Uniqueness for the inverse boundary value problem with singular potentials in 2D

Uniqueness for the inverse boundary value problem with singular potentials in 2D

Emilia Bl˚ asten

Leo Tzou

Jenn-Nan Wang

Abstract

In this paper we consider the inverse boundary value problem for the Schr¨odinger equation with potential in L^p class, p > 4/3. We show that the potential is uniquely determined by the boundary measurements.

1 Introduction

In this work we study the inverse boundary value problem for the Schr¨odinger equation with singular potentials in the plane. Let Ω ⊂ R²be an open bounded domain with Lipschitz boundary ∂Ω. Let q ∈ L^p(Ω) with p > 1 and assume that 0 is not a Dirichlet eigenvalue of the Schr¨odinger operator ∆ + q in Ω. Then the Dirichlet-Neumann map Λq : u|∂Ω7→ ∂νu|∂Ω, where u satisfies ∆u + qu = 0 in Ω, is well-defined (see [4, Lemma 5.1.3] for the precise statement). Here we are concerned with the unique determination of q from the knowledge of Λq, namely, whether

Λq₁ = Λq₂⇒ q1= q2. (1)

We will prove that (1) is indeed true for L^p(Ω) potentials with p > 4/3.

Prior to Bukhgeim’s breakthrough paper [7], there were some results on the inverse boundary value problem of recovering potentials in the plane. Local uniqueness results were obtained in [18], [19], [23] and a generic uniqueness result was proved in [20]. The determination of singularities of the potentials were studied in [17], [21], [22]. The inverse boundary value problem for the potentials is closely related to the Calder´on problem. The aim is to recover the conductivity γ by the Dirichlet-to-Neumann map. The global uniqueness question for the Calder´on problem with nonsmooth conductivities in two dimensions is more or less completely resolved, starting from the case γ ∈ W^2,p in [14], γ ∈ W^1,p in [6], γ ∈ L^∞ in [3], γ ∈ W^1,2 (local uniqueness) in [8], and finally, γ ∈ W^1,2 (global uniqueness) in [15].

Our method follows from the strategies introduced by Bukhgeim in [7] where he showed that (1) holds true for C¹potentials. The key ingredient in Bukhgeim’s method is the invention of special complex geometrical optics solutions with non-degenerate singular phases, namely, Φ(z) = (z − z0)², z, z0∈ C. Using this

∗HKUST Jockey Club Institute for Advanced Study, Hong Kong and Department of Math- ematics and Statistics, University of Helsinki, Finland. Email: emilia.blasten@iki.fi

†Faculty of Science, University of Sydney, Sydney, Australia. Email:

leo.tzou@sydney.edu.au

‡Institute of Applied Mathematical Sciences, NCTS, National Taiwan University, Taipei 106, Taiwan. Email: jnwang@math.ntu.edu.tw

type of complex geometrical optics solutions, Bukhgeim was able to prove the global uniqueness by the method of stationary phase. Bukhgeim’s result was later improved to L^p(Ω) with p > 2 in [11] and [5].

In this paper, we push the uniqueness result even further to p > 4/3. To do so, we need to prove the existence of complex geometrical optics solutions with phase Φ for such potentials. In fact, we show that such complex geometrical optics solutions exist for all potentials in L^p(Ω) with p > 1. The improvement relies on a new estimate for the conjugated Cauchy operator (see Lemma 4.3).

Having constructed the complex geometrical optics solutions, we then perform the usual step — substituting such special solutions into Alessandrini’s identity.

In order to obtain the dominating term containing the difference of potentials in the method of stationary phase, we need to derive more refined estimates of terms of various orders in Alessandrini’s identity. In this step, we need to use the fact that the knowledge the DN map improves the integrability of the potential. In other words, Λ_q1 = Λ_q2for q₁, q₂∈ L^p(Ω) with 3/4 < p < 2 implies q₁− q₂∈ L²(Ω) (see [17]).

We would also like to mention another related paper. It was shown in [13, Thm 2.3] that if p > 1 then for every z₀ ∈ Ω there exists a generic set of potentials in L^p for which its value in a neighbourhood of z0 is recoverable.

It was also remarked in [13] that the neighbourhood of z0 also depends on the chosen potential in the generic family which is determined by the choice of z0 ∈ Ω. Though the assumption on the L^p space of the potential is more general, the dependence of the generic set on the choice of the point z0∈ Ω and the dependence of the neighbourhood on the potential makes it unclear how a global identifiability result would follow from [13, Thm 2.3].

Intuitively, the uniqueness of the inverse boundary value problem is strongly related to the unique continuation property. For higher dimensions (n ≥ 3), it is known that the unique continuation holds for any solution u ∈ H^2,

2n n+2

loc (Ω) when q ∈ L^n/2_loc (scale-invariant potentials) [12], where H_loc^2,s(Ω) = {u ∈ L¹_loc(Ω) :

∆u ∈ L^s_loc(Ω)}. In this situation, the global uniqueness of the inverse boundary value problem with q ∈ L^n/2was established in [9] (also see related result in [10]

for n = 3, q ∈ W^−1,3). When n = 2, the unique continuation holds relative to u ∈ H_loc^2,sfor q ∈ L^p_loc with any p > 1, where s = max{1,_p+2^2p } [1]. Prior to [1], a weaker result which stated that the unique continuation property holds relative to u ∈ H_loc^2,2 for q ∈ L^p_loc with p > 4/3 was proved in [16]. Our uniqueness theorem of the inverse boundary value problem is consistent with the unique continuation result relative to u ∈ H_loc^2,2. It remains an interesting problem to close the gap of the uniqueness theorem for the inverse boundary value problem for q ∈ L^p(Ω) with 1 < p ≤ 4/3.

