L2

ä1/2≤ ‖∆1/2τ f‖L2

On E>τ/8 we have the Sobolev embedding

‖Qhf‖p . ‖Qhf‖H1 . ‖∆1/2τ f‖L2 ,

since |pτ | ∼ (τ + |ξ|)2 when ξ ∈ E>τ/8. Combining these estimates gives the claimed inequal-ity.

Now we have essentially proven Theorem 5.4.1:


Proof of Theorem 5.4.1. It suffices to show that the operator

∆−1τ : Lp′(Rn)→ Lp(Rn)

is bounded. We have already shown that ∆−1/2τ is bounded as a map from L2 → Lp, so

it suffices to show that it is bounded as a map from Lp′ → L2 as well. By duality, this is

equivalent to showing that ∆−1/2−τx1 is a bounded map from L

2 → Lp. But this is equivalentto the corresponding bound on ∆−1/2τx1 by reflection invariance.

5.5 Unbounded gradient terms

We now discuss some issues that arise in the unique continuation problem for operators ofthe form ∆ +A · ∇u+ V , where A is unbounded. Suppose for example that we assume thatA is merely bounded in Lqloc for some q < ∞. In order to apply the Carleman method asbefore, we would need to prove an gradient Carleman estimate of the form

‖eτφ∇u‖Lr . ‖eτφ∆u‖Lp′ (5.8)

where 1/p′ − 1/r = 1/q.Barcelo, Kenig, Ruiz and Sogge [BKRS88] observed that this strategy runs into some

problems when q is close to the optimal exponent n. They showed that no gradient Carlemanestimate can hold for linear weights φ = x1 unless r = p

′ = 2. In contrast, they proved uniquecontinuation for A ∈ L(3n−2)/2 and V ∈ Ln/2+ by establishing a gradient Carleman estimatewith the convex weight φ = x1 + x

21. Furthermore, they showed that for any weight φ, the

estimate (5.8) cannot hold uniformly in τ and u unless 1/p′ − 1/r ≤ 2/(3n− 2).The essential problem is that there are two competing phenomena associated with Car-

leman estimates. Suppose that u is localized to a ball of radius r. By using L2 estimates,which are based on localization, we have shown that for τ > r−1, the following estimateholds:

‖eτx1∇u‖L2 . r‖eτx1∆u‖L2 .

This estimate is good because it doesn’t blow up as τ →∞ and gets better when u is highlylocalized. However, it doesn’t allow us to multiply by functions in Lq for q


In [Wol92], Wolff proved WUCP for the operator ∆ + A · ∇ + V assuming only thatA ∈ Ln and V ∈ Ln/2. Wolff’s result relies on an argument that combines the localizationand Fourier restriction aspects of the Carleman estimate. This idea was later adapted byKoch and Tataru to prove SUCP for a more general class of operators (Theorem 5.1.1).

We will briefly describe some of the ideas involved in Wolff’s argument. We would liketo prove an estimate of the form

‖eτx1∇u‖L2(E) ≤ C(τ, E)‖eτx1∆u‖Lp′ (Rn),

where E is a rectangle centered at the origin. We write u = Qlu+Qhu. For Qhu this is justSobolev embedding (as long as C(1, τE) ≥ 1). On the other hand, for Qlu we can just useHölder’s inequality and the finite band property and reduce this to the Kenig-Ruiz-Soggeestimate:

‖Qlu‖H1(E) . |E|1/n‖Qlu‖W 1,p(E). |E|1/n‖∆x1Qlu‖W 1,p′ (Rn). τ |E|1/n‖∆x1u‖Lp′ (Rn)

Thus we have‖eτx1∇u‖L2(E) . (1 + τ |E|1/n)‖eτx1∆u‖Lp′ . (5.10)

This estimate has the correct Lp′

exponents, but blows up as τ → ∞. However, we mayobserve that the estimate is useful when u is highly localized, i.e. when τ ∼ |E|−1/n.

One might hope to patch together these localized Carleman estimates and deduce aunique continuation theorem. Suppose that u satisfies the differential inequality

|∆u| ≤ A|∇u|+ q|u|, (5.11)

with A ∈ Ln nonnegative, and that u vanishes in an open set. By inversion, we may reduceto the case where u : Rn \B(0, 1)→ R satisfies a similar equation and has compact supportin Rn \B(0, 1). We need to show that this is not possible unless u = 0.

Let K be the convex hull of suppu. By multiplying by an exponential weight, we canlocalize the problem to ∂K, as follows: Let χ be a cutoff function such that χ = 1 near ∂K.By (5.11),

|∆(χu)| ≤ A|∇(χu)|+ qχu+ ε, (5.12)where ε . |u|+ |∇u| and supp ε is contained in the interior of K.

Clearly we can replace A by a function which is bounded below by 1 without chang-ing (5.11). Thus we can use the function

F = A|∇(χu)|+ qχu

as a substitute for χu. In particular, we have

suppF = supp(χu).


Instead of applying the Carleman method to χu measured in some Sobolev norm, we replaceχu with F . This allows us to apply the Carleman method in a scale-invariant way.

For any θ ∈ Sn−1, the function x · θ restricted to the convex set K will have its largestvalue on ∂K. It follows that

‖eλx·θε‖Lp ≤ ‖eλx·θF‖Lp

for large λ, sinc

inverse problems with rough data · 2 complex geometrical optics solutions 11 ... the views of the...

Documents