STABILITY ESTIMATES FOR THE INVERSE FRACTIONAL CONDUCTIVITY PROBLEM GIOVANNI COVI JESSE RAILO TEEMU TYNI AND PHILIPP ZIMMERMANN

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STABILITY ESTIMATES FOR THE INVERSE

FRACTIONAL CONDUCTIVITY PROBLEM

GIOVANNI COVI, JESSE RAILO, TEEMU TYNI, AND PHILIPP ZIMMERMANN

Abstract. We study the stability of an inverse problem for the frac-

tional conductivity equation on bounded smooth domains. We obtain

a logarithmic stability estimate for the inverse problem under suitable

a priori bounds on the globally deﬁned conductivities. The argument

has three main ingredients: 1. the logarithmic stability of the related

inverse problem for the fractional Schr¨odinger equation by R¨uland and

Salo; 2. the Lipschitz stability of the exterior determination problem;

3. utilizing and identifying nonlocal analogies of Alessandrini’s work on

the stability of the classical Calder´on problem. The main contribution

of the article is the resolution of the technical diﬃculties related to the

last mentioned step. Furthermore, we show the optimality of the loga-

rithmic stability estimates, following the earlier works by Mandache on

the instability of the inverse conductivity problem, and by R¨uland and

Salo on the analogous problem for the fractional Schr¨odinger equation.

1. Introduction

Stability estimates for inverse problems give important information on

theoretical limitations of diﬀerent imaging techniques appearing in various

medical, engineering, and scientiﬁc applications. They are also useful for

development of numerical methods. A common feature of many inverse

problems is that they are ill-posed, which means that small measurement

errors may lead to large errors in the reconstructed images. One of the most

popular model problems is the inverse conductivity problem, known as the

Calder´on problem [Cal80], where one aims to recover the conductivity γfrom

the voltage/current measurements on the boundary ∂Ω of an object Ω. In

mathematical terms, one deﬁnes the data as a Dirichlet-to-Neumann (DN)

map Λγ:f7→ γ∂νuf|∂Ω, where ∂νis the outer boundary normal derivative,

the electric potential ufis the unique solution of the boundary value problem

div(γ∇u) = 0 in Ω,

u=fon ∂Ω,

and the voltage fis the given Dirichlet boundary condition. The Calder´on

problem asks to recover γfrom the knowledge of Λγ, which corresponds

to knowing the outer normal ﬂuxes (i.e. boundary currents) generated by

imposing diﬀerent boundary voltages f.

Date: October 6, 2022.

2020 Mathematics Subject Classiﬁcation. Primary 35R30; secondary 26A33, 42B37,

46F12.

Key words and phrases. Fractional Laplacian, fractional gradient, Calder´on problem,

conductivity equation.

arXiv:2210.01875v1 [math.AP] 4 Oct 2022

INVERSE FRACTIONAL CONDUCTIVITY PROBLEM 2

The Calder´on problem serves both as a mathematical model for electri-

cal impedance tomography [Uhl14], and more generally as a prototypical

model for inverse problems. In fact, methods and techniques originally

developed for the classical Calder´on problem have applications in a wide

range of other inverse problems, among which the anisotropic Calder´on prob-

lem [APL05, DSFKSU09], hyperbolic problems [RS88, Sun90] and inverse

problems related to the theory of elasticity [NU94]. The work of Sylvester

and Uhlmann proved a fundamental uniqueness theorem for the classical

Calder´on problem in dimension n≥3, using a reduction to an analogous

problem for the Schr¨odinger equation and constructing the so called complex

geometrical optics (or CGO) solutions [SU87]. Nachman established a re-

construction method [Nac88], and Astala–P¨aiv¨arinta showed a fundamental

uniqueness result when n= 2 using methods from complex analysis and a

reduction to the Beltrami equation [AP06]. We recall the stability theorem

of Alessandrini [Ale88], which is an important motivation for our present

work:

Theorem 1.1 (Alessandrini [Ale88, Theorem 1]).Let Ωbe a bounded do-

main in Rn,n≥3, with C∞boundary ∂Ω. Given sand E,s > n/2,E > 0,

let γ1, γ2be any two functions in Hs+2(Ω) satisfying the following conditions

E−1≤γ`(x),for every xin Ω, ` = 1,2.

kγ`kHs+2(Ω) ≤E, ` = 1,2.

