We consider an inverse problem for the fractional Schr¨odinger equation by using monotonicity formulas

MONOTONICITY-BASED INVERSION OF THE FRACTIONAL SCHR ¨ODINGER EQUATION I. POSITIVE POTENTIALS

BASTIAN HARRACH^†, AND YI-HSUAN LIN^‡

Abstract. We consider an inverse problem for the fractional Schr¨odinger equation by using monotonicity formulas. We provide if-and-only-if monotonicity relations between positive bounded potentials and their associated nonlocal Dirichlet-to-Neumann maps. Based on the monotonicity relation, we can prove uniqueness for the nonlocal Calder´on problem in a constructive manner. Sec- ondly, we offer a reconstruction method for an unknown obstacles in a given domain. Our method is independent of the dimension and only requires the background solution of the fractional Schr¨odinger equation.

Key words. Fractional Schr¨odinger equation, monotonicity method, inverse obstacle problem, shape reconstruction, localized potentials, Calder´on’s problem, Runge approximation property

AMS subject classifications. 35R30

1. Introduction. In this article we will give a constructive uniqueness result for the Calder´on problem for the nonlocal fractional Schr¨odinger equation and develop a shape reconstruction method to determine unknown obstacles in a given domain. Let Ω be a bounded open set in Rⁿ with n ∈ N, and q ∈ L^∞+(Ω) be a potential, where L^∞₊(Ω) consists of all L^∞(Ω)-functions with positive essential infima. For s ∈ (0, 1), the (nonlocal) Dirichlet problem for the fractional Schr¨odinger equation is given by

((−∆)^su + qu = 0 in Ω,

u = F in Ωe:= Rⁿ\ Ω, (1.1)

Note that (−∆)^s is a nonlocal operator as s ∈ (0, 1), so that the Dirichlet data is prescribed on the whole complement of Ω and not only on its boundary ∂Ω.

The Dirichlet-to-Neumann (DtN) operator of (1.1) Λ(q) : H(Ωe) → H(Ωe)^∗ is formally given by

Λ(q)F := (−∆)^su|Ωe, where u ∈ H^s(Rⁿ) solves (1.1). (1.2) The precise definition of (−∆)^s, the DtN map Λ(q), and the function spaces are given in Section 2. For further properties for the nonlocal DtN maps, we also refer readers to [20].

The nonlocal fractional Laplacian operator has received considerable attention for its ability to model anomalous stochastic diffusion problems including jumps and longdis- tance interactions, cf., e.g., [7, 53], and the extensive list of references to applications in the introduction of [12]. Accordingly, inverse problems for the fractional Laplacian

†Institute for Mathematics, Goethe-University Frankfurt, Frankfurt am Main, Germany ([email protected])

‡Institute for Advanced Study, The Hong Kong University Science and Technology, Hong Kong ([email protected])

1

operator appear when an imaging domain is being probed by an anomalous diffusion process. The fractional diffusion model is more complicated than in the standard Brownian motion case modeled by the standard Laplacian (s = 1). However, recent works on inverse problems for nonlocal equations (see the references below) indicate that the inverse problem actually becomes easier to solve than in the standard Lapla- cian case. The present work contributes to this by showing that monotonicity-based reconstruction methods that have been developed for standard diffusion processes can also be applied to the fractional diffusion case and that the methods even become simpler and more powerful.

For a list of recent works on inverse problems for nonlocal equations let us refer to [8, 9, 10, 11, 18, 46], and the review of Salo [59]. Stability questions for the fractional Calder´on problem were studied in [55, 57, 58]. Let us point out that the Calder´on problem for the fractional Schr¨odinger equation was first solved by Ghosh, Salo and Uhlmann [20], who showed the global uniqueness result that Λ(q1) = Λ(q2) implies q1= q2. Remarkably, it was recently shown that uniqueness in the fractional Calder´on problem already holds with a single measurement and with data on arbitrary, possibly disjoint subsets of the exterior, cf. Ghosh, R¨uland, Salo, and Uhlmann [19].

Uniqueness proofs for the fractional Calder´on problem strongly rely on a strong uniqueness property, cf. [20, Theorem 1.2] for the fractional Schr¨odinger equation and [18, Theorem 1.2] for the nonlocal variable case. The strong uniqueness property states that if u = (−∆)^su = 0 (for 0 < s < 1) in an arbitrary open set in Rⁿ, then u ≡ 0 in the entire space Rⁿ for any n ∈ N. Note that this property is no longer true for the standard (local) Laplacian case, i.e., for the case s = 1. In fact, via this property, one can also derive a nonlocal Runge type approximation property (see [20, Theorem 1.3] or Theorem 3.4), which states that an arbitrary L² function can be well approximated by solutions of the fractional Schr¨odinger equation. Recently, R¨uland and Salo [56] studied the fractional Calder´on problem under lower regularity assumptions and the stability results for the potentials.

The first effort of this paper is to prove uniqueness for the Calder´on problem in a constructive way. We will derive the following monotonicity formula in Theorem 4.1:

Let q₀, q₁∈ L^∞₊(Ω), then

q1≤ q0 if and only if Λ(q1) ≤ Λ(q0), (1.3) where q1 ≤ q0 is to be understood pointwise almost everywhere in Ω, and Λ(q1) ≤ Λ(q0) is to be understood in the sense of definiteness of quadratic forms (also known as Loewner order), cf. (3.6) in Section 3. Similar monotonicity relations have been widely applied in the study of inverse problems, see [32, 65] for the origins of the monotonicity method combined with the method of localized potentials [65, 22, 31, 23, 2, 32, 25, 3, 33, 28, 48, 66, 14, 15, 34, 64, 68, 4, 21, 30, 26, 29, 24, 61, 69, 16]

for a list of recent and related works. Also note that similar arguments involving monotonicity conditions and blow-up arguments have been used in various ways in the study of inverse problems, see, e.g., [1, 36, 37, 42, 43].

The monotonicity relation (1.3) immediately implies a constructive global uniqueness result for the Calder´on problem, which is our first main result in this paper, cf.

Corollary 4.5: Any q ∈ L^∞₊(Ω) is uniquely determined by Λ(q) by the following formula: For x ∈ Ω a.e.,

q(x) = sup {ψ(x) : ψ positive (density one) simple function, Λ(ψ) ≤ Λ(q)} . (1.4)

This shows that one can recover an unknown potential with positive infimum by comparing the DtN map with that of simple functions.