2 Main results

Theorem 2.1. Let q1, q2 ∈ L^p(Ω) with 4/3 < p < 2. Assume 0 is not a Dirichlet-eigenvalue of either potential and that their Dirichlet-Neumann maps are identical Λ_q1 = Λ_q2. Then q₁= q₂.

Let us fix some notational convention before stating our second theorem.

Throughout this text we shall always assume the following, which we call the usual assumptions: Let Ω, X ⊂ R² be bounded domains and Ω ⊂⊂ X. Fix a

cut-off function χ ∈ C₀^∞(X) such that χ ≡ 1 on Ω. Also, whenever we let a function belong to L^p(Ω) for any p we automatically extend it by zero to L^p(X).

Finally, let τ > 1 and z₀∈ C ≡ R² and set Φ(z) = (z − z₀)². Moreover should ϕj or Sj be mentioned, then they refer to definitions 3.5 and 3.3. We now state the existence of complex geometrical optics solutions.

Theorem 2.2. Let the usual assumptions hold and qj∈ L^p(Ω) with 1 < p < 2.

For j ∈ {1, 2} let β_j = β_j(z₀) be uniformly bounded over a parameter z₀ ∈ C.

Then for any z₀ we may define the function ϕ_j from Definition 3.5 and the series

f_j(z) =

∞

X

m=0

F_j,m(z) = e^{−iτ (Φ+Φ)}+ ϕ_j(z) + S_jϕ_j(z) + . . .

from Definition 3.6. The latter converges uniformly in the variable z ∈ X for τ large enough and

kFj,mk_∞≤ (Cτ^−α)^m

for some C = C(p, χ, qj, sup_z0|βj|), where α > 0 is a constant obtained in Proposition 4.4. Moreover we have fj ∈ W^1,2(X). Lastly, (∆ + q1)(e^{iτ Φ}f1) = 0 and (∆ + q2)(e^{iτ Φ}f2) = 0 in Ω.

3 Notation and CGO solution buildup

In this section we shall start by defining the operators and notation used in the rest of the paper. At the same time we reduce the construction of Bukhgeim- type [7] complex geometrical optics solutions to an integral equation.

We shall use the complex geometrical optics solutions for (∆ + qj)uj = 0 of the form

u1= e^{iτ Φ}f1, u2= e^{iτ Φ}f2.

The special form of f₁, f₂which we are going to use was first defined in [11] and used in [5] for proving uniqueness for the boundary value inverse problem when the potentials are in L^p, p > 2.

Definition 3.1. With the usual assumptions, define the differential operators

D1f = −4e^{−iτ (Φ+Φ)}∂(e^{iτ (Φ+Φ)}∂f ), D₂f = −4e^{−iτ (Φ+Φ)}∂(e^{iτ (Φ+Φ)}∂f ).

Lemma 3.2. Let the usual assumptions hold. For qj∈ L¹(Ω) and fj∈ L^∞(Ω) we have

(∆ + q1)(e^{iτ Φ}f₁) = 0 ⇔ D₁f₁= q1f₁, (∆ + q₂)(e^{iτ Φ}f₂) = 0 ⇔ D₂f₂= q₂f₂, all in Ω.

Proof. Use ∆ = 4∂∂ = 4∂∂ for distributions on Ω.

For inverting the operators D₁ and D₂ we will have to use conjugated ver- sions of the Cauchy operators ∂⁻¹and ∂⁻¹. We have included a short reminder of their properties in Section 6.

Definition 3.3. Let the usual assumptions hold. The we define the operators S₁, S₂ acting on f ∈ L^∞(X) by

S1f = −1

4∂⁻¹ e^{−iτ (Φ+Φ)}χ∂⁻¹(e^{iτ (Φ+Φ)}q1f )
, S2f = −1

4∂⁻¹ e^{−iτ (Φ+Φ)}χ∂⁻¹(e^{iτ (Φ+Φ)}q2f )
.

Remark 3.4. We have DjSjf = qjf in Ω (but not in X \ Ω).

Definition 3.5. Let the usual assumptions hold and qj∈ L¹(Ω). Then for any given z₀∈ C and βj = β_j(z₀), we define functions of z ∈ X by

ϕ1=1

4∂⁻¹ e^{−iτ (Φ+Φ)}χ(β1(z0) − ∂⁻¹q1)
, ϕ₂=1

4∂⁻¹ e^{−iτ (Φ+Φ)}χ(β₂(z₀) − ∂⁻¹q₂)
.

Note that ∂⁻¹q₁ and ∂⁻¹q₂ inside the parenthesis do not depend on z0. For example

ϕ₁(z) = 1 4π

Z e^{−iτ ((z0−z0)2+(z0−z0)2)}χ(z⁰) β₁(z₀) − ∂⁻¹q₁(z⁰)

z − z⁰ dm(z⁰).

