We study global uniqueness in an inverse problem for the fractional semilinear Schr¨odinger equation (−∆)su + q(x, u

SCHR ¨ODINGER EQUATION

RU-YU LAI AND YI-HSUAN LIN

Abstract. We study global uniqueness in an inverse problem for the fractional semilinear Schr¨odinger equation (−∆)^su + q(x, u) = 0 with s ∈ (0, 1). We show that an unknown function q(x, u) can be uniquely determined by the Cauchy data set. In particular, this result holds for any space dimension greater than or equal to 2. Moreover, we demonstrate the comparison principle and provide a L^∞ estimate for this nonlocal equation under appropriate regularity assumptions.

1. Introduction

Let Ω be a bounded domain in Rⁿ, n ≥ 2 with Lipschitz boundary ∂Ω. We study the nonlocal type inverse problem for the fractional semilinear Schr¨odinger equation with the exterior Dirichlet data

(1.1)  (−∆)^su + q(x, u) = 0 in Ω,

u = g in Ωe, where s ∈ (0, 1), g ∈ C₀³(Ω_e), and

Ω_e := Rⁿ\Ω

is the exterior domain of Ω. Here the fractional Laplacian (−∆)^s is defined by (−∆)^su = cn,sP.V.

Z

Rⁿ

u(x) − u(y)

|x − y|^n+2s dy, for u ∈ H^s(Rⁿ), where P.V. is the principal value and

(1.2) c_n,s = Γ(ⁿ₂ + s)

|Γ(−s)|

4^s π^n/2

is a constant that was explicitly calculated in [2]. For the other equivalent definitions of the fractional Laplacian (−∆)^s, we refer readers to [11].

The study of fractional nonlinear Schr¨odinger (FNS) equations arises in the investigation of the quantum effects in Bose-Einstein Condensation [17]. In ideal boson systems, the classical Gross-Pitaevskii (GP) equations can describe condensation of weakly interacting boson atoms at a low temperature where the probability density of quantum particles is conserved. However in the inhomogeneous media with long-range (nonlocal) interactions between particles, this yields the density profile no longer retains its shape as in the classical GP equations. This dynamics is described by the fractional GP equations, known as the FNS equation, in which the turbulence and decoherence emerge. It was observed in [10]

Key words: Calder´on’s problem, partial data, semilinear, fractional Schr¨odinger equation, nonlocal, maximum principle.

1

that the turbulence appears from the nonlocal property of the fractional Laplacian; while the local nonlinearity helps maintain coherence of the density profile.

To study the equation (1.1), we assume that the function q(x, t) : Ω × R → R fulfilling the following conditions:

(1.3) q(x, t) and ∂tq(x, t) are continuous for (x, t) ∈ Ω × R.

Moreover, suppose that there exist constants µ > 0 and δ ∈ (2, (2n − 2s)/(n − 2s)) such that

(1.4)







|q(x, t)| ≤ µ(1 + |t|^δ−1) for all (x, t) ∈ Ω × R, limt→0

q(x, t)

t = 0 uniformly in x ∈ Ω, and there exist constants b₀ ∈ (0, 1) and r > 0 such that

0 < q(x, t)

t ≤ b₀∂_tq(x, t), for any x ∈ Ω, |t| ≥ r.

(1.5)

Meanwhile, we further assume that there is a constant 0 < M₀ < ∞ such that (1.6) 0 ≤ ∂_tq(x, t) ≤ M₀, for any (x, t) ∈ Ω × R.

The condition (1.6) will be utilized to characterize the well-posedness for the linearized equation of (−∆)^su + q(x, u) = 0. For the nonlinear equation (1.1), the weak solution u ∈ H^s(Rⁿ) exists provided that the coefficient q(x, t) satisfies (1.3)-(1.5) is discussed in Section 2. However, very little result is known in general about uniqueness of the weak solution u of (1.1). We would like to point out that the uniqueness up to translations of the nontrivial solution of the fractional nonlinear equation holds for certain nonlinearity q(x, u), we refer to [4] and references therein.