The second main result of this paper is on the shape reconstruction (or inclusion detection) problem for the fractional Schr¨odinger equation. Let q₀∈ L^∞₊(Ω) denote a known reference coefficient, and q₁∈ L^∞₊(Ω) denote an unknown coefficient function that differs from the reference value q₀ in certain regions. We aim to find these anomalous regions (or scatterers), i.e., the support of q1− q0, from the difference of the Dirichlet-to-Neumann-operators Λ(q1) − Λ(q0). We will prove that this can be done without solving the fractional Schr¨odinger equation for potentials other than the reference potentials q0. More precisely, we will show in Theorem 5.5 that

(supp(q1− q0) =T{C ⊆ Ω closed :

∃α > 0 : −αTC≤ Λ(q1) − Λ(q0) ≤ αTC}, (1.5) where TC:= Λ⁰(q0)χC, and Λ⁰(q0) is the Fr´echet derivative of the DtN operator Λ(q).

The test operator T_C can be easily calculated from knowledge of the solution of the fractional Schr¨odinger equation with reference potentials q₀. Under the additional definiteness condition that either q₁≥ q0 or q₁≤ q0 holds almost everywhere we will also show that the inner support of q₁− q₀ fulfills

inn supp(q₁− q₀) =[

{B ⊆ Ω open ball : ∃α > 0 : Λ(q₁) ≤ Λ(q₀) − αT_B}, (1.6) resp.,

inn supp(q1− q0) =[

{B ⊆ Ω open ball : ∃α > 0 : Λ(q1) ≥ Λ(q0) + αTB}, (1.7) cf. Theorem 5.6.

Inverse shape reconstruction problems were intensively studied in the literature, see [38, 49] for the comprehensive introduction and survey. There are several inclusion detection methods, including the enclosure method, the linear sampling method, the probe method and the factorization method, which have been proposed to solve the inclusion detection inverse problem. These methods strongly rely on special solutions of certain differential equations. For example, the special solutions include the com- plex geometrical optics (CGO) solution, the oscillating decaying (OD) solution and the Wolff solution. In [40, 41, 52, 60, 62, 67]. The authors used the CGO solutions to solve the inverse obstacle problems for different mathematical models for the isotropic problems. However, for the general anisotropic medium, we need to utilize more com- plicated special solutions such as the OD solutions, see [44, 47, 50, 51]. The Wolff solutions are used to solve the inverse obstacle problem for the p-Laplace equation, see [5, 6]. Our monotonicity-based approach does not require to construct any special solutions to practically determine the inclusions via the formulas (1.5)–(1.7). The proof of these formulas however relies on so-called localized potentials [17] , i.e., solu- tions of the fractional Schr¨odinger equations with very large energy on a subset of Ω and very low energy elsewhere, see Corollary 3.5.

The structure of this article is given as follows. In Section 2, we provide basic reviews for the fractional Sobolev spaces, the fractional Schr¨odinger equation and the nonlocal DtN map. In Section 3, we demonstrate the monotonicity formulas and construct the localized potentials for our the fractional Schr¨odinger equation. In Section 4, we show the converse results of the monotonicity relations, which gives if-and-only-if relations

3

between the DtN maps and positive potentials. In addition, we provide a constructive global uniqueness proof for the Calder´on problem by proving (1.4). Finally, in Section 5, we characterize the linearized nonlocal DtN map and derive the inclusion detection formulas (1.5)–(1.7).

2. The Dirichlet-to-Neumann operator for the fractional Schr¨odinger equation. In this section, we briefly summarize some fundamental definitions and notations on the fractional Schr¨odinger equation and the associated DtN operator.

For n ∈ N, we denote by

F , F⁻¹: L²(Rⁿ; C) → L²(Rⁿ; C)

the Fourier transform and its inverse on the space of complex-valued L²-functions, and let S(Rⁿ; C) be the Schwartz space of rapidly decreasing complex-valued functions.

For s ∈ (0, 1), the fractional Laplacian is defined by

(−∆)^s: S(Rⁿ; C) → L²(Rⁿ; C), (−∆)^su :=F⁻¹ |ξ|^2sF (u)
. The fractional Laplacian can be extended to an operator

(−∆)^s: L²(Rⁿ; C) → S⁰(Rⁿ; C) by setting

h(−∆)^su, ϕiS⁰×S:= hu, (−∆)^sϕiL² for all u ∈ L²(Rⁿ; C), ϕ ∈ S(Rⁿ; C), and it can be shown that (−∆)^su will be real-valued for real-valued u (see [12, 45, 63]).

Hence, in the following we will always consider the fractional Laplacian as an operator (−∆)^s: L²(Rⁿ) → S⁰(Rⁿ),

and all function spaces in this work are real-valued unless indicated otherwise.

For 0 < s < 1, the L²-based fractional Sobolev space is defined by H^s(Rⁿ) := {u ∈ L²(Rⁿ) : (−∆)^s/2u ∈ L²(Rⁿ)}

and equipped with the scalar product (u, v)_Hs(Rⁿ):=

ˆ

Rⁿ



(−∆)^s/2u · (−∆)^s/2v + uv

dx for all u, v ∈ H^s(Rⁿ).

It can be shown that H^s(Rⁿ) is a Hilbert space (see [12] for instance). Also note that H^s(Rⁿ) obviously contains the rapidly decreasing Schwartz functions S(Rⁿ), and a fortiori all compactly supported C^∞-functions.

For an open set Ω ⊆ Rⁿ we define

H₀^s(Ω) := closure of C_c^∞(Ω) in H^s(Rⁿ).

Note that this space is sometimes denoted as eH^s(Ω) in the literature, but in the context of Dirichlet and Neumann boundary value problems it seems more natural to denote this space by H₀^s(Ω).

We also define the bilinear form Bq(u, w) :=

ˆ

Rⁿ

(−∆)^s/2u · (−∆)^s/2w dx + ˆ

Ω

quw dx, (2.1)

for any u, w ∈ H^s(Rⁿ). We then have the following variational formulation for the fractional Schr¨odinger equation.

Lemma 2.1. Let q ∈ L^∞(Ω) and f ∈ L²(Ω). u ∈ H^s(Rⁿ) solves (in the sense of distributions)

(−∆)^su + qu = f in Ω if and only if u ∈ H^s(Rⁿ) satisfies

Bq(u, w) = ˆ

Ω

f w dx for all w ∈ H₀^s(Ω).

Proof. Note that for u ∈ H^s(Rⁿ) we can interpret (−∆)^su as a distribution on Ω, and a simple computation shows that

h(−∆)^su + qu − f, ψi_D⁰_{(Ω)×D(Ω)}

= ˆ

Rⁿ

(−∆)^s/2u · (−∆)^s/2ψ dx + ˆ

Ω

quψ dx − ˆ

Ω

f ψ dx = 0

for all test functions ψ ∈ D(Ω) = C_c^∞(Ω), so that the assertion follows by continuous extension.