Definition 3.6. Let the usual assumptions hold. For j ∈ {1, 2} define

fj= e^{−iτ (Φ+Φ)}+

∞

X

m=0

S_j^mϕj

where Sj is as in Definition 3.3 and ϕj as in Definition 3.5. For convenience we write

F_j,0= e^{−iτ (Φ+Φ)}, F_j,m= S_j^m−1ϕ_j when m ∈ N, m ≥ 1. Hence f^j=P∞

m=0Fj,m.

We have made enough definitions now to show the structure of the complex geometrical optics solutions. Given z0 ∈ C and βj = βj(z0) we can show formally that if

u1= e^{iτ Φ}f1, u2= e^{iτ Φ}f2, then (∆ + qj)uj = 0 in Ω. This follows from writing

fj = e^{−iτ (Φ+Φ)}+ ϕj+ Sj(fj− e^{−iτ (Φ+Φ)}),

applying Dj and noting that Dj(e^{−iτ (Φ+Φ)}) = 0, DjSj is the multiplication operator by q_j in Ω, and D_jϕ_j = q_je^{−iτ (Φ+Φ)} in Ω. Then D_jf_j = q_jf_j and Lemma 3.2 gives the rest. For proving actual existence and estimates, see the proof at the end of the next section.

4 Estimates for conjugated operators and CGO existence

In this section we will start by showing that a fundamental operator of the form a 7→∂⁻¹(e^{−iτ (Φ+Φ)}a) has decay properties as τ → ∞. Section 7 contains the technical cut-off function estimates. Once estimates for this fundamental operator have been shown we can prove the required estimates for Sj from Definition 3.3. At the end of this section all the details for proving the existence of complex geometrical optics solutions for L^p-potentials with 1 < p < 2, i.e.

Theorem 2.2, will be given. From now on, we define p∗ satisfying the relation 1

p =1 2 + 1

p∗. Thus, we have p∗ > 2.

Definition 4.1. Let the usual assumptions hold. Define the operator T by

T a = ∂⁻¹(e^{−iτ (Φ+Φ)}a).

Lemma 4.2. Let the usual assumptions hold and T as in Definition 4.1. Then, for p∗ > 2 we can extend T to a mapping W₀^1,p∗(X) → L^∞(X) with norm estimate

kT ak_L∞(X)≤ Cτ^−1/p∗kak_W1,p∗(X),

where W₀^1,p∗(X) is the completion of C₀^∞(X) under the W^1,p∗-norm.

Proof. Let ψ ∈ C₀^∞(R²) be a test function supported in B(¯0, 2) with 0 ≤ ψ ≤ 1 and ψ ≡ 1 in B(¯0, 1). Write ψ_τ(z) = ψ(τ^1/2(z−z₀)). Let h(z) = (1−ψ_τ(z))/(z−

z₀). By integration by parts (Lemma 6.3) we have

∂⁻¹(e^{−iτ (Φ+Φ)}a) = ∂⁻¹ e^{−iτ (Φ+Φ)}ψτa

− 1

2iτ e^{−iτ (Φ+Φ)}ha − ∂⁻¹(e^{−iτ (Φ+Φ)}∂ha) − ∂⁻¹(e^{−iτ (Φ+Φ)}h∂a)
.

Then recall that by Lemma 6.4 we have ∂⁻¹: L^p∗(X) → W^1,p∗(X), the latter of which is embedded into L^∞(X) since p∗ > 2. Hence, by taking the L^∞(X)-norm we have

kT ak_∞≤ C kψτak_p∗+ τ⁻¹(khak_∞+ ∂ha

_p∗+ h∂a

_p∗)
.

The claim follows from H¨older’s inequality and lemmas 7.1 and 7.3 after esti- mating

kT ak_∞≤ C kψτk_p∗+ τ⁻¹(khk_∞+ ∂h

_p∗)
(kak_∞+ ∂a

_p∗) and noting that τ^−1/2 ≤ τ^−1/p∗since τ > 1.

Lemma 4.3. Let the usual assumptions hold and T be as in Definition 4.1.

Assume that 2 < p∗ < ∞ and 1/2 + 1/p∗ ≥ 1/q > 1/2. Then we can extend T to a mapping W₀^1,q(X) → L^p∗(X) with norm

kT ak_Lp∗(X)≤ Cτ^{1/q−1−1/p∗}kak_W1,q(X)

where C = C(p, q, X).

Proof. Let ψ ∈ C₀^∞(R²) be a test function supported in B(0, 2) with 0 ≤ ψ ≤ 1 and ψ ≡ 1 in B(0, 1). For τ > 0 and z₀ ∈ R² write ψ_τ(z) = ψ(τ^1/2(z − z₀)).

Let h(z) = (1 − ψτ(z))/(z − z₀). Integration by parts gives

∂⁻¹(e^{−iτ (Φ+Φ)}a) = ∂⁻¹ e^{−iτ (Φ+Φ)}ψτa

− 1

2iτ e^{−iτ (Φ+Φ)}ha − ∂⁻¹(e^{−iτ (Φ+Φ)}∂ha) − ∂⁻¹(e^{−iτ (Φ+Φ)}h∂a)
by Lemma 6.3.

Sobolev embedding and Lemma 6.4 imply that ∂⁻¹ : L^p(X) → L^p∗(X).

Taking the L^p∗(X)-norm gives

kT ak_p∗≤ C kψτak_p+ τ⁻¹(khak_p∗+ ∂ha

_p+ h∂a

_p)
.