We consider the nonlocal inverse problem with related nonlocal Cauchy data set, instead of the Dirichlet to Neumann (DN) map, Λ_q : u|_Ωe → (−∆)^su|_Ωe defined in [6], due to the lack of uniqueness of solutions for (1.1). The Cauchy data set is defined by

C_q^Ωe =(u|_Ωe, N_q^su|_Ωe) : u ∈ H^s(Rⁿ) is a solution of (1.1) , where

N_q^su(x) := c_n,s Z

Ω

u(x) − u(y)

|x − y|^n+2s dy

stands for the nonlocal Neumann derivative and the constant c_n,s is the same as (1.2).

Note that when the equation (1.1) has a unique solution, the inverse problem is to recover q(x, u) from the DN map. For more details about the nonlocal Neumann derivative N_q^s, DN map Λq, and their connection, we refer to (2.2) and [3, 6].

In this paper, we focus on the Calder´on problem for the fractional semilinear Schr¨odinger equation, that is, to recover the coefficient q(x, u) from the collected external data set C_q^Ωe. It is the nonlinear and nonlocal analogue of well-known Calder´on problem, the math- ematical model of electrical impedance tomography. As a noninvasive type of medical imaging, the electrical conductivity of the object is inferred from voltage and current measurements collected only on the surface of the object. There are several aspects in the Calder´on problem, including uniqueness, stability estimates, reconstruction, and numerical algorithms for the known conductivity. For the classical semilinear Schr¨odinger equation

−∆u + q(x, u) = 0, global uniqueness of an inverse problem with the related DN map on

full boundary ∂Ω is due to [9, 15] when n ≥ 3 and to [8, 15] when n = 2. We refer to the survey paper [16] for recent developments in inverse boundary value problems for linear and nonlinear elliptic equations. The uniqueness result has been studied for fractional Schr¨odinger equation in [6] and for variable coefficients nonlocal elliptic operators in [5].

The stability estimate for the fractional Schr¨odinger equation was shown in [13].

For each coefficient q(x, u), we define a set A_q ⊂ Rⁿ× R by

A_q := {(x, u) ∈ Ω × R : there exists a solution u = u(x) of (1.1)}.

The main result in this paper is stated in Theorem 1.1, which solves the uniqueness for the fractional semilinear inverse problem with partial data for arbitrary dimension n ≥ 2.

Theorem 1.1. Let Ω ⊂ Rⁿ be a bounded Lipschitz domain and 0 < s < 1. Suppose that q₁(x, t) and q₂(x, t) satisfy the conditions (1.3)-(1.6). Let W ⊂ Ω_e be an arbitrary open set. Suppose that the partial Cauchy data sets C_q^W₁ = C_q^W₂, that is,

(1.7) (u₁|_W, N_q^s₁u₁|_W) = (u₂|_W, N_q^s₂u₂|_W) ,

where u_j are solutions to (−∆)^su_j+ q_j(x, u_j) = 0 in Ω with u_j = g in Ω_e for j = 1, 2, for any g ∈ C₀³(W ). Then A_q1 = A_q2 and

q₁(x, u(x)) = q₂(x, u(x)) in A_q1.

The proof of Theorem 1.1 starts by showing the comparison principle for the linear fractional Schr¨odinger equation and the L^∞estimate for the related solutions. The estimate plays a crucial role in the linearization argument that reduces the inverse problem for the fractional nonlinear equation to the inverse problem for the fractional linear equation.

Notice that there exists a strong uniqueness property for the fractional Laplacian, that is, for any u in Rⁿ satisfies u|_W = (−∆)^su|_W = 0 in some open set W , then u is identically zero in Rⁿ (see [6, Theorem 1.2]).

The paper is organized as follows. In section 2, we introduce fundamental tools of the fractional semilinear equation. The comparison principle and the estimate for solutions are discussed in section 3. In section 4, we prove Theorem 1.1 by utilizing the linearization scheme.

2. Preliminaries

First, let us begin with the fractional Sobolev spaces. Let 0 < s < 1, H^s(Rⁿ) = W^s,2(Rⁿ) is the L²-based fractional Sobolev space with norm

kuk_H^s_(Rⁿ₎= kuk_L²_(Rⁿ₎+ k(−∆)^s/2uk_L²_(Rⁿ₎. Let O ⊂ Rⁿ be an open set (not necessarily bounded), then we define

H^s(O) = {u|_O: u ∈ H^s(Rⁿ)}

and

H₀^s(O) = closure of C_c^∞(O) in H^s(O).

Next, we characterize the existence of solutions to our main nonlocal problem (1.1).