The following estimate holds 1

kγ1−γ2kL∞(Ω) ≤CEω(kΛγ1−Λγ2kH1/2(∂Ω)→H−1/2(∂Ω)),

where the function ωis such that

ω(t)≤ |log t|−δ,for every t, 0<t<1/e,

and δ,0< δ < 1, depends only on nand s.

Mandache showed that the logarithmic stability estimates are optimal

up to the constants C, δ [Man01]. The works of Alessandrini and Man-

dache therefore show that the classical Calder´on problem is ill-posed and

furthermore accurately characterize this phenomenon. Mandache’s work

was recently systematically studied and extended by Koch, R¨uland and Salo

[KRS21] to many diﬀerent settings. For the other recent works on the sta-

bility of the classical Calder´on problem, we point to the following works

[CDR16, CS14], where stability under partial data is obtained, and stability

for recovery of anisotropic conductivies is considered. Under certain a priori

assumptions, such as piecewise constant conductivities, the stronger result

of Lipschitz stability holds [AV05]. Lipschitz stability is also possible with a

ﬁnite number of measurements [AS22]. In a diﬀerent direction, we mention

[AN19] for an application of stability to the statistical Calder´on problem.

In the present work, we study the stability properties of an inverse prob-

lem for a nonlocal analogue of the classical Calder´on problem. There has

been growing interest towards establishing the theory of inverse problems for

elliptic nonlocal variable coeﬃcient operators. Other recent studies include

1Given a bounded linear mapping A:X→Ybetween two Banach spaces, we denote

its operator norm by kAkX→Y.

INVERSE FRACTIONAL CONDUCTIVITY PROBLEM 3

inverse problems for the fractional powers of elliptic second order operators

[GU21] and inverse problems for source-to-solutions maps related to frac-

tional geometric operators on manifolds [FGKU21, QU22]. We note that

the exterior value inverse problems considered in [GU21] for the operators

(div(γ∇))s, 0 <s<1, generated by the heat semigroups, give another

possibility to deﬁne a nonlocal conductivity equation which is presumably

diﬀerent from the equation we study here.

Let s∈(0,1) and consider the exterior value problem for the fractional

conductivity equation

divs(Θγ∇su) = 0 in Ω,

u=fin Ωe,

(1)

where Ωe:=Rn\Ω is the exterior of the domain Ω and Θγ:R2n→Rn×nis

the matrix deﬁned as Θγ(x, y):=γ1/2(x)γ1/2(y)1n×n. We say u∈Hs(Rn)

is a (weak) solution of (1) if u−f∈e

Hs(Ω) and

Bγ(u, φ) := Cn,s

2ˆR2n

γ1/2(x)γ1/2(y)

|x−y|n+2s(u(x)−u(y))(φ(x)−φ(y)) dxdy = 0

holds for all φ∈C∞

c(Ω). For all f∈X:=Hs(Rn)/e

Hs(Ω) in the abstract

trace space there is a unique weak solutions uf∈Hs(Rn) of the fractional

conductivity equation (1). The fractional conductivity operator converges in

the sense of distributions to the classical conductivity operator when applied

to suﬃciently regular functions when s↑1 [Cov20, Lemma 4.2].

The exterior DN map Λγ:X→X∗is deﬁned by

hΛγf, gi:=Bγ(uf, g).

The inverse problem for the fractional conductivity equation asks to recover

the conductivity γfrom Λγ, which maps as Λγ:Hs(Ωe)→H−s

Ωe(Rn) in

the case of Lipschitz domains. We deﬁne mγ:=γ1/2−1 and call it the

background deviation of γ. Let Ω ⊂Rnbe bounded in one direction and

n≥1. (We suppose additionally that 0 < s < 1/2 when n= 1.) The

uniqueness properties of this inverse problem are studied extensively in the

recent literature and the following list summarizes these advances:

•Global uniqueness. If W⊂Ωeis an open nonempty set such that

γi|Ware continuous a.e., and mi∈H2s, n

2s(Rn)∩Hs(Rn), j= 1,2,

then γ1=γ2if and only if Λγ1f|W= Λγ2f|Wfor all f∈C∞

c(W)

[CRZ22]. This result generalizes and expands the scope of the earlier

works [Cov20, RZ22b], which solved the inverse problem in certain

special cases by means of the fractional Liouville transformation.

