Uniqueness in an inverse problem of fractional elasticity

Giovanni Covi, Maarten de Hoop, Mikko Salo

## Abstract

We study an inverse problem for fractional elasticity. In analogy to the
classical problem of linear elasticity, we consider the unique recovery of the
Lamé parameters associated to a linear, isotropic fractional elasticity
operator from fractional Dirichlet-to-Neumann data. In our analysis we make
use of a fractional matrix Schrödinger equation via a generalization of the
so-called Liouville reduction, a technique classically used in the study of
the scalar conductivity equation. We conclude that unique recovery is possible
if the Lamé parameters agree and are constant in the exterior, and their
Poisson ratios agree everywhere. Our study is motivated by the significant
recent activity in the field of nonlocal elasticity.

###### Contents

  1. 1 Introduction
    1. 1.1 Motivation and connection to the literature
    2. 1.2 Organization of the rest of the article
  2. 2 Preliminaries
    1. 2.1 Tensor products and contractions
    2. 2.2 Fractional Sobolev spaces
    3. 2.3 The fractional Laplacian and other nonlocal operators
    4. 2.4 Hölder spaces
    5. 2.5 Fundamentals of the classical theory of linear elasticity
  3. 3 The isotropic fractional elasticity operator
  4. 4 Well-posedness and the DN map
  5. 5 The fractional Liouville reduction
  6. 6 The Alessandrini identity
  7. 7 Runge approximation property and proof of the main Theorem

##  Introduction

We consider an inverse problem for a fractional elasticity operator of the
form

$\mathbf{E}^{s}u:=(\nabla\cdot)^{s}(\mathcal{C}(x,y)\nabla^{s}u)$

for a fixed $s\in(0,1)$. The fractional gradient $\nabla^{s}$ and the
fractional divergence $(\nabla\cdot)^{s}$ appearing in the definition of
$\mathbf{E}^{s}$ go back to [16] and they will be defined in detail in Section
2. They can be thought of as nonlocal counterparts of the classical gradient
and divergence operators, with the expected properties
$(\nabla^{s})^{*}=(\nabla\cdot)^{s}$ and
$(\nabla\cdot)^{s}\nabla^{s}=(-\Delta)^{s}$. The tensor
$\mathcal{C}(x,y):=C^{1/2}(x)C^{1/2}(y)$ is itself a nonlocal counterpart of
the classical stiffness tensor $C$. Section 3 is dedicated to the detailed
definition of $\mathbf{E}^{s}$.

We make the assumption that $C$ is isotropic, that is, its properties are
completely described by two Lamé parameters $L,M$ (see Section 2.5). The
scalar functions $L,M$ are assumed to be constant outside a domain of interest
$\Omega$, and unknown within it. The goal of the inverse problem we wish to
study is to recover $L,M$ (and thus $C$) within $\Omega$ from measurements
performed on its exterior
$\Omega_{e}:=\mathbb{R}^{n}\setminus\overline{\Omega}$. To this end, we start
by considering the direct problem

$\begin{array}[]{rll}\mathbf{E}^{s}u&=0&\quad\mbox{ in }\Omega\\\
u&=f&\quad\mbox{ in }\Omega_{e}\end{array},$

for which we prove well-posedness in a weak sense for any sufficiently regular
exterior datum $f$. This means that given any $f$ defined on $\Omega_{e}$
(e.g. we might have $f\in C^{\infty}_{c}(\Omega_{e})$) there is one and only
one solution $u_{f}\in H^{s}(\mathbb{R}^{n})$ to the direct problem. Building
on this, we define the exterior measurements as a nonlocal Dirichlet-to-
Neumann (DN) map $\Lambda_{L,M}$, associating each exterior datum $f$ to the
corresponding nonlocal Neumann data $\mathbf{E}^{s}u_{f}|_{\Omega_{e}}$. It is
clear that $\Lambda_{L,M}$ carries information about the Lamé parameters
$L,M$, as these are involved in the definition of the unique solution $u_{f}$.
More specifically, we would like to recover $L,M$ within $\Omega$ from partial
data, that is, from the knowledge of $\Lambda_{L,M}f|_{W_{2}}$ for all $f\in
C^{\infty}_{c}(W_{1})$, where $W_{1},W_{2}$ are non-empty, open and disjoint
subsets of $\Omega_{e}$. This means that the exterior data $f$ will be
supported in $W_{1}$, while the measurements will be performed on $W_{2}$
only. These restrictions represent the physical situation in which not all of
the exterior is accessible for measurement. We thus ask the following inverse
problem:

Q: _Does $\Lambda_{L_{1},M_{1}}f|_{W_{2}}=\Lambda_{L_{2},M_{2}}f|_{W_{2}}$ for
all $f\in C^{\infty}_{c}(W_{1})$ imply that $L_{1}=L_{2}$ and $M_{1}=M_{2}$
within $\Omega$?_

We build on the work of [27]. We will first use the assumption on the DN maps
in order to obtain an integral identity, the so called Alessandrini identity,
relating the difference of the DN maps to the differences of the Lamé
parameters via some special solutions to the direct problem. Then, we shall
test the Alessandrini identity with aptly chosen solutions in order to deduce
the desired result. Such solutions will be produced by means of a Runge
approximation property, which we will prove for our equation. The proof of the
Runge approximation property will itself rely on a unique continuation
property (UCP), which is the key point of the technique.