Lemma 2.1. (Existence of weak solutions) Let s ∈ (0, 1) and q(x, t) be a scalar-valued function satisfying (1.3)-(1.5). For any g ∈ C₀³(Ω_e), there exists at least one solution u ∈ H^s(Rⁿ) of the nonlocal Dirichlet problem (1.1).

Proof. For any g ∈ C₀³(Ω_e), define the function eg to be an extension of g by

eg :=

(0 in Ω, g in Ω_e.

It is easy to see that eg ∈ C₀²(Rⁿ) ⊂ H²(Rⁿ). Now, consider a function w := u −eg and we have the fact q(x, u(x)) = q(x, w(x)) for x ∈ Ω since eg = 0 in Ω. Then we can rewrite the equation (1.1) as

(2.1)  (−∆)^sw + q(x, w) + h(x) = 0 in Ω, w = 0 in Ω_e,

where h(x) = (−∆)^seg(x) ∈ L²(Rⁿ) (see [6, Remark 2.2]). Therefore, by using [1, Theorem 11.2], one can see that there exists a weak solution w ∈ H₀^s(Ω) to the equation (2.1). This implies that there exists a solution u = w +eg ∈ H^s(Rⁿ) of (1.1) such that u = g in Ωe.  Remark 2.1. In fact, from [1, Theorem 11.2], there exist infinitely many weak solutions {w_k}_k∈N ⊂ H₀^s(Ω) of (2.1) such that

Z Z

Rⁿ×Rⁿ

|w_k(x) − w_k(z)|²

|x − z|^n+2s dxdz → ∞ as k → ∞.

We only choose w<sub>`</sub> ∈ H<sub>0</sub><sup>s</sup>(Ω) for some ` ∈ N to be a weak solution of (2.1). Hence, for this semilinear nonlocal problem, it is more natural to formulate the Calder´on problem by using the characterization of the Cauchy data set.

In this paper, our exterior Dirichlet data g are given in C₀³(Ωe) ⊂ H^2s(Rⁿ), so that (−∆)^su ∈ H^−s(Rⁿ), where u ∈ H^s(Rⁿ) is a solution of (1.1). By using the relation

(−∆)^su|_Ωe = N_q^su|_Ωe− mu|_Ωe + (−∆)^s(E₀g)|_Ωe (2.2)

(see Lemma 3.2 in [6]), where m ∈ C^∞(Ω_e) is defined by m(x) = c_n,sR

Ω 1

|x−y|^n+2sdy and E0 is a zero extension in Ω such that E0g(x) = g(x) for x ∈ Ωe, E0g(x) = 0 for x ∈ Ω.

Therefore, N_q^su ∈ H^−s(Rⁿ) and the Cauchy data (u|Ωe, N_q^su|Ωe) can be regarded as in the function space H^s(Ω_e) × H^−s(Ω_e) (indeed, u|_Ωe ∈ H^2s(Ω_e) ⊂ H^s(Ω_e)).

3. L^∞-estimate of weak solutions

In this section, we offer a L^∞-estimate for the solution of the fractional Schr¨odinger equation under suitable regularity assumptions. This estimate will be used in the lin- earization scheme of the inverse problem for the fractional semilinear equation. The result of this section is motivated by [12] in which the author considers elliptic integro-differential operators.

3.1. Comparison principle. We begin by proving the maximum principle for the frac- tional Schr¨odinger equation. The definition of weak solutions is stated as follows.

Definition 3.1. The function u ∈ H^s(Rⁿ) is called a weak solution of the fractional Schr¨odinger equation (−∆)^su + au = f in Ω with u = g in Ωe if

Z

Rⁿ

(−∆)^s/2u · (−∆)^s/2φdx + Z

Ω

auφdx = Z

Ω

f φdx

with u − g ∈ eH^s(Ω) for any φ ∈ C_c^∞(Ω). Here eH^s(Ω) is the closure of C_c^∞(Ω) in H^s(Rⁿ).

The comparison principle can be derived directly from the following maximum principle.

Proposition 3.1 (Maximum principle). Let Ω ⊂ Rⁿ be a bounded domain and a(x) ∈ L^∞(Ω) be a nonnegative potential. Let u ∈ H^s(Rⁿ) be a weak solution of

(3.1)  (−∆)^su + a(x)u = f in Ω,

u = g in Ω_e.