Again, recall that W^1,q(X) ,→ L^q∗(X) where 1/q∗ = 1/q − 1/2. H¨older’s inquality gives

kψτak_p≤ kψτk_r

1kak_q∗, 1

r1

=1 p− 1

q∗ = 1 + 1 p∗ −1

q, khak_p∗≤ khk_r

2kak_q∗, 1

r₂ =1 2 + 1

p∗ −1 q, ∂ha

_p≤ ∂h

_r

1kak_q∗, h∂a

_p≤ khk_r

2

∂a _q. Lemmas 7.1 and 7.3 then give

kψτak_p≤ Cτ−1−1/p∗+1/qkak_q∗, khak_p∗ ≤ Cτ^{−1/p∗+1/q}kak_q∗, ∂ha

_p≤ Cτ^{−1/p∗+1/q}kak_q∗

h∂a

_p≤ Cτ^{−1/p∗+1/q} ∂a

_q, which implies the claim.

Proposition 4.4. Let the usual assumptions hold and let q1, q2∈ L^p(Ω) where 1 < p < 2. Then we can extend S1 and S2 from Definition 3.3 to the following maps with corresponding norm estimates

S_j: L^∞(X) → L^p∗(X), kSjf k_p∗≤ Cτ^−1/2kf k_∞, Sj : L^∞(X) → L^∞(X), kSjf k_∞≤ Cτ^−αkf k_∞,

where C = C(p, χ) kqjk_pand 0 < α < 1/p with α = α(p). If in addition p > 4/3 then we have the extension

Sj : L^p∗(X) → L^p∗/2(X), kSjf k_p∗/2≤ Cτ^−1/2kf k_p∗.

Proof. We shall prove the claim for j = 1. The other case follows similarly. Us- ing the notation from Lemma 4.3 we can write −4S1f = T (χ∂⁻¹(e^{iτ (Φ+Φ)}q1f )).

The lemma combined with Lemma 6.4 gives us kS1f k_q∗≤ Cτ^−1/2

χ∂⁻¹(e^{iτ (Φ+Φ)}q1f )

_W1,q ≤ Cτ^−1/2kq1f k_q

whenever 2 < q∗ < ∞ and 1/q = 1/2 + 1/q∗. For the first estimate choose q = p, q∗ = p∗, and for the third one 1/q = 1/p + 1/p∗, q∗ = p∗/2. H¨older’s inequality implies the rest.

The second claim follows by interpolation. Let 2 < Q < ∞ and 1 < q < p.

If q1∈ L^Q(X) then by Lemma 4.2 kS₁f k_∞≤ Cτ^−1/Q

χ∂⁻¹(e^{iτ (Φ+Φ)}q₁f ) _W1,Q

and Lemma 6.4 gives the bound Cτ^−1/Qkq1k_Qkf k_∞· The latter lemma and Sobolev embedding imply that ∂⁻¹ : L^q → L^q∗ and ∂⁻¹ : L^q∗ → L^∞. Hence kS1f k_∞≤ C kq1k_qkf k_∞. Since q < p < Q and 1/Q > 0 interpolation gives us the second estimate with some α > 0.

Lemma 4.5. Let the usual assumptions hold and qj ∈ L^p(Ω) with 1 < p < 2.

Then ϕj, the function of z ∈ X given by Definition 3.5, is in L^∞(X) with norm kϕjk_∞≤ Cτ^−α

for α > 0 as in Proposition 4.4, and is in L^p∗(X) satisfying kϕjk_p∗≤ Cτ^−1/2,

where C = C(p, χ)(kqjk_p+ |βj(z0)|).

Proof. Note that ϕ₁=¹₄β₁(z₀)∂⁻¹ e^{−iτ (Φ+Φ)}χ
+S1(e^{−iτ (Φ+Φ)}) and use Propo- sition 4.4 and lemmas 4.2 and 4.3.

Proof of Theorem 2.2. By Proposition 4.4 and Lemma 4.5 we have

kSjf k_∞≤ C(p, χ) kqjk_pτ^−αkf k_∞, kϕjk_∞≤ C(p, χ)(kqjk_p+ |βj(z0)|)τ^−α where these norms are over the variable z ∈ X. Hence we get kFj,mk_∞ ≤ C^mτ^−mαfor some C = C(p, χ, qj, sup_z0|βj(z0)|) when m ≥ 0. If τ > C^1/αthen the series for fj converges in L^∞(X).

The following observations, which are each easy to check inD⁰(X), imply that Djfj = qjfj in Ω. Note that βj are functions of the parameter z0 but constant in the variable z. Recall the definitions 3.1, 3.3 and 3.5 of Dj, Sj and ϕ_j. Then

• f_j= e^{−iτ (Φ+Φ)}+ ϕ_j+ S_j(f_j− e^{−iτ (Φ+Φ)}),

• D_j(e^{−iτ (Φ+Φ)}) = 0,

• DjSjf = qjf in Ω,

• Djϕj= qje^{−iτ (Φ+Φ)} in Ω.

Lemma 3.2 shows that we indeed get solutions to (∆ + qj)uj= 0.

Recall that 1/p = 1/2 + 1/p∗. Then by Lemma 6.4 we have ∂⁻¹, ∂⁻¹ : L^p∗ → W^1,p∗ which embeds to W^1,2 locally since p∗ > 2. Moreover by Sovolev embedding we have ∂⁻¹, ∂⁻¹ : L^p → L^p∗. Hence by the first item above fj ∈ W^1,2(X).