However, as it will be clarified in Remark 7.3, the approach described above
does not directly lead to the desired result. It is instead necessary to first
reduce the given problem to a more manageable one of Schrödinger type. We do
so by means of the so called _fractional Liouville reduction_ , which is
reminiscent of the techniques used for both the classical and fractional
conductivity equations (see [14] for the latter). After the reduction, the
direct problem reads

$\begin{array}[]{rll}(-\Delta)^{s-1}Dw-w\cdot Q&=0&\quad\mbox{ in }\Omega\\\
w&=g&\quad\mbox{ in }\Omega_{e}\end{array},$

where $D$ is a fixed differential operator of second order, $Q$ is a new
potential containing information relative to the Lamé parameters $L,M$, the
new exterior datum $g$ is computed from $f$, and the new solution $w$ is
computed from $u$ (see Section 5 for the details).

The reduction from the fractional elasticity equation to matrix Schrödinger
equation is possible for arbitrary Lamé coefficients. However, in order to
make a reduction on the level of exterior measurements and to derive a
suitable integral identity, we need to assume that the Poisson ratio is a
fixed (unknown) function:

######  Definition 1.1.

If $(L_{1},M_{1}),(L_{2},M_{2})$ are two couples of Lamé parameters, we write
$(L_{1},M_{1})\sim(L_{2},M_{2})$ if and only if they coincide on $\Omega_{e}$
and $\nu_{1}=\nu_{2}$ on ${\mathbb{R}}^{n}$, where the Poisson ratio $\nu$
relative to a couple of Lamé parameters $(L,M)$ is $\nu:=\frac{L}{(n-1)L+2M}$
(see Section 2.5 for the precise definition).

As it turns out, if the Poisson ratio is a fixed function, then the technique
described above will work for the transformed problem and will allow us to
determine the matrix potential $Q$ from the nonlocal DN map. We can then
recover the Lamé parameters $(L,M)$ from $Q$ and prove the following
uniqueness result.

######  Theorem 1.2.

Let $\Omega,W_{1},W_{2}\subset\mathbb{R}^{n}$ be bounded open sets such that
$W_{1},W_{2}\subset\Omega_{e}$, and assume $s\in(0,1)$. Let $(L_{1},M_{1})$
and $(L_{2},M_{2})$ be two couples of Lamé parameters satisfying assumptions
(A1)-(A3). If $(L_{1},M_{1})\sim(L_{2},M_{2})$ and

$\Lambda_{L_{1},M_{1}}f|_{W_{2}}=\Lambda_{L_{2},M_{2}}f|_{W_{2}}\qquad\mbox{
for all }f\in C^{\infty}_{c}(W_{1}),$

then $(L_{1},M_{1})=(L_{2},M_{2})$.

Here assumptions (A1)-(A3) state that $L,M$ are $(2s+\varepsilon)$-Hölder
regular, constant outside of $\Omega$ and enjoy a certain positivity condition
(see Section 3).

###  1.1 Motivation and connection to the literature

We consider nonlocal elasticity, that is, space-nonlocality, when nonlocal
stress is defined as the Riesz fractional integral of the strain field in
space. There exist several complex phenomena, occurring possibly in damage
zones [36], that cannot be addressed by classical local continua. For a recent
overview, we refer to [23]. While the notion of nonlocal elasticity dates back
to the work of Mindlin [47, 48] (who considered a variational formulation
resulting in stable and well-posed solutions of boundary value problems,
enabling the removal of singularities from dislocations and cracks), we focus
on a fractional generalization of the classical Eringen integral model [21,
22] of nonlocality. That is, we consider nonlocal elasticity based on the so-
called fractional (or generalized) linear “gradient”-elasticity model [8, 58]

$\boldsymbol{\sigma}=C:[\boldsymbol{\epsilon}+a_{s}(-\Delta)^{s}\boldsymbol{\epsilon}],\quad
a_{s}=\ell_{s}^{2},$

where $(-\Delta)^{s}$ is a fractional Laplacian in the Riesz form,
$s\in\mathbb{R}^{+}\setminus\mathbb{Z}$, $\ell_{s}$ is a material parameter
signifying an internal length scale, $\boldsymbol{\sigma}$ is the stress
tensor, and $\boldsymbol{\epsilon}$ is the strain tensor. Assuming that the
stiffness tensor $C$ is isotropic and has constant coefficients results in the
following fractional elasticity operator $\mathbf{T}^{s}$ from [58]:

$\mathbf{T}^{s}u:=\nabla\cdot(C(1+a_{s}(-\Delta)^{s})\nabla
u)=\nabla\cdot(C\nabla u)+a_{s}\nabla\cdot(C(-\Delta)^{s}\nabla u).$

The operator $\mathbf{E}^{s}$ which we study in the present paper generalizes
$\mathbf{T}^{s}$ to the case of variable coefficients (see section 2.3 for the
precise definitions of the fractional operators used here). In order to see
this, compute