This is a technique used to reduce the fractional conductivity equa-

tion to the fractional Schr¨odinger equation introduced in [GSU20],

which is in turn better understood.

•Low regularity uniqueness. If W⊂Ωeis an open nonempty set such

that γi|Ware continuous a.e., and mi∈Hs,n/s(Rn), then γ1=γ2

if and only if Λγ1f|W= Λγ2f|Wfor all f∈C∞

c(W) [RZ22c]. This

uses a general UCP result for the fractional Laplacians in [KRZ22].

•Counterexamples for disjoint measurement sets. For any nonempty

open disjoint sets W1, W2⊂Ωewith dist(W1∪W2,Ω) >0 there

INVERSE FRACTIONAL CONDUCTIVITY PROBLEM 4

exist two diﬀerent conductivities γ1, γ2∈L∞(Rn)∩C∞(Rn) such

that γ1(x), γ2(x)≥γ0>0, m1, m2∈Hs,n/s(Rn)∩Hs(Rn), and

Λγ1f|W2= Λγ2f|W2for all f∈C∞

c(W1) [RZ22c]. See the original

work on the construction of counterexamples with H2s, n

2s(Rn) regu-

larity assumptions in [RZ22a], with some limitations in the cases of

unbounded domains when n= 2,3.

In this article, we obtain a quantitative stability estimate for the global

inverse fractional conductivity problem on bounded smooth domains with

full data. This is based on one of the possible global uniqueness proofs

presented in [CRZ22, RZ22c]. There remain some nontrivial challenges in

order to obtain a quantitative version of the partial data uniqueness re-

sults in [CRZ22, RZ22c], as well as to remove the regularity/boundedness

assumptions of the domain even for the full data case.

We will next recall two earlier stability results related to the fractional

Calder´on problems. The ﬁrst one considers the stable recovery of γin the

exterior, based on [CRZ22, Proposition 1.4]. The second one considers the

stability of the analogous inverse problem for the fractional Schr¨odinger

equation (−∆)s+qdue to R¨uland and Salo [RS20, Theorem 1.2]. The

uniqueness properties of the Calder´on problem for large classes of fractional

Schr¨odinger type equations have been extensively studied starting from the

seminal work of [GSU20]. These include perturbations to the fractional pow-

ers of elliptic operators [GLX17], ﬁrst order perturbations [CLR20], nonlin-

ear perturbations [LL22], higher order equations with local perturbations

[CMR21, CMRU22], quasilocal perturbations [Cov21], and general theory

for nonlocal elliptic equations [RS20, RZ22b].

In particular, the following results are needed in our proofs:

Theorem 1.2 ([RZ22c, Remark 3.3]).Let Ω⊂Rnbe a domain bounded

in one direction and 0<s<1. Assume that γ1, γ2∈L∞(Rn)satisfy

γ1(x), γ2(x)≥γ0>0, and are continuous a.e. in Ωe. There exists a

constant C > 0depending only on ssuch that 2

kγ1−γ2kL∞(Ωe)≤CkΛγ1−Λγ2k∗.

Given a Sobolev multiplier q∈M(Hs→H−s) (cf. [RS20, CMRU22]), we

deﬁne the following bilinear form

Bq(u, v):=ˆRn

(−∆)s/2u(−∆)s/2v dx +hqu, vi, u, v ∈Hs(Rn),

related to the fractional Schr¨odinger operator (−∆)s+q.

Theorem 1.3 ([RS20, Theorem 1.2]).Let Ω⊂Rnbe a bounded smooth

domain, 0< s < 1, and W1, W2⊂Ωbe nonemtpy open sets. Assume that

for some δ, M > 0the potentials q1, q2∈Hδ, n

2s(Rn)have the bounds

kqjkHδ, n

2s(Ω) ≤M, j = 1,2.