5 Proof of the main result

We will prove Theorem 2.1 in this section. The proof will be split into several lemmas. By Alessandrini’s identity

Z

Ω

(q1− q2)u1u₂dx = 0

for solutions u_j to (∆ + q_j)u_j= 0 in Ω. For any parameter z₀∈ C let uj be the complex geometrical optics solution given by Theorem 2.2. Recall that they are defined in X ⊃ Ω but are solutions only in Ω. Then the product u₁u₂ will be a series of terms, and these will have to be estimated carefully. Lemmas 5.1 – 5.6 deal with this. The main proof follows.

Lemma 5.1. Let the usual assumptions hold and 1 < p < 2 with q₁, q₂∈ L^p(Ω).

Let f₁, f₂∈ L^∞(X) be as in Theorem 2.2 and set u₁= e^{iτ Φ}f₁ and u₂= e^{iτ Φ}f₂. Then

2τ π

Z

(q1− q2)u1u2dx =

∞

X

k+l=0

2τ π

Z

(q1− q2)e^{iτ (Φ+Φ)}F1,kF2,ldx

where k, l ≥ 0 and the sum converges in the L^∞(X)-norm with respect to z0. Proof. We can estimate

Z ^∞

X

k+l=N

(q₁− q2)e^{iτ (Φ+Φ)}F_1,kF_2,ldx     

≤

Z ^∞

X

k+l=N

|q1− q2| |F1,k| |F2,l| dx

and use the z0-independent estimates for Fj,mfrom Theorem 2.2 to see that the remainder tends to zero as N → ∞. Hence the sum can be taken out of the integral and the claim follows.

Lemma 5.2. Let the usual assumptions hold and q1, q2 ∈ L^p(Ω) with 4/3 <

p < 2. For j ∈ {1, 2} and m ∈ N take Fj,m as in Definition 3.6.

Then if q₁− q2∈ L²(Ω) we have 

   τ

Z

(q1− q2)e^{iτ (Φ+Φ)}F1,kF2,ldx    

≤ C^k+lτ^{−(k+l−2)α}.

when k+l ≥ 3. Here C = C(p, χ, kq1k_p, kq₂k_p, kq₁− q2k₂)(1+|β1(z0)|+|β2(z0)|) and α > 0 is as in Proposition 4.4.

Proof. We may assume that k ≥ l. By H¨older’s inequality the integral can be estimated with

Cτ kq₁− q2k₂kF1,kk_p∗/2kF2,lk_∞

because p > 4/3 imples p∗ > 4 for 1/p = 1/2 + 1/p∗ and then 1/2 + 2/p∗ ≤ 1.

Proposition 4.4 and Lemma 4.5 imply the following estimates kF_j,0k_∞= 1, kF_j,1k_∞≤ Cτ^−α, kF_j,1k_p∗≤ Cτ^−1/2, kFj,m+1k_∞≤ Cτ^−αkFj,mk_∞,

kF_j,m+1k_p∗≤ Cτ^−1/2kF_j,mk_∞, kFj,m+1k_p∗/2≤ Cτ^−1/2kFj,mk_p∗

for j ∈ {1, 2} and m = 1, 2, . . .. These imply

kF1,kk_p∗/2 ≤ C^kτ^{−1−(k−2)α}, kF2,lk_∞≤ C^lτ^−lα for k ≥ 2, l ≥ 0. The claim is direct consequence.

From Lemma 5.2, we can see that the higher order terms decay in τ whenever k + l ≥ 3. A more refined estimate shows that the term of k + l = 2 also decays.

Lemma 5.3. Let the usual assumption hold and q1, q₂∈ L^p(Ω) with 4/3 < p <

2. For j ∈ {1, 2} and m ∈ N let Fj,m be as in Definition 3.6. Assume that q₁− q₂∈ L²(Ω). For k + l = 2, we have

    τ

Z

(q1− q2)e^{iτ (Φ+Φ)}F1,kF2,ldx    

≤ Cτ^1/p−3/4,

where the constant C is of the form C(p, χ, kq₁k_p, kq₂k_p, kq₁− q₂k₂)(1+|β₁(z₀)|+

|β₂(z₀)|).

Proof. We can assume that k ≥ l. There are two cases: k = 2, l = 0 and k = l = 1. Start with the first one. The integral with F1,2F2,0 is

−1 4τ

Z

(q1− q2)∂⁻¹ e^{−iτ (Φ+Φ)}χ∂⁻¹(e^{iτ (Φ+Φ)}q1ϕ1)
dx

by Definition 3.3. We have q1− q2 ∈ L²(Ω) and hence we should take the L^r∗(X)-norm of the remaining factor for any r∗ ≥ 2.

We note that ϕ1∈ L^p∗(X) and q1∈ L^p(X). Hence their product is in L^q(X) with 1/q = 1/p+1/p∗ = 2/p−1/2. Choose 1/r∗ = 1/p−1/4. Then 2 < r∗ < ∞ and 1/2 + 1/r∗ ≥ 1/q > 1/2 since 4/3 < p < 2. Hence by Lemma 4.3

∂⁻¹ e^{−iτ (Φ+Φ)}χ∂⁻¹(e^{iτ (Φ+Φ)}q1ϕ1)

_r∗≤ Cτ^{1/q−1−1/r∗}kq1k_pkϕ1k_p∗

and the exponential is 1/q − 1 − 1/r∗ = 1/p − 5/4. Recall that kϕ₁k_p∗≤ Cτ^−1/2 by Lemma 4.5. The claim for k = 2, l = 0 follows.