Suppose 0 ≤ f ∈ L^∞(Ω) in Ω and 0 ≤ g ∈ L^∞(Ω_e) in Ω_e. Then u ≥ 0 in Ω.

Proof. If u ∈ H^s(Rⁿ) is a weak solution of (3.1), by the weak formulation, we have (3.2)

Z

Rⁿ

(−∆)^s/2u · (−∆)^s/2φdx + Z

Ω

auφdx = Z

Ω

f φdx, for any φ ∈ eH^s(Ω). Note that

Z

Rⁿ

(−∆)^s/2u · (−∆)^s/2φdx = Z Z

R²ⁿ

(u(x) − u(z))(φ(x) − φ(z))

|x − z|^n+2s dxdz

= Z Z

R²ⁿ\(Ωe×Ωe)

(u(x) − u(z))(φ(x) − φ(z))

|x − z|^n+2s dxdz, where we have used φ ≡ 0 in Ω_e.

Next, we write u = u⁺− u⁻in Rⁿ, where u⁺ = max{u, 0} and u⁻ = max{−u, 0}, Notice that u⁺, u⁻ ∈ H^s(Rⁿ) due to u ∈ H^s(Rⁿ). Recall u = g ≥ 0 in Ω_e, then we can take φ := u⁻ ∈ eH^s(Ω) as a test function. We assume that u⁻ is not identically zero, and we want to prove that it will lead to a contradiction.

Since f ≥ 0 and φ = u⁻ ≥ 0, the right hand side of (3.2) (3.3)

Z

Ω

f φdx ≥ 0.

On the other hand, one has Z Z

R²ⁿ\(Ωe×Ωe)

(u(x) − u(z))(φ(x) − φ(z))

|x − z|^n+2s dxdz

= Z Z

Ω×Ω

(u(x) − u(z))(u⁻(x) − u⁻(z))

|x − z|^n+2s dxdz + 2

Z

Ω

Z

Ωe

(u(x) − g(z))u⁻(x)

|x − z|^n+2s dzdx

=I + II, where

I :=

Z Z

Ω×Ω

(u(x) − u(z))(u⁻(x) − u⁻(z))

|x − z|^n+2s dxdz, II := 2

Z

Ω

Z

Ωe

(u(x) − g(z))u⁻(x)

|x − z|^n+2s dzdx.

To estimate I, since (u⁺(x) − u⁺(z))(u⁻(x) − u⁻(z)) ≤ 0, we obtain I ≤ −

Z Z

Ω×Ω

(u⁻(x) − u⁻(z))²

|x − z|^n+2s dxdz < 0.

(3.4)

The last strict inequality holds because u⁻ can not be a constant in Ω. If u⁻ is a constant, which means u ≡ −c₀ is a negative constant in Ω (for some constant c₀ > 0). By the definition of the fractional Laplacian, one can see that for x ∈ Ω,

(−∆)^su(x) = c_n,sP.V.

Z

Rⁿ

−c₀− u(z)

|x − z|^n+2sdz

= c_n,s Z

Ωe

−c₀− g(z)

|x − z|^n+2sdz < 0,

since g(z) ≥ 0 for z ∈ Ω_e. Therefore, by using (3.1) and a ≥ 0 in Ω, we know that 0 ≤ f = (−∆)^su + au < 0 in Ω,

which leads to a contradiction. Hence, u⁻ can not be a constant.

For II, since g(z) ≥ 0 in Ω_e and u(x)u⁻(x) ≤ 0 in Ω, we deduce that II ≤ 0. Therefore, Z Z

R²ⁿ\(Ωe×Ωe)

(u(x) − u(z))(φ(x) − φ(z))

|x − z|^n+2s dxdz < 0.

which contradicts to (3.2) (because f ≥ 0 in Ω and g ≥ 0 in Ω_e).  With the maximum principle, the comparison principle for the fractional Schr¨odinger equation follows immediately.

Corollary 3.2 (Comparison principle). Let u₁ and u₂ be weak solutions of

 (−∆)^su₁ + a(x)u₁ = f₁ in Ω,

u₁ = g₁ in Ω_e, and  (−∆)^su₂+ a(x)u₂ = f₂ in Ω, u₂ = g₂ in Ω_e, respectively. Suppose that f₁ ≥ f₂ in Ω and g₁ ≥ g₂ in Ω_e. Then u₁ ≥ u₂ in Ω.