We now introduce the Dirichlet trace operator in abstract quotient spaces.

Lemma 2.2. With Ωe= Rⁿ\ Ω, we define

γΩ_e: H^s(Rⁿ) → H(Ωe) := H^s(Rⁿ)/H₀^s(Ω), u 7→ u + H₀^s(Ω).

Then, for all u, v ∈ H^s(Rⁿ),

γΩ_eu = γΩ_ev implies that u(x) = v(x) for x ∈ Ωe a.e.

Proof. If γΩ_eu = γΩ_ev, then u − v ∈ H₀^s(Ω), so that there exists a sequence (φk)_k∈N⊂ C_c^∞(Ω) with φk → u − v in H^s(Rⁿ). In particular, this implies that φk|Ω_e = 0 and φk→ u − v in L²(Rⁿ), so that it follows that u|Ω_e = v|Ω_e as L²(Ωe)-functions.

For the sake of readability we will write u|Ω_e instead of γΩ_eu in the following. Also note that Lemma 2.2 implies that for two functions F, G ∈ C_c^∞(Ωe), F = G if and only if F − G ∈ H₀^s(Ω), so that we can identify C_c^∞(Ωe) with its image in the quotient space H(Ωe) and thus consider C_c^∞(Ωe) as a subspace of H(Ωe).

We can now state the following result on the solvability of the Dirichlet problem and the definition of Neumann boundary values.

Lemma 2.3. Let q ∈ L^∞+(Ω).

5

(a) For every F ∈ H(Ωe) and f ∈ L²(Ω), we have that u ∈ H^s(Rⁿ) solves the Dirichlet problem

(−∆)^su + qu = f in Ω, u|Ω_e= F, (2.2) if and only if u = u⁽⁰⁾+ u^{(F )}, where u^{(F )}∈ H^s(Rⁿ) fulfills u^{(F )}|Ω_e = F (for F ∈ C_c^∞(Ωe) we can simply choose u^{(F )}:= F ), and u⁽⁰⁾∈ H₀^s(Ω) solves

Bq(u⁽⁰⁾, w) = −Bq(u^{(F )}, w) + ˆ

Ω

f w dx for all w ∈ H₀^s(Ω).

The Dirichlet problem (2.2) is uniquely solvable and the solution u ∈ H^s(Rⁿ) depends linearly and continuously on F ∈ H(Ωe) and f ∈ L²(Ω).

(b) For a solution u ∈ H^s(Rⁿ) of (−∆)^su + qu = 0, we define the Neumann exterior data N u ∈ H(Ωe)^∗ by

hN u, Gi :=Bq(u, v^(G)), where v^(G)∈ H^s(Rⁿ) fulfills v^(G)|Ω_e = G, where H(Ωe)^∗is the dual space of H(Ωe) and h·, ·i := h·, ·i_H(Ω

e)^∗×H(Ωe). Then the Dirichlet-to-Neumann operator

Λ(q) : H(Ω_e) → H(Ω_e)^∗, F 7→ N u,

is a symmetric linear bounded operator, where u solves (2.2) with f = 0.

Proof. Obviously Bq is a coercive, symmetric, and continuous bilinear form on the Hilbert space H₀^s(Ω). Thus the assertion follows from a standard application of the Lax-Milgram theorem and the equivalence result in Lemma 2.1.

Remark 2.4. If u ∈ H^2s(Rⁿ) solves

(−∆)^su + qu = 0 in Ω, then a computation as in the proof of Lemma 2.1 shows that

hN u, F i = ˆ

Ω_e

(−∆)^su · v^{(F )}dx

where v^{(F )}|Ωe = F . This motivates to formally write the Neumann boundary values as

N u = (−∆)^su|_Ωe.

Note that N u = (−∆)^su|_Ωe ∈ L²(Ω_e) rigorously holds under the additional smooth- ness condition u ∈ H^2s(Rⁿ) (not only u ∈ H^s(Rⁿ)) but no regularity assumptions on the open set Ω ⊆ Rⁿ is required. Alternatively, it is also possible to justify the notation N u = (−∆)^su|Ω_e without the additional smoothness (i.e., for u ∈ H^s(Rⁿ)) when regularity assumptions on Ω are imposed, cf. [20].

For a more in-depth discussion on the spaces of Dirichlet and Neumann boundary values for Lipschitz domains we refer readers to [20]. For the results in this work it suffices to use the abstract quotient space definitions given above, which also has the advantage that we can treat arbitrary open sets Ω without any boundary regularity assumptions.

3. Monotonicity relations and localized potentials for the fractional Schr¨odinger equation. In this section we will derive monotonicity and localized potentials results for the fractional Schr¨odinger equation

(−∆)^su + qu = 0 in Ω.

3.1. Monotonicity relations. We will first show that increasing the coefficient q increases the Dirichlet-Neumann-operator in the sense of quadratic forms.

Lemma 3.1. (Monotonicity relations) Let n ∈ N, Ω ⊆ Rⁿ be an open set, and s > 0.

For j = 0, 1, let qj∈ L^∞₊(Ω) and uj∈ H^s(Rⁿ) be solutions of ((−∆)^su_j+ q_ju_j = 0 in Ω,

uj|Ω_e = F, (3.1)

where F ∈ H(Ωe). Then we have the following monotonicity relations h(Λ(q1) − Λ(q0)) F, F i ≤

ˆ

Ω

(q1− q0)|u0|²dx, (3.2)

and

h(Λ(q1) − Λ(q0))F, F i ≥ ˆ

Ω

(q1− q0) |u1|²dx. (3.3) Moreover, we have

h(Λ(q1) − Λ(q0))F, F i ≥ ˆ

Ω

q0

q₁(q1− q0) |u0|²dx (3.4) and

h(Λ(q₁) − Λ(q₀))F, F i ≤ ˆ

Ω

q1

q0

(q₁− q₀) |u₁|²dx. (3.5)

Proof. The proof is similar to [31, Lemma 2.1]. Note also that the idea goes back to Ikehata, Kang, Seo, and Sheen [35, 39], and that similar results have been obtained and used in many other works on the monotonicity and factorization method.