$\mathbf{E}^{s+1}u:=(\nabla\cdot)^{s+1}(C^{1/2}(x):C^{1/2}(y)\nabla^{s+1}u)=\nabla\cdot\\{(\nabla\cdot)^{s}C^{1/2}(x):C^{1/2}(y)\nabla^{s}\nabla
u\\},$

where the relations $\nabla^{s+1}=\nabla^{s}\nabla$ and the corresponding one
for the fractional divergence are definitions, in accordance to [10]. If the
coefficients are assumed to be constant, a straightforward computation shows
that

$\mathbf{E}^{s+1}u=\nabla\cdot((\nabla\cdot)^{s}C\nabla^{s}\nabla
u)=\nabla\cdot(C(\nabla\cdot)^{s}\nabla^{s}\nabla
u)=\nabla\cdot(C(-\Delta)^{s}\nabla u),$

and thus

$\mathbf{T}^{s}u=a_{s}\mathbf{E}^{s+1}u+\nabla\cdot(C\nabla u).$

In an alternative introduction of a model for fractional elasticity directly
through a kernel, one develops a fractional Taylor series using the Caputo
fractional derivative (involving a left-sided Riemann-Liouville fractional
integral) of its Fourier transform [52, 34]. The physical basis of such an
introduction is an assumption pertaining to fractional spatial dispersion for
a nonlocal elastic continuum.

Zorica and Oparnica [61] presented time-fractional wave equations that model
hereditary viscoelastic behavior and space-fractional wave equations
associated with certain nonlocal elasticity models. For a number of fractional
wave equations, the authors provided mathematical evidence of energy
dissipation and conservation.

An inverse problem with nonlocal elasticity was considered by Askes and
Aifantis [1]. For an important overview of general fractional derivative
equations, containing fractional time and space derivatives, and inverse
problems examining ill-posedness in space dimension $1$, we refer to Jin and
Rundell [33].

The theory of linear elasticity has given rise to classical inverse problems
which are related to our question, and can rather be considered the main
inspiration of our study. We briefly present the main concepts of the
classical theory of linear elasticity in Section 2.5 (see also [41]). The
operator of classical elasticity is

$\mathbf{E}u:=\nabla\cdot(C\nabla u),$

where $C$ is the stiffness tensor. Since $\mathbf{E}$ is a local operator, the
measurements involved in the inverse problems are performed on the boundary
$\partial\Omega$ of the domain $\Omega$ rather than in its exterior
$\Omega_{e}$. Most of the results concern the isotropic case where $C$ depends
on two scalar Lamé parameters $L$ and $M$. The main uniqueness results for
this inverse problem in three dimensions are in [50, 51] and [19, 20]. They
state that if $M$ is sufficiently close to a constant, or if either $L$ or $M$
is a fixed (unknown) function, then the Lamé parameters are uniquely
determined by the boundary measurements. In [20] one also finds the classical
analogue of Theorem 1.2 stating that the Lamé parameters are determined if the
Poisson ratio is a fixed function. The two-dimensional case is considered in
[32]. Many other different aspects of this inverse problem have been
considered, such as inclusion detection [2], identification of the elastic
moduli in beams, plates and other geometric configurations [3, 12, 19, 25],
the linearized problem [29, 30, 31], uniqueness [6, 50, 51], and
identification of residual stresses [4, 24, 45, 53], among others. We refer to
the survey [5] for many more results.

The inverse problem we study is related to the fractional Calderón problem,
which was introduced in the seminal paper [27] as a nonlocal counterpart to
the classical Calderón problem arising in electric impedance tomography.
Uniqueness was achieved in the case of bounded potentials and fractional
exponent $s\in(0,1)$ [27] also with a single measurement [26], and later
extended to rough potentials [54] and all positive fractional exponents
$s\in\mathbb{R}^{+}\setminus\mathbb{Z}$ [10]. Perturbed versions of the same
problem were studied e.g. in [9] for $s\in(1/2,1)$ and first order
perturbations and [11] for $s\in\mathbb{R}^{+}\setminus\mathbb{Z}$ and general
high order local perturbations. Nonlocal perturbations were studied in some
specific cases in [7, 15, 44]. Uniqueness has also been studied in numerous
other settings, including the fractional magnetic Schrödinger equation [13,
10, 38, 37, 39] and the fractional heat equation [43, 55]. A very recent
fractional elasticity equation with constant principal coefficients [40] and
the fractional conductivity equation [14] bear a strong connection with the
present study. Moreover, the fractional Schrödinger equation was studied in
the semilinear setting [42, 38, 37, 39]. We refer to the surveys [57, 56] for
more information about the fractional Calderón problem.

###  1.2 Organization of the rest of the article

The remaining part of the paper is organized as follows. Section 2 contains
preliminaries from functional analysis, classical elasticity theory and
nonlocal vector calculus, as well as the definitions of the notations used in
the article. Section 3 defines and describes the main object of study of the
paper, the fractional elasticity operator $\mathbf{E}^{s}$. The Dirichlet
problem related to such operator is studied in Section 4, and in Section 5 it
is shown to be equivalent to a Dirichlet problem for the fractional
Schrödinger equation. An integral identity relating coefficients and measured
data, the so called Alessandrini identity, is shown to hold in Section 6. In
Section 7 we prove the Runge approximation property and eventually the main
Theorem 1.2.