Proof. Let u := u₁− u₂ and apply proposition 3.1, then we complete the proof. Further-

more, one can conclude that u₁ ≥ u₂ in Rⁿ. 

Remark 3.1. From the above comparison principle, once the solution exists, the unique- ness will automatically hold for the fractional linear Schr¨odinger equation (3.1).

3.2. L^∞ bounds for solutions. The main goal of this section is stated as follows.

Proposition 3.3. Suppose f ∈ L^∞(Ω) and g ∈ L^∞(Ωe). Let u be a solution to (3.1), then the following L^∞ estimate

kuk_L^∞_(Ω)≤ kgk_L^∞_(Ωe)+ Ckf k_L^∞_(Ω), (3.5)

holds for some constant C > 0 independent of u, f, and g.

In order to derive (3.5), we need to construct a barrier function for the fractional Schr¨odinger equation.

Lemma 3.4 (Barrier). Let Ω be a bounded Lipschitz domain in Rⁿ and a(x) ∈ L^∞(Ω) is a nonnegative potential. Then there exists a function ϕ ∈ C_c^∞(Rⁿ) such that

(3.6)







(−∆)^sϕ + a(x)ϕ ≥ 1 in Ω, ϕ ≥ 0 in Rⁿ, ϕ ≤ C in Ω, where C > 0 is a constant depending on n, s, and Ω.

Proof. Let B_R be an arbitrarily large ball such that Ω b BRand η ∈ C_c^∞(B_R) be a smooth cutoff function such that

0 ≤ η ≤ 1 in Rⁿ, η ≡ 1 in Ω.

For any x ∈ Ω, it is clear that η(x) = 1 that is the maximum value of η. Thus, one has 2η(x) − η(x + y) − η(x − y) ≥ η(x) − η(x + y) ≥ 0.

(3.7)

Recall that for any function u in the Schwartz space, we can also represent the fractional Laplacian as (see [2] for instance)

(−∆)^su(x) = c_n,s 2

Z

Rⁿ

2u(x) − u(x + y) − u(x − y)

|y|^n+2s dy for all x ∈ Rⁿ,

where c_n,s is the constant in (1.2). For any x ∈ Ω, from (3.7) and by using the change of variables z = x + y, one has

(−∆)^sη + a(x)η ≥ 1 2c_n,s

Z

Rⁿ

η(x) − η(z)

|x − z|^n+2sdz + a(x)η

≥ 1 2c_n,s

Z

Rⁿ\B_R

1

|x − z|^n+2sdz

≥ λ

for some constant λ > 0. Here we have utilized the fact that η(x)−η(z) = 1 for z ∈ Rⁿ\B_R, then we have

Z

Rⁿ\B_R

1

|x − z|^n+2sdz = Z

Rⁿ\(x+B_R)

1

|y|^n+2sdy ≥ Z

Rⁿ\B_2R

1

|y|^n+2sdy = c > 0,

where c > 0 is a positive constant. Now, we let ϕ(x) := η(x)/λ, then we complete the

proof. 

It remains to prove the L^∞ bound for the solution u.

Proof of Proposition 3.3. Let

v(x) := kgk_L^∞_(Ωe)+ kf k_L^∞_(Ω)ϕ(x),

where ϕ(x) is the barrier given by Lemma 3.4. From a(x) ≥ 0 and (3.6), we deduce that (−∆)^su + a(x)u = f ≤ (−∆)^sv + a(x)v in Ω

and g ≤ v in Ωe. Applying the comparison principle in Corollary 3.2, we obtain u ≤ kgk_L^∞_(Ωe)+ Ckf k_L^∞_(Ω) in Ω,

where we use ϕ ≤ C in Ω. Similarly, the same argument will hold for −u, which finishes the proof.

 4. Proof of Theorem 1.1

In this section, we apply the linearization scheme to transfer the inverse problem for the nonlocal semilinear Schr¨odinger equation to the inverse problem for the nonlocal linear equation.