From the definition of the Dirichlet-Neumann-operator in Lemma 2.3 we have that hΛ(q0)F, F i =Bq₀(u0, u0), and

hΛ(q₁)F, F i =Bq1(u₁, u₁) =Bq1(u₁, u₀) We thus obtain

0 ≤Bq₁(u1− u0, u1− u0) =Bq₁(u1, u1) − 2Bq₁(u1, u0) +Bq₁(u0, u0)

= − hΛ(q₁)F, F i +Bq1(u₀, u₀)

=h(Λ(q0) − Λ(q1))F, F i +Bq₁(u0, u0) −Bq₀(u0, u0)

=h(Λ(q0) − Λ(q1))F, F i + ˆ

Ω

(q1− q0)|u0|²dx,

7

which shows the first assertion (3.2). Interchanging q0and q1in the above calculation also yields (3.3).

In addition, when qj ∈ L^∞₊(Ω), one can see

h(Λ(q1) − Λ(q0))F, F i =Bq₀(u0− u1, u0− u1) − ˆ

Ω

(q0− q1)|u1|²dx

= ˆ

Rⁿ

|(−∆)^s/2(u1− u0)|²dx + ˆ

Ω

q0(u1− u0)²+ (q1− q0)|u1|²
dx

≥ ˆ

Ω

q0(u1− u0)²+ (q1− q0)|u1|²
dx = ˆ

Ω

q1u²₁− 2q0u1u0+ q0u²₀
dx

= ˆ

Ω

q₁

 u₁−q0

q₁u₀

² dx +

ˆ

Ω

 q₀−q₀²

q₁



|u₀|²dx

≥ ˆ

Ω

q0

q1

(q1− q0) |u0|²dx,

which proves (3.4), and interchanging q₀and q₁ yields (3.5).

For two functions q₀, q₁∈ L^∞₊(Ω), we write q₀≥ q₁if q₀(x) ≥ q₁(x) almost everywhere in Ω. For two operators Λ(q₀), Λ(q₁) : H(Ω_e) → H(Ω_e)^∗ we write Λ(q₀) ≥ Λ(q₁) if

h(Λ(q0) − Λ(q₁))F, F i ≥ 0 for all F ∈ H(Ω_e). (3.6) With respect to these partial orders, we have the following monotonicity property of the Dirichlet-Neumann-operators.

Corollary 3.2. For any q0, q1∈ L^∞₊(Ω),

q0≥ q1 implies Λ(q0) ≥ Λ(q1). (3.7)

Proof. This follows immediately from Lemma 3.1.

3.2. Localized potentials . In this subsection we will show the existence of localized potentials solutions of the fractional Schr¨odinger equation that have an ar- bitrarily high energy on some part of the imaging domain and an arbitrarily low energy on another part. These localized potentials will allow us to control the energy terms in Lemma 3.1 and show a converse of the monotonicity relation (3.7).

For the fractional Schr¨odinger equation, the existence of localized potentials is a simple consequence from the unique continuation and Runge approximation result shown by Ghosh, Salo and Uhlmann [20], see also [18] for further discussions and [29] for the connection between Runge approximation properties and localized potentials. We use the following unique continuation result from Ghosh, Salo and Uhlmann [20].

Theorem 3.3. [20, Theorem 1.2] Let n ∈ N, and 0 < s < 1. If u ∈ H^r(Rⁿ) for some r ∈ R, and both u and (−∆)^su vanish in the same arbitrary non-empty open set in Rⁿ, then u ≡ 0 in Rⁿ.

Ghosh, Salo and Uhlmann [20, Theorem. 1.3] also showed that this unique continu- ation result implies the Runge-type approximation property that any L²(Ω)-function can be approximated by solutions of the fractional Schr¨odinger equation with exte- rior Dirichlet data supported on an arbitrarily small open set. Since our formulation

slightly differs from [20], and the proof if short and simple, we give the proof for the sake of completeness.

Theorem 3.4. Let n ∈ N, Ω ⊆ Rⁿ be an open set, 0 < s < 1, q ∈ L^∞₊(Ω), and O ⊆ Ωe= Rⁿ\ Ω be open. For every f ∈ L²(Ω) there exists a sequence Fk ∈ C_c^∞(O), so that the corresponding solutions u^k∈ H^s(Rⁿ) of

(−∆)^su^k+ qu^k = 0 in Ω, u^k|Ω_e = Fk, fulfill that u^k|Ω→ f in L²(Ω).

Proof. For F ∈ C_c^∞(O) let S(F ) := u ∈ H^s(Rⁿ) denote the solution of

(−∆)^su + qu = 0 in Ω, u|Ω_e = F, (3.8) i.e. (see Lemma 2.3) u = u⁽⁰⁾+F where u⁽⁰⁾∈ H₀^s(Ω) solvesBq(u⁽⁰⁾, w) = −Bq(F, w) for all w ∈ H₀^s(Ω).

The assertion follows if we can show that the space of all such solutions {S(F )|Ω: F ∈ C_c^∞(O)} ⊆ L²(Ω)

has trivial L²(Ω)-orthogonal complement. To that end let f ∈ {S(F )|Ω: F ∈ C_c^∞(O)}^⊥⊆ L²(Ω) and v ∈ H^s(Rⁿ) solve

(−∆)^sv + qv = f in Ω, v|_Ωe= 0, i.e., v ∈ H₀^s(Ω) and Bq(v, w) = ´

Ωf w|Ωdx for all w ∈ H₀^s(Ω). Then for all F ∈ C_c^∞(O), the solution u = S(F ) of (3.8) fulfills

0 = ˆ

Ω

f udx = ˆ

Ω

f (u⁽⁰⁾+ F )dx = ˆ

Ω

f u⁽⁰⁾dx =Bq(v, u⁽⁰⁾) = −Bq(v, F ).

Using Lemma 2.1 with O instead of Ω this yields that (−∆)^sv + qv = 0 in O.

Since also v|O= 0 (cf. Lemma 2.2), it follows from Theorem 3.3 that v ≡ 0 in Rⁿand thus f = 0 which proves the assertion.

Theorem 3.4 implies the existence of a localized potentials sequence.

Corollary 3.5. Let n ∈ N, Ω ⊆ Rⁿ be an open set, 0 < s < 1, q ∈ L^∞₊(Ω), and O ⊆ Ωe = Rⁿ\ Ω be an arbitrary open set. For every measurable set M ⊆ Ω with positive measure, there exists a sequence F^k∈ C_c^∞(O), so that the corresponding solutions u^k ∈ H^s(Rⁿ) of

(−∆)^su^k+ qu^k = 0 in Ω, u^k|_Ωe = F^k, for all k ∈ N (3.9) fulfill that

ˆ

M

|u^k|²dx → ∞ and ˆ

Ω\M

|u^k|²→ 0 as k → ∞.

9

Proof. Using Theorem 3.4 there exists a sequence eF^k so that the corresponding solutionsue^k|_Ω converge against 1

|M |χ_M in L²(Ω), and thus keu^kk²_L2(M )=

ˆ

M

|eu^k|²dx → 1, and keu^kk²_L2(Ω\M )= ˆ

Ω\M

|ue^k|²dx → 0.