### Acknowledgments

Giovanni Covi was supported by an Alexander-von-Humboldt postdoctoral
fellowship. Maarten de Hoop was supported by the Simons Foundation under the
MATH + X program, the National Science Foundation under grant DMS-2108175, and
the corporate members of the Geo-Mathematical Imaging Group at Rice
University. Mikko Salo was partly supported by the Academy of Finland (Centre
of Excellence in Inverse Modelling and Imaging, grant 284715) and by the
European Research Council under Horizon 2020 (ERC CoG 770924).

##  Preliminaries

In this section we recall the definitions of relevant function spaces, define
some concepts from mathematical physics related to the problem of elasticity,
and establish some useful notations.

###  2.1 Tensor products and contractions

We will make wide use of the concepts of tensor product and contraction,
mainly with respect to vectors. This will let us write our equations in a more
understandable way. We let $\mathbb{N}$ be the set of _strictly_ positive
integers.

######  Definition 2.1 (Tensor product).

Let $m,n\in\mathbb{N}$, and assume $a,b$ are multi-indices belonging to
$\mathbb{N}^{m},\mathbb{N}^{n}$ respectively. Consider two tensors
$A_{\alpha_{1},...,\alpha_{m}}$ and $B_{\beta_{1},...,\beta_{n}}$, with
$\alpha_{i}\in\\{1,...,a_{i}\\}$ for $i=1,...,m$ and
$\beta_{j}\in\\{1,...,b_{j}\\}$ for $j=1,...,n$ respectively. The _tensor
product_ $A\otimes B$ is the new tensor of elements

$(A\otimes
B)_{\alpha_{1},...,\alpha_{m},\beta_{1},...,\beta_{n}}=A_{\alpha_{1},...,\alpha_{m}}B_{\beta_{1},...,\beta_{n}}.$

In particular, if $m=n=1$ (and thus
$A\in{\mathbb{R}}^{a},B\in{\mathbb{R}}^{b}$ are vectors) the tensor product
$A\otimes B$ is just the $a\times b$ matrix of elements

$(A\otimes B)_{\alpha,\beta}=A_{\alpha}B_{\beta},\quad\mbox{for
}\alpha=1,...,a\mbox{ and }\beta=1,...,b.$

######  Definition 2.2 (Tensor contraction of order $k$).

Let $m,n,a,b,A,B$ be as in Definition 2.1. Assume that $k\in\mathbb{N}$ is
such that $k\leq\min\\{m,n\\}$, with $a_{m+\ell-k}=b_{\ell}$ for all
$\ell\in\\{1,...,k\\}$. We define the _$k$ -th contraction_ of tensors $A$ and
$B$ as the new tensor given by

$(A\cdot_{k}B)_{\alpha_{1},...,\alpha_{m-k},\beta_{k+1},...,\beta_{n}}=A_{\alpha_{1},...,\alpha_{m-k},\gamma_{1},...,\gamma_{k}}B_{\gamma_{1},...,\gamma_{k},\beta_{k+1},...,\beta_{n}},$

where we assume the Einstein summation convention on repeated indices. The
ranges of the $\alpha,\beta$ indices appearing in the above formula are the
same as in the definitions of $A,B$, while the $\gamma_{\ell}$ index has range
in $\\{1,...,b_{\ell}\\}$ for $\ell=1,...,k$.

In the present work, we apply the above definition only with $k=1$ and $k=2$,
so we use the standard symbols $\cdot$ and $:$ in place of the more general
$\cdot_{1}$ and $\cdot_{2}$. It is useful to observe that when $A,B$ are
vectors or matrices the above contraction operator $\cdot$ coincides with the
usual scalar product and matrix multiplication.

The definitions are immediately extended to functions. If $A,B$ are tensor-
valued functions defined on some set $\Omega$, then we let

$\displaystyle A\otimes B:x$ $\displaystyle\mapsto A(x)\otimes
B(x),\quad\mbox{for all }x\in\Omega,$ $\displaystyle A\cdot_{k}B:x$
$\displaystyle\mapsto A(x)\cdot_{k}B(x),\quad\mbox{for all }x\in\Omega.$

Accordingly, if $V,W$ are sets of tensor-valued functions defined on some set
$\Omega$, then we let $V\otimes W$ be the new set of functions on $\Omega$
given by

$V\otimes W:=\\{v\otimes w,v\in V,w\in W\\}.$

######  Remark 2.3.

This should not be confused with the familiar Cartesian product $V\times W$.
For example, if $a,b\in{\mathbb{N}}$ and $V,W$ are sets of vector-valued
functions $v:\Omega\mapsto{\mathbb{R}}^{a}$ and
$w:\Omega\mapsto{\mathbb{R}}^{b}$ respectively, then the elements of $V\times
W$ map $\Omega$ to ${\mathbb{R}}^{a+b}$, while the ones of $V\otimes W$ map
$\Omega$ to ${\mathbb{R}}^{a\times b}$. In the present work we shall often
have $a=2$ and $V=\\{v\\}$ for some fixed function
$v:\Omega\rightarrow\mathbb{R}^{2}$.