In the case k = l = 1 note that β₁(z₀) − ∂⁻¹q₁ ∈ W^1,p(X) by Lemma 6.4.

Note that 4/3 < p < 2 implies 1/2 + 1/4 ≥ 1/p > 1/2. Then by Lemma 4.3 kϕ1k₄≤ Cτ^{1/p−1−1/4}

β1(z0) − ∂⁻¹q1

_W1,p ≤ Cτ^1/p−5/4(|β1(z0)| + kq1k_p).

When k = l = 1, the absolute value of the integral in the lemma statement becomes

    τ

Z

(q1− q2)e^{iτ (Φ+Φ)}ϕ1ϕ2dx    

≤ kq1− q2k₂τ kϕ1k₄kϕ2k₄

by H¨older’s inequality. The claim follows since τ^2/p−6/4< τ^1/p−3/4when τ > 1 and p > 4/3.

We recall the method of stationary phase and its convergence in the L²-sense before proceeding to deal with terms of order one and zero in the Alessandrini identity.

Lemma 5.4. For z₀∈ C, Φ(z) = (z − z0)² and τ ∈ R define the operator Ef (z0) =2τ

π Z

e^{−iτ (Φ+Φ)}f (z)dm(z)

for f ∈ C₀^∞(C). Here dm(z) is the two-dimensional Lebesgue measure in C.

Then E can be extended to a unitary operator on L²(C) such that

τ →±∞lim kEf − f k₂= 0.

Proof. Consider the function z 7→ 2τ exp(−i(z²+z²))/π defined on C ≡ R². Its Fourier transform is exp(i(ξ² + ξ²)/(16τ )) by for example [5]. We have Ef = ^2τ_πe^−i(z2+z2)∗ f and henceF {Ef} (ξ) = e^{iξ2 +ξ16τ2}f (ξ). Parseval’s theoremˆ implies the unitary extension to L²(C). When τ → ±∞ the exponential tends to 1 pointwise. Dominated convergence and Parseval’s theorem imply the second claim.

The following way of dealing with the first order terms comes from [11, 5].

Lemma 5.5. Let the usual assumptions hold and q1, q2 ∈ L^p(Ω) with 4/3 <

p < 2. For j ∈ {1, 2} and m ∈ N, let F^j,m be as in Definition 3.6. Moreover let βj∈ L^∞(X) with respect to the z0-variable. Then

τ →∞lim

2τ π

Z

(q1− q2)e^{iτ (Φ+Φ)}F1,kF2,ldx ₂

≤ C

β2− ∂⁻¹q2

_p∗kq1− q2k_p for k = 1, l = 0, where the L²(X)-norm is taken over the variable z₀ and C = C(p, χ). A similar bound holds for k = 0 and l = 1.

Proof. Recall that ϕ2 = ¹₄∂⁻¹ e^{−iτ (Φ+Φ)}χ(β2(z0) − ∂⁻¹q2)
and hence the in- tegral becomes

τ 2π

Z

(q₁− q2)∂⁻¹ e^{−iτ (Φ+Φ)}χ(β₂(z₀) − ∂⁻¹q₂)
dx when k = 1, l = 0. By Fubini’s theorem this is equal to

− τ 2π

Z

e^{−iτ (Φ+Φ)}χ(β₂(z₀) − ∂⁻¹q₂)∂⁻¹(q₁− q2)dx, and using the stationary phase operator of Lemma 5.4 it is equal to

1

4E χ∂⁻¹q₂∂⁻¹(q1− q2)
(z0) −1

4β₂(z0)E χ∂⁻¹(q1− q2)
(z0).

We have ∂⁻¹(q₁− q₂) ∈ L^p∗(X) where p∗ > 4 since p > 4/3 (e.g. Lemma 6.4 and Sobolev embedding). Similarly χ∂⁻¹q2∈ L^p∗(C). Their product is in L²(C) since χ has compact support. Hence the operator E is being applied to L²(C)-functions above. Since z⁰ 7→ β2(z0) is uniformly bounded, the above converges to

1

4χ(∂⁻¹q₂− β2)∂⁻¹(q1− q2)

in the L²(C)-norm with respect to z⁰ as τ → ∞ by Lemma 5.4. The claim follows from the norm estimates at the beginning of this paragraph.

Lemma 5.6. Let the usual assumptions hold and q₁− q₂∈ L²(Ω). Then

τ →∞lim

2τ π

Z

(q₁− q₂)e^{iτ (Φ+Φ)}F_1,0F_2,0dx − (q₁− q₂) ₂

= 0,

where Fj,0 are as in Definition 3.6 and the norm is taken with respect to the variable z0∈ C.

Proof. This follows directly from Fj,0 = exp(−iτ (Φ + Φ)) and the stationary phase Lemma 5.4.

We are ready to prove uniqueness for the inverse problem with potential in L^p, 4/3 < p < 2.

Proof of Theorem 2.1. In view of Green’s identity and the symmetry of the DN map, we can see that the condition of identical Dirichlet-Neumann maps imply

that Z

Ω

(q1− q2)u1u2dx = 0

for any solution u_j ∈ W^1,2(Ω) to (∆ + q_j)u_j = 0 in Ω. We also note that Theorem 3 in [17] implies that q₁− q₂ ∈ H^t(Ω) for any t < 3 − 4/p. Hence q1− q2∈ L²(Ω).