Without loss of generality, we can assume for all k ∈ N that ue^k 6≡ 0. Moreover, by possibly removing a sufficiently small open ball from M (so that the measure of M remains positive), we can assume that Ω \ M contains an open set. Thus, kue^(k)k_L2(Ω\M )> 0 follows from Theorem 3.3. Setting

F^k := Fe^k keu^kk^1/2_L2(Ω\M )

the sequence of corresponding solutions u^k ∈ H^s(Rⁿ) of (3.9) has the desired property that

ku^kk²_L2(M )= kue^kk²_L2(M )

keu^kkL²(Ω\M )

→ ∞, and ku^kk²_L2(Ω\M )= keu^kk_L2(Ω\M )→ 0,

as k → ∞.

The following lemma will also be useful for applying localized potentials in next sec- tions.

Lemma 3.6. Let n ∈ N, Ω ⊆ Rⁿ be a bounded open set, 0 < s < 1, and q0, q1 ∈ L^∞₊(Ω). Set D := supp(q0− q1). There exist constants c, C > 0 so that, for all F ∈ H(Ωe), the solutions u0, u1∈ H^s(Rⁿ) of

(−∆)^su_j+ q_ju_j= 0 in Ω, u_j|Ωe= F (j = 0, 1) fufill

cku₁k_L2(D)≤ ku₀k_L2(D)≤ Cku₁k_L2(D). (3.10)

Proof. For the difference w := u0− u1∈ H₀^s(Ω) we have that 0 =Bq₀(u0, w) =Bq₁(u1, w) for all w ∈ H₀^s(Ω).

With the coercivity constant α > 0 ofBq1 we can estimate

αku1− u0k²_Hs(Ω)≤Bq₁(u1− u0, u1− u0) = −Bq₁(u0, u1− u0)

=Bq₀(u0, u1− u0) −Bq₁(u0, u1− u0)

= ˆ

Ω

(q0− q1)u0(u1− u0) dx

≤ kq0− q1k_L^∞_(Ω)ku0k_L2(D)ku1− u0k_Hs(Ω).

Hence,

ku1k_L2(D)− ku0k_L2(D)≤ ku1− u0k_L2(D)≤ ku1− u0k_Hs(Ω)

≤ 1

αkq0− q1k_L∞(Ω)ku0k_L2(D), which shows that

ku₁k_L2(D) ≤ Cku₀k_L2(D) with C := 1 + 1

αkq₀− q₁k_L∞(Ω). The other inequality follows from interchanging q₀ and q₁.

4. Converse monotonicity relations and the nonlocal Calder´on prob- lem. This section contains the first main result of this work. We will show that the monotonicity relation between the coefficients and the Dirichlet-to-Neumann opera- tors holds in both directions, and use this to give a monotonicity-based, constructive proof of Ghosh, Salo and Uhlmann’s uniqueness result for the Calder´on problem for the fractional Schr¨odinger equation [20].

Theorem 4.1. Let n ∈ N, Ω ⊆ Rⁿ be a bounded open set, and 0 < s < 1. For any two potentials q0, q1∈ L^∞₊(Ω), we have that

q₀≤ q1 if and only if Λ(q₀) ≤ Λ(q₁). (4.1)

Proof. From Corollary 3.2, we know that q₀ ≤ q1 implies Λ(q₀) ≤ Λ(q₁). Hence, it remains to show Λ(q₀) ≤ Λ(q₁) implies that q₀ ≤ q1 a.e. in Ω. We will prove this via contradiction and assume that q₀ ≤ q1 is not true a.e. in Ω. Then there exists δ > 0 and a measurable set M ⊂ Ω with positive measure such that q₀− q₁≥ δ on M . Using the sequence of localized potentials from Corollary 3.5 for the coefficient q0, and the monotonicity inequality (3.2) from Lemma 3.1, we obtain

(Λ(q1) − Λ(q0))F^k, F^k ≤ ˆ

Ω

(q1− q0)|u^k₀|²dx

≤ kq1− q0k_L^∞_{(Ω\M )}ku^(k)₀ k²_L2(Ω\M )− δku^k₀k²_L2(M )

→ −∞, as k → ∞.

This shows that Λ(q0) 6≤ Λ(q1).

Theorem 4.1 implies global uniqueness for the fractional Calder´on problem. But let us stress again, that this has already been proven in [20] and we have used the results from [20] in our proof of the existence of localized potentials.

Corollary 4.2. Let n ∈ N, Ω ⊆ Rⁿ be a bounded open set, and 0 < s < 1. For any two potentials q0, q1∈ L^∞₊(Ω),

q0= q1 if and only if Λ(q0) = Λ(q1).

Proof. This follows immediately from Theorem 4.1.

Moreover, Theorem 4.1 suggests the constructive uniqueness result that q can be reconstructed from Λ(q) by taking the supremum over an appropriate class of test

11

functions ψ with Λ(ψ) ≤ Λ(q). For a rigorous formulation of this result we have to be attentive to the somewhat subtle fact that function values on null sets might still affect the supremum when the supremum is taken over uncountably many functions.

Recall that a point x ∈ Rⁿ is called of density one for E if

r→0lim

|B(x, r) ∩ E|

|B(x, r)| = 1,

where B(x, r) stands for the ball of radius r and centered at x. We therefore define the space of density one simple functions

Σ :=n

ψ =Pm

j=1a_jχ_Mj : a_j∈ R, Mj ⊆ Ω is a density one seto ,

where we call a subset M ⊆ Ω a density one set if it is non-empty, measurable and has Lebesgue density 1 in all x ∈ M . Σ+ ⊆ Σ denotes the subset of density one simple functions with positive essential infima on Ω (i.e., where all coefficients aj are positive).

Lemma 4.3.

(a) Density one sets have positive measure.

(b) Finite intersections of density one sets are density one sets.

(c) If ψ ∈ Σ is nonzero at some point ˆx ∈ Ω, then there exists a density one set M containing ˆx so that ψ(x) = ψ(ˆx) for all x ∈ M .

Proof. Assertion (a) is obvious. To prove (b), let M1, M2 ⊆ Ω be density one sets, and let x ∈ M1∩ M2. Then

r→0lim

|B(x, r) \ (M1∩ M2)|

|B(x, r)| ≤ lim

r→0

|B(x, r) \ M1|

|B(x, r)| + lim

r→0

|B(x, r) \ M2|

|B(x, r)| = 0, which shows that

lim

r→0

|B(x, r) ∩ M1∩ M2|

|B(x, r)| = 1.