The following Lemma collects some elementary properties of tensor products and
contractions, which can be easily proved using the index notation:

######  Lemma 2.4.

Let $m,n,p\in{\mathbb{N}}$, and assume that $a,b,c$ are multi-indices
belonging to ${\mathbb{N}}^{m},{\mathbb{N}}^{n},{\mathbb{N}}^{p}$
respectively. Consider three tensors

$\displaystyle A_{\alpha_{1},...,\alpha_{m}},$ $\displaystyle\quad\mbox{ with
}\alpha_{i}\in\\{1,...,a_{i}\\}\mbox{ for }i=1,...,m,$ $\displaystyle
B_{\beta_{1},...,\beta_{n}},$ $\displaystyle\quad\mbox{ with
}\beta_{j}\in\\{1,...,b_{j}\\}\mbox{ for }j=1,...,n,$ $\displaystyle
C_{\gamma_{1},...,\gamma_{p}},$ $\displaystyle\quad\mbox{ with
}\gamma_{k}\in\\{1,...,c_{k}\\}\mbox{ for }k=1,...,p.$

The following equalities hold whenever the tensor contractions are well-
defined:

  1. (i)

$(A\otimes B)\cdot_{n}C=(A\otimes C)\cdot_{n}B\quad\mbox{ for }n=p,$

  2. (ii)

$A\cdot_{k}(B\otimes C)=(A\cdot_{k}B)\otimes C\quad\mbox{ for all
}k\in{\mathbb{N}},$

  3. (iii)

$(A\cdot_{k}B)\cdot_{q}C=A\cdot_{k}(B\cdot_{q}C)\quad\mbox{ for all
}k,q\in{\mathbb{N}}\mbox{ with }n\geq k+q,$

  4. (iv)

$A\cdot_{n+p}(B\otimes C)=(A\cdot_{p}C)\cdot_{n}B$.

If $A,B$ are tensor-valued functions of $x\in\Omega$, then the equalities

  1. (v)

$\nabla(A\otimes B)=\nabla A\otimes B+A\otimes\nabla B\quad\mbox{ for }m=a=1$,

  2. (vi)

$\nabla(A\cdot_{m}B)=\nabla A\cdot_{m}B+\nabla B\cdot_{m}A\quad\mbox{ for
}m=n$,

  3. (vii)

$\nabla\cdot(A\otimes B)=A\otimes\nabla\cdot B+B^{T}\cdot\nabla A\quad\mbox{
for }m=a=1,\;n=2$.

also hold whenever the tensor contractions are well-defined. $\square$

######  Remark 2.5.

For all vectors $v$ we use the convention $(\nabla
v)_{ij}:=\partial_{i}v_{j}$.

###  2.2 Fractional Sobolev spaces

Let $r\in\mathbb{R}$, and assume that $\Omega,F\subset\mathbb{R}^{n}$
respectively are an open and a closed set. We indicate by
$H^{r}(\mathbb{R}^{n})$ the usual $L^{2}$-based Bessel potential spaces. These
are endowed with the norm

$\|u\|_{H^{r}}:=\|(1+|\xi|^{2})^{r/2}\hat{u}(\xi)\|_{L^{2}({\mathbb{R}}^{n})}.$

Here by
$\hat{u}(\xi)=\mathcal{F}u(\xi):=\int_{\mathbb{R}^{n}}e^{-ix\cdot\xi}u(x)\,dx$
we indicate the Fourier transform. Using the same notation as in [46] and
[27], we also define the following fractional Sobolev spaces:

$\displaystyle H^{r}(\Omega)$ $\displaystyle:=\\{u|_{\Omega},u\in
H^{r}(\mathbb{R}^{n})\\},$ $\displaystyle\widetilde{H}^{r}(\Omega)$
$\displaystyle:=\,\mbox{closure of }C^{\infty}_{c}(\Omega)\mbox{ in
}H^{r}(\mathbb{R}^{n}),$ $\displaystyle H^{r}_{F}(\mathbb{R}^{n})$
$\displaystyle:=\\{u\in H^{r}(\mathbb{R}^{n}):\mbox{ supp}(u)\subset F\\}.$

In particular, $H^{r}(\Omega)$ is endowed with the quotient norm
$\|u\|_{H^{r}(\Omega)}:=\inf\\{\|w\|_{H^{r}}:w\in
H^{r}(\mathbb{R}^{n}),w|_{\Omega}=u\\}$. One sees that the following
inclusions and identities

$\widetilde{H}^{r}(\Omega)\subseteq
H^{r}(\Omega)\,,\quad\widetilde{H}^{r}(\Omega)\subseteq
H^{r}_{\overline{\Omega}}\,,\quad H^{r}_{F}\subseteq H^{r}\,,$
$(\widetilde{H}^{r}(\Omega))^{*}=H^{-r}(\Omega)\,,\quad(H^{r}(\Omega))^{*}=\widetilde{H}^{-r}(\Omega)$

hold. If in addition $\Omega$ is known to be Lipschitz, then
$\widetilde{H}^{r}(\Omega)=H^{r}_{\overline{\Omega}}$ for all
$r\in\mathbb{R}$, as shown in [11, 46].