Suppose also that zero is not a Dirichlet eigenvalue for the exterior value

problem

(2) (−∆)su+qju= 0,in Ω

2Here and in the rest of paper we use the notation kAk∗:= kAkHs(Ωe)→(Hs(Ωe))∗.

INVERSE FRACTIONAL CONDUCTIVITY PROBLEM 5

with u|Ωe= 0, for j= 1,2. Then one has 3

kq1−q2kLn

2s(Ω) ≤ω(kΛq1−Λq2ke

Hs(W1)→(

Hs(W2))∗),

where Λqj:X→X∗with Λqjf, g:= Bqj(uf, g)is the DN map related to

the exterior value problem for equation (2), and ωis a modulus of continuity

satisfying

ω(x)≤C|log x|−σ,0< x ≤1

for some Cand σdepending only on Ω,n,s,W1,W2,δand M.

Lemma 1.4 (Liouville reduction, [RZ22c, Lemma 3.9]).Let 0< s <

min(1, n/2). Assume that γ∈L∞(Rn)with conductivity matrix Θγand

background deviation msatisﬁes γ(x)≥γ0>0and m∈Hs,n/s(Rn). Let

qγ:= (−∆)sm

γ1/2. Then there holds

hΘγ∇su, ∇sφiL2(R2n)=h(−∆)s/2(γ1/2u),(−∆)s/2(γ1/2φ))iL2(Rn)

+hqγ(γ1/2u),(γ1/2φ)i

for all u, φ ∈Hs(Rn).

In the sequel, we will call qγabove a potential.

1.1. Main results. We next state our main result, whose proof is based on

a reduction to Theorems 1.2 and 1.3.

Theorem 1.5. Let 0< s < min(1, n/2), > 0and assume that Ω⊂Rn

is a smooth bounded domain. Suppose that the the conductivities γ1, γ2∈

L∞(Rn)with background deviations m1, m2fulﬁll the following conditions:

(i) γ0≤γ1(x), γ2(x)≤γ−1

0for some 0< γ0<1,

(ii) m1, m2∈H4s+2, n

2s(Rn)and there exists C1>0such that

(3) kmikH4s+2, n

2s(Rn)≤C1

for i= 1,2,

(iii) m1−m2∈Hs(Rn)and there exists C2>0

(4) k(−∆)smikL1(Ωe)≤C2

for i= 1,2.

If θ0∈(max(1/2,2s/n),1) and there holds kΛγ1−Λγ2k∗≤3−1/δ for some

0< δ < 1−θ0

2, then we have

kγ1/2

1−γ1/2

2kLq(Ω) ≤ω(kΛγ1−Λγ2k∗)

for all 1≤q≤2n

n−2s, where ω(x)is a logarithmic modulus of continuity

satisfying

ω(x)≤C|log x|−σ,for 0< x ≤1,

for some constants σ, C > 0depending only on s, , n, Ω, C1, C2, θ0and γ0.

Remark 1.6. We make several comments about Theorem 1.5 and its as-

sumptions to clarify some interesting points:

3Note kAke

Hs(W1)→(

Hs(W2))∗= sup{ |hAu1, u2i| ;kujkHs(Rn)= 1, uj∈C∞

c(Wj)}.

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STABILITYESTIMATESFORTHEINVERSEFRACTIONALCONDUCTIVITYPROBLEMGIOVANNICOVI,JESSERAILO,TEEMUTYNI,ANDPHILIPPZIMMERMANNAbstract.Westudythestabilityofaninverseproblemforthefrac-tionalconductivityequationonboundedsmoothdomains.Weobtainalogarithmicstabilityestimatefortheinverseproblemundersuitableaprioribou...

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STABILITY ESTIMATES FOR THE INVERSE FRACTIONAL CONDUCTIVITY PROBLEM GIOVANNI COVI JESSE RAILO TEEMU TYNI AND PHILIPP ZIMMERMANN

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