For the last assertion, let ψ ∈ Σ. Then ψ =Pm

j=1ajχM_j with aj∈ R and density one sets Mj ⊆ Ω. Then ψ is constant on the intersection of all Mj containing ˆx, so that (c) follows from (b).

Lemma 4.3(c) shows that we might interpret the density one simple functions as simple functions where function values that are only attained on a null set are replaced by zero. Note also, that the Lebesgue’s density theorem implies that every measurable set agrees almost everywhere with a density one set (see Corollary 3 in Section 1.7 of [13] for instance), and thus every simple function agrees with a density one simple function almost everywhere in Ω.

As before, we write ψ ≤ q if ψ(x) ≤ q(x) almost everywhere in Ω. We then have the following variant of the simple function approximation theorem.

Lemma 4.4. For each function q ∈ L^∞+(Ω) we have that

q(x) = sup{ψ(x) : ψ ∈ Σ₊, ψ ≤ q} almost everywhere in Ω.

Proof. By the standard simple function approximation theorem [54], there exists a sequence (ψk)_k∈N of simple functions with ψk ≤ q and kψk − qk_L∞(Ω) ≤ 1/k.

q ∈ L^∞₊(Ω) implies that ψk∈ L^∞₊(Ω) for almost all k ∈ N, and by changing the values of the countably many functions ψk on a null set we can assume that ψk ∈ Σ+. This shows that

q(x) = lim

k→∞ψ_k(x) ≤ sup{ψ(x) : ψ ∈ Σ₊, ψ ≤ q} almost everywhere in Ω.

To show equality, it suffices to show that for each δ > 0 the set M := {x ∈ Ω : q(x) + δ < sup{ψ(x) : ψ ∈ Σ₊, ψ ≤ q}}

is a null set. To prove this, assume that M is not a null set for some δ > 0. By removing a null set from M , we can assume that M is a density one set and that q(x) > 0 for all x ∈ M . By using the Lusin’s theorem (see [54] for instance), all measurable function are approximately continuous at almost every point, M must contain a point ˆx in which q is approximately continuous, and thus the set

M⁰:= {x ∈ Ω : q(x) ≤ q(ˆx) + δ/3}

has density one in ˆx. (see [13]). Removing a null set, we can assume that M⁰ is a density one set still containing ˆx.

Moreover, by the definition of M , there must exist a ψ ∈ Σ₊ with ψ ≤ q and

q(ˆx) +2

3δ ≤ ψ(ˆx).

Since q(ˆx) > 0, there exists a density one set M⁰⁰containing ˆx where ψ(x) = ψ(ˆx) for all x ∈ M⁰⁰.

We thus have that

q(x) + δ/3 ≤ q(ˆx) +2

3δ ≤ ψ(ˆx) = ψ(x) for all x ∈ M⁰∩ M⁰⁰,

with density one sets M⁰ and M⁰⁰ that both contain ˆx, so that their intersection possesses positive measure. But this contradicts that q(x) ≥ ψ(x) almost everywhere, and thus shows that M is a null set for all δ > 0, and hence

q(x) ≥ sup{ψ(x) : ψ ∈ Σ+, ψ ≤ q} almost everywhere in Ω.

Corollary 4.5. Let n ∈ N, Ω ⊆ Rⁿ be an open set, and 0 < s < 1. A potential q ∈ L^∞₊(Ω) is uniquely determined by Λ(q) via the following formula

q(x) = sup{ψ(x) : ψ ∈ Σ₊, Λ(ψ) ≤ Λ(q)} almost everywhere in Ω.

Proof. This follows immediately from Theorem 4.1 and Lemma 4.4.

13

5. Shape reconstruction by linearized monontonicity tests . The results in Section 4 show that the coefficient q in the fractional Schr¨odinger equation

(−∆)^su + qu = 0 in Ω

can be reconstructed from the Dirichlet-to-Neumann operator Λ(q) by comparing Λ(q) with the DtN map Λ(ψ) of (density one) simple functions ψ. A practical implemen- tation of these monotonicity tests would require solving the fractional Schr¨odinger equation for each utilized simple function ψ.

In this section we will study the shape reconstruction problem of determining regions where a coefficient function q ∈ L^∞₊(Ω) changes from a known reference function q0 ∈ L^∞₊(Ω) (e.g., q0 may describe a background coefficient, and q1 denotes the coefficient function in the presence of anomalies or scatterers). We will show that the support of q1− q0 can be reconstructed with linearized monotonicity tests [32, 14].

These linearized tests only utilize the solution of the fractional Schr¨odinger equation with the reference coefficient function q0 ∈ L^∞₊(Ω). They do not require any other special solutions of the equation.

5.1. Linerization of the Dirichlet-to-Neumann operator. We start by showing Fr´echet differentiability of the Dirichlet-to-Neumann operator.

Lemma 5.1. Let n ∈ N, Ω ⊆ Rⁿ be a bounded open set, and 0 < s < 1. The Dirichlet-to-Neumann operator

Λ : D(Λ) := L^∞+(Ω) ⊂ L^∞(Ω) → L(H(Ω_e), H(Ω_e)^∗), q 7→ Λ(q), is Fr´echet differentiable. At q ∈ L^∞₊(Ω) its derivative is given by

Λ⁰(q) : L^∞(Ω) → L(H(Ω_e), H(Ω_e)^∗), r 7→ Λ⁰(q)r, h(Λ⁰(q)r)F, Gi : =

ˆ

Ω

rS_q(F )S_q(G)dx for all r ∈ L^∞(Ω), F, G ∈ H(Ω_e),

where Sq : H(Ωe) → H^s(Rⁿ), F 7→ u, is the solution operator of the Dirichlet problem (−∆)^su + qu = 0 in Ω and u|Ω_e= F.

Proof. Let q ∈ L^∞₊(Ω). Λ⁰(q) is a linear bounded operator since Sq is linear and bounded, cf. Lemma 2.3. For sufficiently small r ∈ L^∞(Ω), so that q + r ∈ L^∞₊(Ω), we obtain from the monotonicity relations (3.2) and (3.4) in Lemma 3.1 that for all F ∈ H(Ωe),

0 ≥ h(Λ(q + r) − Λ(q) − Λ⁰(q)r) F, F i ≥ ˆ

Ω

 q q + rr − r



|uq|²dx,

where u_q = S_q(F ).