We shall also use products of Sobolev spaces. If $r\in\mathbb{R}$, the space
$H^{r}(\mathbb{R}^{n})\times H^{r}(\mathbb{R}^{n})$ consists of all the
vectors $u=(u_{1},u_{2})$ with $u_{1},u_{2}\in H^{r}(\mathbb{R}^{n})$, endowed
with the norm

$\|u\|^{2}_{H^{r}(\mathbb{R}^{n})\times
H^{r}(\mathbb{R}^{n})}:=\|u_{1}\|^{2}_{H^{r}(\mathbb{R}^{n})}+\|u_{2}\|^{2}_{H^{r}(\mathbb{R}^{n})}.$

###  2.3 The fractional Laplacian and other nonlocal operators

Let $s\in\mathbb{R}^{+}\setminus\mathbb{Z}$ and $u\in\mathcal{S}$, the set of
Schwartz functions. The fractional Laplacian of $u$ can be defined as

$(-\Delta)^{s}u:=\mathcal{F}^{-1}(|\xi|^{2s}\hat{u}(\xi)),$

which makes $(-\Delta)^{s}$ a continuous map from $\mathcal{S}$ to
$L^{\infty}$ (see [27]). The definition above can be uniquely extended in such
a way that $(-\Delta)^{s}$ acts as a continuous operator
$(-\Delta)^{s}:H^{r}(\mathbb{R}^{n})\rightarrow H^{r-2s}(\mathbb{R}^{n})$ for
all $r\in\mathbb{R}$. Further extensions are possible to Sobolev spaces of
negative exponent and to $L^{p}$-based Sobolev spaces ([10, 27]). It is
possible to give many other definitions of the fractional Laplacian, which at
least in the case $s\in(0,1)$ can be shown to be equivalent to ours ([35]). In
particular, the fractional Laplacian can be defined as a singular integral by

$(-\Delta)^{s}u(x):=C_{n,s}\,PV\int_{\mathbb{R}^{n}}\frac{u(x)-u(y)}{|x-y|^{n+2s}}dy,$

where $C_{n,s}:=\frac{4^{s}\Gamma(n/2+s)}{\pi^{n/2}|\Gamma(-s)|}$.

The fractional Laplacian presents a nonlocal behaviour, as made evident by the
following property it enjoys:

######  Theorem 2.6 (UCP for the fractional Laplacian).

Let $s\in\mathbb{R}^{+}\setminus\mathbb{Z}$, and assume $u\in
H^{r}(\mathbb{R}^{n})$ for some $r\in\mathbb{R}$. If $u=(-\Delta)^{s}u=0$ in a
non-empty open set $V$, then $u\equiv 0$ in $\mathbb{R}^{n}$.

The proof of this theorem can be found in [10], where the authors explore also
the case of negative exponent $s$. The main case $s\in(0,1)$ upon which the
proof is based was proved in [27]. One more property of the fractional
Laplacian we shall use is the following fractional Poincaré inequality.

######  Theorem 2.7.

Let $n\geq 1$, $s\geq t\geq 0$, $K\subset{\mathbb{R}}^{n}$ a compact set and
$u\in H_{K}^{s}({\mathbb{R}}^{n})$. There exists a constant
$\widetilde{c}=\widetilde{c}(n,K,s)>0$ such that

$\left\lVert(-\Delta)^{t/2}u\right\rVert_{L^{2}({\mathbb{R}}^{n})}\leq\widetilde{c}\left\lVert(-\Delta)^{s/2}u\right\rVert_{L^{2}({\mathbb{R}}^{n})}.$

Many variously flavoured proofs of the above statement can be found in [10].

Next, we shall define nonlocal counterparts to the gradient and divergence
operators as in [14, 10]. These are special instances of the general nonlocal
vector calculus operators introduced in [16, 17]. For $u\in
C^{\infty}_{c}(\mathbb{R}^{n},\mathbb{R}^{n})$ and $s\in(0,1)$ the fractional
gradient is

$\nabla^{s}u(x,y):=(u(y)-u(x))\otimes\zeta(x,y),\quad\mbox{ where
}\zeta(x,y):=\frac{C_{n,s}^{1/2}}{\sqrt{2}}\frac{x-y}{|x-y|^{n/2+s+1}}$

and since one sees that
$\|\nabla^{s}u\|_{L^{2}(\mathbb{R}^{2n})}\leq\|u\|_{H^{s}(\mathbb{R}^{n})}$,
the definition is extended to act as $H^{s}(\mathbb{R}^{n})\rightarrow
L^{2}(\mathbb{R}^{2n})$ by density. Many more properties of the fractional
gradient can be found in [13]. The fractional divergence $(\nabla\cdot)^{s}$
is defined as the adjoint of $\nabla^{s}$, that is