Let ε > 0 and take βj∈ C₀^∞(X) such that

kβ₁− ∂⁻¹q₁k_Lp∗(X)< ε, kβ₂− ∂⁻¹q₂k_Lp∗(X)< ε (2) which is possible since ∂⁻¹q₁, ∂⁻¹q₂ ∈ L^p∗ by Sobolev embedding and Lemma 6.4. Let z₀∈ C and from now β1and β₂shall be evaluated at z₀if not mentioned otherwise, and note that they are uniformly bounded. Then, given τ > 1 large enough let u₁ = e^{iτ Φ}f₁ and u₂ = e^{iτ Φ}f₂ be the solutions in the variable z with parameter z₀, given by Theorem 2.2. They are in W^1,2(X) and satisfy (∆ + qj)uj = 0 in Ω. By Lemma 5.1 we have

2τ π

Z

(q1− q2)u1u2dx =

∞

X

k+l=0

2τ π

Z

(q1− q2)e^{iτ (Φ+Φ)}F1,kF2,ldx.

In view of lemmas 5.2 and 5.3

∞

X

k+l=2

2τ π     Z

(q₁− q₂)e^{iτ (Φ+Φ)}F_1,kF_2,ldx    

≤ Cτ^1/p−3/4+

∞

X

k+l=3

C^k+lτ^{−(k+l−2)α}

= C τ^1/p−3/4+

∞

X

N =3

(N + 1)(Cτ )^{−(N −2)α}

!

for any z0 ∈ R², and where C = Cp,q₁,q₂,Ω(1 + kβ1k_∞+ kβ2k_∞). Recall that p > 4/3 so the first exponent is negative. Note that for τ sufficiently large, (Cτ )^−α< 1 so the sum can be rewritten as

∞

X

N =3

(N + 1)((Cτ )^−α)^{N −2}=

∞

X

N =1

(N + 1)((Cτ )^−α)^N + 2

∞

X

N =1

((Cτ )^−α)^N

= 1

(1 − (Cτ )^−α)² − 1 + 2(Cτ )^−α 1 − (Cτ )^−α,

which tends to zero as τ → ∞. Hence the sum of the terms with k + l ≥ 2 in the original sum tends to zero when β_j are fixed.

For the terms with k + l ∈ {0, 1} we will use lemmas 5.5 and 5.6. By them

τ →∞lim

1

X

k+l=0

2τ π

Z

(q1− q2)e^{iτ (Φ+Φ)}F1,kF2,ldx − (q1− q2) L²(X)

≤ C(kβ1− ∂⁻¹q₁kL^p∗(X)+ kβ₂− ∂⁻¹q₂kL^p∗(X)) ≤ 2Cε

where the L²(X)-norm is taken with respect to z0 and this time C does not depend on β1or β2. We can redo this whole argument for any ε > 0, and thus by Alessandrini’s identity

kq1− q2k_L2(Ω)≤ lim

τ →∞

q1− q2−

∞

X

k+l=0

2τ π

Z

(q1− q2)e^{iτ (Φ+Φ)}F1,kF2,ldx L²(Ω)

the latter of which can be made as small as we please by choosing β1, β2. The claim follows.

6 Appendix 1: Cauchy operator and integration by parts

We define the two fundamental tools for solving the two-dimensional inverse problem of the Schr¨odinger operator in this section: the Cauchy operators and an integration by parts formula for the Cauchy operator conjugated by an ex- ponential. These were used by Bukhgeim [7] for solving the problem.

Definition 6.1. Let u ∈E⁰(R²) be a compactly supported distribution. Then we define the Cauchy operators by

∂⁻¹u = 1

πz∗ u, ∂⁻¹u = 1 πz ∗ u.

Remark 6.2. The notations∂⁻¹and ∂⁻¹cause no problems because 1/(πz) and 1/(πz) are the fundamental solutions to the operators ∂ = (∂1+ i∂2)/2 and

∂ = (∂₁− i∂2)/2.

Lemma 6.3. Let τ > 0, z₀ ∈ C and Φ(z) = (z − z0)². Let ψ ∈ C₀^∞(R²) with ψ ≡ 1 in a neighbourhood of 0, and write

ψ_τ(z) = ψ(τ^1/2(z − z0)), h(z) = 1 − ψ_τ(z) z − z0

. Then for a ∈ C₀^∞(R²) we have the integration by parts formula

∂⁻¹(e^{−iτ (Φ+Φ)}a) = ∂⁻¹ e^{−iτ (Φ+Φ)}ψτa

− 1

2iτ e^{−iτ (Φ+Φ)}ha − ∂⁻¹(e^{−iτ (Φ+Φ)}∂ha) − ∂⁻¹(e^{−iτ (Φ+Φ)}h∂a)
.

If we had set h(z) = (1 − ψτ(z))/(z − z0) instead then

∂⁻¹(e^{−iτ (Φ+Φ)}a) = ∂⁻¹ e^{−iτ (Φ+Φ)}ψa

− 1

2iτ e^{−iτ (Φ+Φ)}ha − ∂⁻¹(e^{−iτ (Φ+Φ)}∂ha) − ∂⁻¹(e^{−iτ (Φ+Φ)}h∂a)
.

Proof. The proof follows by differentiating e^{−iτ (Φ+Φ)}ha and noting that by Re- mark 6.2 the operators∂⁻¹∂ and ∂⁻¹∂ are the identity on compactly supported distributions.