Using that Λ(q), Λ(q + r), and Λ⁰(q)r are symmetric operators, it follows that kΛ(q + r) − Λ(q) − Λ⁰(q)rk_L(H(Ωe),H(Ω_e)^∗)

= sup

kF k_H(Ωe)=1

|h(Λ(q + r) − Λ(q) − Λ⁰(q)r) F, F i|

≤ sup

kF k_H(Ωe)=1

ˆ

Ω

q q + rr − r

|uq|²dx ≤

r² q + r

_L∞(Ω)

sup

kF k_H(Ωe)=1

kSq(F )k²_L2(Ω)

≤ krk_L∞(Ω)

r q + r

_L∞(Ω)

kSqk_L(H(Ωe),Hs(Rⁿ)),

which shows

krk_{L∞ (Ω)}lim →0

kΛ(q + r) − Λ(q) − Λ⁰(q)rk_{L(H(Ωe),H(Ωe)}∗)

krk_L^∞_(Ω) = 0.

Remark 5.2. Using the Fr´echet derivative, the monotonicity relations (3.2) and (3.4) in Lemma 3.1 can be written as follows. For all q₀, q₁∈ L^∞₊(Ω)

Λ⁰(q0)(q1− q0) ≥ Λ(q1) − Λ(q0) ≥ Λ⁰(q0) q0

q₁(q1− q0)

 .

We also have an analogue of the monotonicity result in Theorem 4.1.

Theorem 5.3. Let n ∈ N, Ω ⊆ Rⁿ be a bounded open set, and 0 < s < 1. Then for all q ∈ L^∞₊(Ω) and r0, r1∈ L^∞(Ω),

r0≤ r1 if and only if Λ⁰(q)r0≤ Λ⁰(q)r1.

Proof. If r0≤ r1then Λ⁰(q)r0≤ Λ⁰(q)r1follows immediately from the characterization of Λ⁰(q) in Lemma 5.1. The converse follows from the same localized potentials argument as in the proof of Theorem 4.1.

Note that this implies uniqueness of the linearized fractional Calder´on problem:

Corollary 5.4. Let n ∈ N, Ω ⊆ Rⁿ be a bounded open set, and 0 < s < 1. For all q ∈ L^∞₊(Ω), the Fr´echet derivative Λ⁰(q) is injective, i.e.

Λ⁰(q)r = 0 if and only if r = 0.

Proof. This follows immediately from Theorem 5.3.

5.2. Reconstructing the support of a coefficient change. In this subsec- tion, let n ∈ N, Ω ⊆ Rⁿbe a bounded open set, and 0 < s < 1. As in the introduction, let q₀ ∈ L^∞₊(Ω) denote a known reference coefficient, and q₁ ∈ L^∞₊(Ω) denote an un- known coefficient function that differs from the reference value q₀ in certain regions.

We aim to find these anomalous regions (or scatterers), i.e., the support of q₁− q₀, from the difference of the Dirichlet-to-Neumann-operators Λ(q₁) − Λ(q₀).

15

To that end, we introduce, for a measurable subset M ⊆ Ω, the testing operator TM : H(Ωe) → H(Ωe)^∗ by setting TM := Λ⁰(q0)χM. i.e.,

hTMF, Gi :=

ˆ

M

Sq₀(F )Sq₀(G)dx for all F, G ∈ H(Ωe), (5.1) where, as in Lemma 5.1, Sq₀ : H(Ωe) → H^s(Rⁿ), F 7→ u0, denotes the solution operator of the reference Dirichlet problem

(−∆)^su0+ q0u0= 0 in Ω and u0|Ω_e= F.

The following theorem shows that we can find the support of q − q₀ by shrinking closed sets, cf. [32, 16].

Theorem 5.5. For each closed subset C ⊆ Ω,

supp(q₁− q0) ⊆ C if and only if ∃α > 0 : −αTC ≤ Λ(q1) − Λ(q₀) ≤ αT_C. Hence,

supp(q₁− q₀) =\

{C ⊆ Ω closed : ∃α > 0 : −αT_C ≤ Λ(q₁) − Λ(q₀) ≤ αT_C}.

Proof.

(a) Let supp(q₁− q₀) ⊆ C. Then every sufficiently large α > 0 fulfills q1≤ q0+ αχC.

Using Theorem 4.1 and Remark 5.2, we thus obtain

Λ(q₁) ≤ Λ(q₀+ αχ_C) ≤ Λ(q₀) + Λ⁰(q₀)αχ_C= Λ(q₀) + αT_C. Moreover, for sufficiently small β > 0 we also have that

q1≥ q0+ (β − q0)χC and q0≥ β and thus (using Theorems 4.1, 5.3, and Remark 5.2)

Λ(q1) − Λ(q0) ≥ Λ(q0+ (β − q0)χC) − Λ(q0)

≥ Λ⁰(q0)

 q0

q0+ (β − q0)χC

(β − q0)χC



≥ −Λ⁰(q₀)

 q²₀

q₀+ (β − q₀)χ_Cχ_C



≥ −1

βkq0k²_L∞(Ω)Λ⁰(q0)χC, which shows that

Λ(q1) − Λ(q0) ≥ −αTC

is also fulfilled for sufficiently large α > 0.

(b) To show the converse implication, let α > 0 fulfill

−αTC≤ Λ(q1) − Λ(q₀) ≤ αT_C. Then we obtain using Remark 5.2

Λ⁰(q0)(−αχC) = −αTC≤ Λ(q1) − Λ(q0) ≤ Λ⁰(q0)(q1− q0), so that it follows from Theorem 5.3 that

q₁− q0≥ −αχC,

and in particular q1− q0≥ 0 almost everywhere on Ω \ C.

Likewise we obtain using Remark 5.2

Λ⁰(q0)(αχC) = αTC≥ Λ(q1) − Λ(q0) ≥ Λ⁰(q0) q0

q₁(q1− q0)

 , so that it follows from Teorem 5.3 that

q0

q1

(q1− q0) ≤ αχC.

Since q1, q0 ∈ L^∞₊(Ω) this yields that q1− q0 ≤ 0 almost everywhere on Ω \ C. Hence, q1 = q0 almost everywhere in the open set Ω \ C and thus supp(q1− q0) ⊆ C.

In the definite case that either q1 ≥ q0 or q1 ≤ q0 holds almost everywhere in Ω, we can also use the union of small test balls to characterize the so-called inner support of q1− q0. The inner support inn supp(r) of a measurable function r : Ω → R is defined as the union of all open sets U on which the essential infimum of |κ| is positive, cf.

[32, Section 2.2].

Theorem 5.6.

(a) Let q₁≤ q₀. For every open set B ⊆ Ω and every α > 0 (1) q1≤ q0− αχB implies Λ(q1) ≤ Λ(q0) − αTB. (2) Λ(q1) ≤ Λ(q0) − αTB implies B ⊆ inn supp(q1− q0).