$\langle(\nabla\cdot)^{s}v,u\rangle:=\langle v,\nabla^{s}u\rangle$

for all $v\in L^{2}(\mathbb{R}^{2n}),u\in H^{s}(\mathbb{R}^{n})$. Thus
$(\nabla\cdot)^{s}:L^{2}(\mathbb{R}^{2n})\rightarrow H^{-s}(\mathbb{R}^{n})$.
It is useful to keep in mind that for more regular functions one has

$(\nabla\cdot)^{s}v(x)=\frac{C_{n,s}^{1/2}}{\sqrt{2}}\int_{\mathbb{R}^{n}}\frac{v(x,y)+v(y,x)}{|x-y|^{n/2+s+1}}\cdot(x-y)dy.$

More generally, we define the fractional gradient of order
$s\in\mathbb{R}^{+}\setminus\mathbb{Z}$ as $\nabla^{s}:=\nabla^{s-\lfloor
s\rfloor}\nabla^{\lfloor s\rfloor}$, and a corresponding relation holds by
definition for the fractional divergence. Most importantly, one sees that the
property

$(\nabla\cdot)^{s}\nabla^{s}u=(-\Delta)^{s}u$

holds in $H^{s}(\mathbb{R}^{n})$. Finally, we define the $j$-th fractional
derivative as

$\partial^{s}_{j}u(x,y):=\nabla^{s}u(x,y)\cdot
e_{j}=\frac{C_{n,s}^{1/2}}{\sqrt{2}}\frac{u(y)-u(x)}{|x-y|^{n/2+s+1}}(x_{j}-y_{j}).$

In particular this definition implies that
$\partial^{s}_{j}u\cdot\partial^{s}_{i}v=\partial^{s}_{i}u\cdot\partial^{s}_{j}v$
holds for all vectors $u,v$ and all $i,j\in\\{1,...,n\\}$. This property,
which of course does not hold in the classical case $s=1$, will be fundamental
for our arguments.

###  2.4 Hölder spaces

Let $r\in(0,1)$. Following [59], we define the Hölder space
$C^{r}(\mathbb{R}^{n})$ as the set of bounded functions $u$ such that

$|u(x)-u(y)|\leq C|x-y|^{r}$

for all $x,y\in\mathbb{R}^{n}$. If $k\in{\mathbb{N}}$, we let
$C^{k}(\mathbb{R}^{n})$ be the set of all bounded continuous functions $u$
such that $D^{\beta}u$ is bounded and continuous for all multi-indexes
$|\beta|\leq k$. Finally, if $r\in{\mathbb{R}}^{+}\setminus{\mathbb{Z}}^{+}$
we define $C^{r}(\mathbb{R}^{n})$ to be the set of all functions $u\in
C^{\lfloor r\rfloor}(\mathbb{R}^{n})$ with $D^{\beta}u\in C^{r-\lfloor
r\rfloor}(\mathbb{R}^{n})$ for all multi-indexes $|\beta|=\lfloor r\rfloor$.

Next, we shall list some properties of Hölder spaces which will be useful in
our arguments. It follows immediately from the definition that for all
$r,r^{\prime}\in{\mathbb{R}}^{+}$ one has the embedding

$r\leq r^{\prime}\Rightarrow C^{r^{\prime}}(\mathbb{R}^{n})\subseteq
C^{r}(\mathbb{R}^{n}).$ (1)

The Hölder space $C^{r}({\mathbb{R}}^{n})$ is clearly closed under composition
with smooth functions, that is

$u\in C^{r}({\mathbb{R}}^{n}),\,F\in C^{\infty}({\mathbb{R}})\Rightarrow
F(u)\in C^{r}({\mathbb{R}}^{n}).$ (2)

It is also known that $C^{r}({\mathbb{R}}^{n})$ is an algebra for all
$r\in\mathbb{R}^{+}\setminus{\mathbb{Z}}^{+}$, which means

$u,v\in C^{r}({\mathbb{R}}^{n})\Rightarrow uv\in C^{r}({\mathbb{R}}^{n}).$ (3)

Let $m\in\mathbb{R}$ and assume $\Psi\in OPS^{m}_{1,0}$, i.e. $\Psi$ is a
pseudodifferential operator with symbol in the Hörmander class $S^{m}_{1,0}$.
If both $r,r-m\in{\mathbb{R}}^{+}\setminus{\mathbb{Z}}^{+}$, then

$\Psi:C^{r}({\mathbb{R}}^{n})\rightarrow C^{r-m}({\mathbb{R}}^{n}).$ (4)

For later purposes, we also show that the operator
$(-\Delta)^{s-1}\partial_{i}\partial_{j}$ maps $C^{r}({\mathbb{R}}^{n})$ to
$C^{r-2s}({\mathbb{R}}^{n})$. We write the symbol as

$\left\lvert\xi\right\rvert^{2s-2}\xi_{i}\xi_{j}=\psi(\xi)\left\lvert\xi\right\rvert^{2s-2}\xi_{i}\xi_{j}+(1-\psi(\xi))\left\lvert\xi\right\rvert^{2s-2}\xi_{i}\xi_{j}$

where $\psi\in C^{\infty}_{c}(\mathbb{R}^{n})$ satisfies $\psi=1$ near $0$,
and note that the second symbol has the right mapping properties by (4). The
first symbol gives rise to a convolution operator $u\mapsto k*u$, where