4.1. Linearization. This linearization method was used in the local type inverse problem, see [7, 9, 14, 15].

Theorem 4.1. Let n ≥ 2 and 0 < s < 1. Let g and h be in C₀³(Ω_e) and η be in R. Suppose that u_g+ηh is the solution of (1.1) with u_g+ηh = g + ηh in Ω_e. Suppose that u^∗ is the unique solution of the linearized equation

(4.1)  (−∆)^su^∗+ ∂_tq(x, u_g)u^∗ = 0 in Ω, u^∗ = h in Ω_e, then we have

η→0lim

u_g+ηh − u_g η − u^∗

H^s(Rⁿ)

= 0.

Proof. First, by using (1.6), i.e., 0 ≤ ∂_tq(x, t) ≤ M₀ for (x, t) ∈ Ω × R gives the well- posedness of the fractional linear Schr¨odinger equation, i.e., for a given function h ∈ C₀³(Ωe) ⊂ H^s(Ωe), there exists a unique weak solution u^∗ ∈ H^s(Rⁿ) such that u^∗ solves (4.1).

Next, consider

w := u_g+ηh− u_g

η ∈ H^s(Rⁿ), for η ∈ R, then w is a solution of

 (−∆)^sw + Q(x)w = 0 in Ω, w = h in Ωe, where

Q(x) = Z 1

0

∂_tq(x, ξu_g+ηh(x) + (1 − ξ)u_g(x))dξ ≥ 0.

Let v := w − u^∗, then v solves

(4.2)  (−∆)^sv + Q(x)v = − (Q(x) − ∂_tq(x, u_g)) u^∗ in Ω, v = 0 in Ω_e. By multiplying v on both sides of (4.2), we can see that

kvkH^s(Rⁿ) ≤ Ck (Q(x) − ∂tq(x, ug)) u^∗k_L²_(Rⁿ₎

≤ CkQ(x) − ∂_tq(x, u_g)k_L^∞_(Ω)ku^∗k_L²_(Rⁿ₎, (4.3)

for some constant C > 0 independent of v and u^∗. Now, since u_g+ηh− u_g ∈ H^s(Rⁿ) solves

 (−∆)^s(u_g+ηh− u_g) + Q(x)(u_g+ηh− u_g) = 0 in Ω, u_g+ηh − u_g = ηh in Ω_e, where h ∈ C₀³(Ωe) ⊂ L^∞(Ωe). By the L^∞ estimate (3.5), we have

kug+ηh− ugk_L^∞_(Ω) ≤ |η|khk_L^∞_(Ωe) → 0, as η → 0.

By the continuity of ∂_tq(x, t), it implies that

k∂_tq(x, ξu_g+ηf + (1 − ξ)u_g) − ∂_tq(x, u_g)k_L^∞_(Ω) → 0

as η → 0 that leads to

kQ(x) − ∂_tq(x, u_g)k_L^∞_(Ω) ≤ Z 1

0

k∂_tq(x, ξu_g+ηf + (1 − ξ)u_g) − ∂_tq(x, u_g)k_L^∞_(Ω)dξ → 0, whenever η → 0. Combining with (4.3), the proof is complete.

 4.2. Proof of Theorem 1.1. Now, we are ready to prove our main theorem.

Proof of Theorem 1.1. For any g ∈ C₀³(W ) ⊂ C₀³(Ω_e), let u⁽¹⁾g ∈ H^s(Rⁿ) be a solution of (

(−∆)^su⁽¹⁾g + q₁(x, u⁽¹⁾g ) = 0 in Ω, u⁽¹⁾g = g in Ω_e.

From the Cauchy data assumption (1.7), there exists a solution u⁽²⁾g ∈ H^s(Rⁿ) such that u⁽²⁾g = u⁽¹⁾g , N_q^s₂u⁽²⁾g = N_q^s₁u⁽¹⁾g in W and u⁽²⁾g solves

(

(−∆)^su⁽²⁾g + q₂(x, u⁽²⁾g ) = 0 in Ω, u⁽²⁾g = g in Ω_e.