Hence,

inn supp(q₁− q0) =[

{B ⊆ Ω open ball : ∃α > 0 : Λ(q1) ≤ Λ(q₀) − αT_B}.

(b) Let q1≥ q0. For every open set B ⊆ Ω and every α > 0

(1) q1≥ q0+ αχB implies Λ(q1) ≥ Λ(q0) + ˜αTB with ˜α := _inf(q^inf(q0)α

0)+α. (2) Λ(q₁) ≥ Λ(q₀) + αT_B implies B ⊆ inn supp(q − q₀).

Hence,

inn supp(q1− q0) =[

{B ⊆ Ω open ball : ∃α > 0 : Λ(q1) ≥ Λ(q0) + αTB}.

Proof.

(a) If q1≤ q0− αχB, then we obtain using Theorem 5.3, and Remark 5.2 that Λ(q1) − Λ(q0) ≤ Λ⁰(q0)(q1− q0) ≤ −αΛ⁰(q0)χB = −αTB.

17

On the other hand, if Λ(q) ≤ Λ(q0) − αTB then we obtain from Remark 5.2 that

−αΛ⁰(q0)χB= −αTB ≥ Λ(q1) − Λ(q0) ≥ Λ⁰(q0) q0

q₁(q1− q0)



so that it follows from Theorem 5.3 that

−αχB≥ q₀ q1

(q₁− q0).

Hence, q0−q1≥ _sup(q^inf(q1)

0)α almost everywhere on B and thus B ⊆ inn supp(q1− q₀).

(b) If q₁ ≥ q₀+ αχ_B, then we obtain using Theorems 4.1, 5.3, and Remark 5.2 that

Λ(q1) − Λ(q0) ≥ Λ(q0+ αχB) − Λ(q0)

≥ Λ⁰(q0)

 q0

q0+ αχB

αχB



= Λ⁰(q0)



1 − α q0+ α

 αχB



≥ Λ⁰(q0)



1 − α

inf(q0) + α

 αχB



= inf(q0)α inf(q0) + αTB. On the other hand, if Λ(q₁) ≥ Λ(q₀) + αT_B then we obtain from Remark 5.2 that

αΛ⁰(q0)χB = αTB ≤ Λ(q1) − Λ(q0) ≤ Λ⁰(q0)(q1− q0), so that it follows from Theorem 5.3 that

αχ_B≤ q₁− q₀, and thus B ⊆ inn supp(q1− q0).

6. Discussion and Outlook. We have shown an if-and-only-if monotonicity re- lation between a positive potential q ∈ L^∞₊(Ω) in the fractional Schr¨odinger equation, and the associated Dirichlet-to-Neumann operator Λ(q) (cf. Theorem 4.1)

q0≤ q1 if and only if Λ(q0) ≤ Λ(q1).

From this we obatined a constructive uniqueness result for the Calder´on problem for the fractional Schr¨odinger equation. The potential is uniquely determined by the simple reconstruction formula (cf. Corollary 4.5)

q(x) = sup {ψ(x) : ψ positive (density one) simple function, Λ(ψ) ≤ Λ(q)} . Let us give some remarks on a possible practical implementation of our results. First of all, let us stress that the localized potentials used in this work can be created with Dirichlet data supported in arbitrarily small open subsets ∅ 6= O ⊆ Ωe, cf. Corol- lary 3.5. Hence, all results in this work remain valid if the full data DtN is replaced by the partial data DtN

Λ(q) : H₀^s(O) → H^−s(O),

where H₀^s(O) is the closure of C_c^∞(O) in H^s(Rⁿ), and H^−s(O) := H₀^s(O)⁰.

For a numerical implementation, one could choose a family of characteristic functions χ1, . . . , χM for disjoint density one sets (e.g., a pixel partition) P1, . . . , PM ⊆ Ω, M ∈ N, and determine

αm:= sup{α ∈ R : Λ(α^mχm) ≤ Λ(q)}.

Then, ψ = PM

m=1α_mχ_m is the largest piecewise-constant function (on the given partition) with ψ ≤ q. Analogously, one could obtain a piecewise-constant function approximation q from above. A numerical implementation of this approach would be computationally rather expensive as it requires solving the fractional Schr¨odinger equation for a large number of sets P_m and contrast levels α (though these solutions could be precomputed in advance).

A computationally more efficient approach can be used for detecting regions where the potential q differs from a known reference function q0. The support of this change can be determined by shrinking closed sets according to the formula (cf. Theorem 5.5)

supp(q − q0) =\

{C ⊆ Ω closed : ∃α > 0 : −αTC ≤ Λ(q) − Λ(q0) ≤ αTC}, where the operator TCcan be calculated from integrating the solution of the fractional Schr¨odinger equation for the reference potential q0 over the set C, and no other PDE solutions are required for this approach. Moreover, the inner support of the potential change can be calculated by comparing Λ(q) − Λ(q₀) with T_B for open balls B, cf. Theorem 5.6.

Algorithms based on linearized monotonicity tests have been successfully applied to the standard Laplacian case (s = 1), cf. the works cited in the introduction. Among these works, let us mention the recent papers [15, 29] that show reconstructions on simulated and real-life measurement data, and discuss practical implementation issues and the regularization of measurement errors.

For the standard Laplacian case, monotonicity-based reconstruction methods have recently been extended to the Schr¨odinger (or Helmholtz) equation with general (not- necessarily positive) potential function q ∈ L^∞(Ω), cf. [29, 21], and monotonicity arguments were also used to prove stability results, cf. [27, 24, 61]. It can be expected that these results can also be extended to the fractional diffusion case, and we aim to study these questions in future research.

Acknowledgment. Yi-Hsuan Lin is partially supported by MOST of Taiwan under the project 160-2917-I-564-048.

REFERENCES

[1] G. Alessandrini. Singular solutions of elliptic equations and the determination of conductivity by boundary measurements. J. Differential Equations, 84(2):252–272, 1990.

[2] L. Arnold and B. Harrach. Unique shape detection in transient eddy current problems. Inverse Problems, 29(9):095004, 2013.

[3] A. Barth, B. Harrach, N. Hyv¨onen, and L. Mustonen. Detecting stochastic inclusions in elec- trical impedance tomography. Inverse Problems, 33(11):115012, 2017.

[4] T. Brander, B. Harrach, M. Kar, and M. Salo. Monotonicity and enclosure methods for the p-Laplace equation. SIAM J. Appl. Math., 78(2):742–758, 2018.

[5] T. Brander, M. Kar, and M. Salo. Enclosure method for the p-Laplace equation. Inverse Problems, 31(4):045001, 2015.

19