Lemma 6.4. Let X ⊂ R² be a bounded domain and 1 < p < ∞. Then the Cauchy operators ∂⁻¹ and ∂⁻¹ are bounded L^p(X) → W^1,p(X).

Proof. If f ∈ L^p(X) we extend it by zero to R²\ X to create a compactly supported distribution and thus ∂⁻¹f is well defined by Definition 6.1. The convolution kernel 1/(πz) is locally integrable, so by Young’s inequality

∂⁻¹f

_Lp(X)≤ C kf k_Lp(X),

because in essence∂⁻¹f has the same values in X as the convolution of f with the kernel χX−X(z)/(πz), where X − X = {z ∈ R²| z = z1− z2, zj ∈ R²}.

For the derivatives note that by Remark 6.2 we have ∂∂⁻¹f = f . On the other hand ∂∂⁻¹f = Πf which is the Beurling transform, and hence bounded L^p(X) → L^p(X). For reference see for example Section 4.5.2 in [2] or [24] for a more classical approach.

7 Appendix 2: Cut-off function estimates

This section contains all the technical cut-off function construction and norm estimates used in the paper.

Lemma 7.1. Let ψ ∈ C₀^∞(R²). For z0 ∈ R² and τ > 0 write ψτ(z) = ψ(τ^1/2(z − z0)). Then, given any vector v ∈ C², we have

kψτk_Lp(R²)= kψk_Lp(R²)τ^−1/p, kv · ∇ψτk_Lp(R²)= kv · ∇ψk_Lp(R²)τ^1/2−1/p for 1 ≤ p ≤ ∞.

Proof. This follows directly from the scaling properties and translation invari- ance of L^p-norms in R².

Lemma 7.2. Let τ > 0 and set R²τ= R²\ B(0, τ^−1/2). Then

z^−a _Lp

(R²τ)=

 2π ap − 2

^1/p

τ^a/2−1/p for a > 0 and 2/a < p ≤ ∞.

Proof. This is a direct computation using the polar coordinates integral trans- form R

R²τ. . . dz =R∞ τ^−1/2

R

S¹. . . dσ(θ)rdr, with z = rθ.

Lemma 7.3. Let ψ ∈ C₀^∞(R²) be a test function supported in B(0, 2) with 0 ≤ ψ ≤ 1 and ψ ≡ 1 in B(0, 1). For τ > 0 and z0 ∈ R² write ψτ(z) = ψ(τ^1/2(z − z0)). Let h(z) = (1 − ψτ(z))/(z − z0). Then

khk_Lp(R²)≤ Cpτ^1/2−1/p

for C_p< ∞ when 2 < p ≤ ∞ and for any complex vector v ∈ C² we have kv · ∇hk_Lp(R²)≤ Cψ,p,vτ^1−1/p

for Cψ,p,v< ∞ when 1 ≤ p ≤ ∞. The same conclusions hold if we had defined h by dividing 1 − ψτ by z − z0 instead of its complex conjugate.

Proof. For the first claim note that |h(z)| ≤ |z − z0|⁻¹ and supp h ⊂ R²τ+ z0= R²\ B(z0, τ^−1/2). Hence khk_Lp(R²)≤

z⁻¹ _Lp

(R²τ)and Lemma 7.2 takes care of the first estimate.

For the second estimate

v · ∇h(z) = v · ∇ψτ(z)

z − z₀ −1 − ψτ(z) (z − z₀)². The L^p-norm of the first term is bounded by kv · ∇ψ_τk_Lp

z⁻¹ _L∞

(R²τ) which is at most Cψ,p,vτ^1−1/p according to lemmas 7.1 and 7.2. The second term is supported in R²\ B(z0, τ^−1/2) and bounded pointwise by |z − z0|⁻². Hence, as in the first paragraph, it has the required bound.

Acknowledgements

Leo Tzou was partially supported by Australian Research Council DP190103451 and DP190103302. Jenn-Nan Wang was supported in part by MOST 105-2115- M-002-014-MY3.

References

[1] W. Amrein, A. Berthier, and V. Georgescu. L^p-inequalities for the laplacian and unique continuation. Annales de l’institut Fourier, 31(3):153–168, 1981.

[2] K. Astala, T. Iwaniec, and G. Martin. Elliptic partial differential equa- tions and quasiconformal mappings in the plane, volume 48 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 2009.

[3] K. Astala and L. P¨aiv¨arinta. Calder´on’s inverse conductivity problem in the plane. Ann. of Math. (2), 163(1):265–299, 2006.

[4] E. Bl˚asten. On the Gel’fand-Calder´on inverse problem in two dimensions.

Doctoral thesis, University of Helsinki, Finland, 2013.

[5] E. Bl˚asten, O. Y. Imanuvilov, and M. Yamamoto. Stability and unique- ness for a two-dimensional inverse boundary value problem for less regular potentials. Inverse Problems and Imaging, 9(3):709–723, 2015.

[6] R. Brown and G. Uhlmann. Uniqueness in the inverse conductivity prob- lem for nonsmooth conductivities in two dimensions. Communications in Partial Differential Equations, 22(5-6):1009–1027, 1997.

[7] A. L. Bukhgeim. Recovering a potential from Cauchy data in the two- dimensional case. J. Inverse Ill-Posed Probl., 16(1):19–33, 2008.