For any h ∈ C₀³(W ) ⊂ C₀³(Ω_e), using the Cauchy data assumption again, there are solutions u^(j)_g+ηh ∈ H^s(Rⁿ) of

( (−∆)^su^(j)_g+ηh+ q_j(x, u^(j)_g+ηh) = 0 in Ω, u^(j)_g+ηh = g + ηh in Ω_e, for j = 1, 2, such that

N_q^s₁u⁽¹⁾_g+ηh = N_q^s₂u⁽²⁾_g+ηh in W ⊂ Ω_e

for any η ∈ R. We differentiate the above equation with respect to η at η = 0 and use (2.2) with Theorem 4.1, then we obtain that

(4.4) N_q^s₁˙u⁽¹⁾_g,h = N_q^s₂˙u⁽²⁾_g,h in W, where ˙u⁽¹⁾_g,h and ˙u⁽²⁾_g,h are solutions of

(−∆)^s˙u⁽¹⁾_g,h + ∂_tq₁(x, u⁽¹⁾_g ) ˙u⁽¹⁾_g,h = 0 in Ω with ˙u⁽¹⁾_g,h = h in Ω_e, (4.5)

and

(−∆)^s˙u⁽²⁾_g,h + ∂_tq₂(x, u⁽²⁾_g ) ˙u⁽²⁾_g,h = 0 in Ω with ˙u⁽²⁾_g,h = h in Ω_e. (4.6)

Recall that ∂_tq_j ∈ L^∞ and ∂_tq_j ≥ 0 in Ω. Thus, by using the well-posedness of the linearized fracional Schr¨odinger equation (see Section 2 in [6]), nonlocal DN maps Λ_∂tqj exist and are defined by

Λ∂tqj(x,u^(j)g ) : ˙u^(j)_g,h|_Ωe → (−∆)^s˙u^(j)_g,h|_Ωe, for j = 1, 2.

For a fixed g ∈ C₀³(W ), since (4.4) holds for any h ∈ C₀³(W ), by using (2.2), we derive Λ_∂

tq1(x,u⁽¹⁾g )h|_W = Λ_∂

tq2(x,u⁽²⁾g )h|_W for any h ∈ C₀³(W ).

Thus, we can use global uniqueness of the fractional linear Schr¨odinger equation (see The- orem 1.1 in [6]), then we can conclude

(4.7) ∂tq1(x, u⁽¹⁾_g ) = ∂tq2(x, u⁽²⁾_g ) in Ω.

Via (4.7), we know that ˙u⁽¹⁾_g,h and ˙u⁽²⁾_g,h solve (4.5) and (4.6) (with the same coefficients), respectively. From the well-posedness for the fractional linear Schr¨odinger equation again, one can see that the weak solution will be unique, that is,

˙u⁽¹⁾_g,h = ˙u⁽²⁾_g,h in H^s(Rⁿ).

In particular, we take the original g by ηg and h by g, then we have

˙u⁽¹⁾_ηg,g = ˙u⁽²⁾_ηg,g in H^s(Rⁿ), for any η ∈ R.

By using Theorem 4.1, this implies that d

dηu⁽¹⁾_ηg = d

dηu⁽²⁾_ηg in H^s(Rⁿ) for any η ∈ R.

Hence, there exists a function ψ = ψ(x) ∈ H^s(Rⁿ) independent of η such that (4.8) u⁽¹⁾_g = u⁽²⁾_g + ψ in H^s(Rⁿ)

for all g ∈ C₀³(Ω_e).

Now, by using the assumption on Cauchy data (1.7),

{(u⁽¹⁾_g |_W, N_q^s₁u⁽¹⁾_g |_W)} = {(u⁽²⁾_g |_W, N_q^s₂u⁽²⁾_g |_W)}, we can obtain ψ ∈ H^s(Rⁿ) such that

(4.9) ψ = (−∆)^sψ = 0 in W ⊂ Ω_e.

We apply Theorem 1.2 in [6], then the function ψ ≡ 0 in Rⁿ. Finally, we substitute u⁽¹⁾g (x) = u⁽²⁾g (x) for x ∈ Rⁿ into the fractional semilinear Schr¨odinger equation

(−∆)^su⁽¹⁾_g + q₁(x, u⁽¹⁾_g ) = (−∆)^su⁽²⁾_g + q₂(x, u⁽²⁾_g ) in Ω, which implies

q₁(x, u) = q₂(x, u) for x ∈ Ω.

This completes the proof of our main result. 

Acknowledgment. Both authors would like to thank Camelia Pop for helpful discus- sions. The second author was supported in part by MOST of Taiwan 160-2917-I-564-048.

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School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA E-mail address: [email protected]

Department of Mathematics, University of Washington, Seattle, WA 98195, USA and Institute for Advanced Study, Hong Kong University Science and Technology, Hong Kong SAR

E-mail address: [email protected]