$k=F^{-1}(\psi)*F^{-1}(\left\lvert\xi\right\rvert^{2s-2}\xi_{i}\xi_{j}).$

Now the first function in the convolution is Schwartz and the second one is
homogeneous of order $-n-2s$ and smooth outside of $0$ [28], which proves that
$k\in L^{1}(\mathbb{R}^{n})$. Since $\psi(\xi)$ is compactly supported, the
Fourier characterization of Hölder spaces [60] implies that $u\mapsto k*u$ is
bounded between any two Hölder spaces. This proves that whenever
$r,r-2s\in\mathbb{R}^{+}\setminus\mathbb{Z}^{+}$, one has

$(-\Delta)^{s-1}\partial_{i}\partial_{j}:C^{r}({\mathbb{R}}^{n})\to
C^{r-2s}({\mathbb{R}}^{n}).$ (5)

###  2.5 Fundamentals of the classical theory of linear elasticity

This section is a brief introduction to some of the fundamental concepts in
the theory of linear elasticity. Our main reference in this respect is
Landau’s book [41]; however, we do not necessarily restrict our discussion to
the case $n=3$.

The theory of elasticity studies the mechanics of the deformations of
continuous media. Let $\Omega\subset\mathbb{R}^{n}$ be an open, bounded set
representing a physical body. When forces are applied to $\Omega$, the body
answers to that by changing shape and volume, i.e. each point $x\in\Omega$ is
transferred to a new location $x^{\prime}\in\mathbb{R}^{n}$. Thus it is
possible to define a _displacement vector field_ $u$ such that
$u(x)=x^{\prime}-x$ for all $x\in\Omega$.

By computing the change in distance between two points which were originally
close to each other, one sees that the new infinitesimal distance
$dl^{\prime}$ is related to the original one $dl$ by

$dl^{\prime 2}-dl^{2}=2u_{ik}dx_{i}dx_{k},$

where

$u_{ik}:=\frac{1}{2}\left(\frac{\partial u_{i}}{\partial x_{k}}+\frac{\partial
u_{k}}{\partial x_{i}}+\frac{\partial u_{l}}{\partial x_{i}}\frac{\partial
u_{l}}{\partial x_{k}}\right)$

is called the _strain tensor_. It is customary in the case of small
deformations to neglect the second order terms of the strain tensor and use
instead what is known as _Cauchy’s strain tensor_ , _linear strain tensor_ or
_small strain tensor_ :

$\varepsilon_{ik}:=\frac{1}{2}\left(\frac{\partial u_{i}}{\partial
x_{k}}+\frac{\partial u_{k}}{\partial x_{i}}\right)\approx u_{ik}.$

One immediately sees that the tensor $\varepsilon$ is symmetric.

When deformed, the body leaves its equilibrium state and some _internal
forces_ (or _stresses_) $F$ are generated, which attempt to return the body to
its original undeformed state. The resultant of such forces acting on a region
$\Omega^{\prime}\subset\Omega$ can be computed as $\int_{\Omega^{\prime}}FdV$.
Assuming that such forces do not act on a distance, one should be able to
express their resultant as an integral over $\partial\Omega^{\prime}$. This
suggests that $F$ should be of the form $\nabla\cdot\sigma$, where the new
tensor $\sigma$ is called _stress tensor_. Further investigation of the total
moment of the forces acting on $\Omega^{\prime}$ reveals that the stress
tensor $\beta$ should also be symmetric.

Because we are interested in a theory of _linear_ elasticity, we shall assume
a linear relationship between the stress tensor $\sigma$, representing the
forces acting on $\Omega$, and the strain tensor $\varepsilon$, which
represents the resulting deformation of the body. This gives rise to the
following _Hooke’s law_ (or _constitutive equation of linear elasticity_)

$\sigma_{ij}=C_{ijlm}\varepsilon_{lm},$

where the new fourth-order tensor $C$, which completely describes the elastic
behaviour of the body, is called _elasticity_ or _stiffness tensor_. Using
_Newton’s second law_ , we can eventually define the _operator of classical
elasticity_ $\mathbf{E}$ as

$\mathbf{E}u:=\nabla\cdot(C(\nabla u+\nabla u^{T})).$ (6)

One can also associate a _potential energy_ $U$ to $\mathbf{E}$:

$\displaystyle U(u)$
$\displaystyle:=\frac{1}{2}\int_{\mathbb{R}^{n}}C_{ijlm}(x)\varepsilon_{ij}(x)\varepsilon_{lm}(x)\,dx.$

This quantity is often assumed to be non-negative, with $U(u)=0$ holding if
and only if $u=0$.

Let us now assume that the material is _isotropic_ , i.e. it is completely
characterized by properties which are independent of direction. In this case
the elasticity tensor can be expressed as

$C_{ijlm}=\lambda\delta_{ij}\delta_{lm}+\mu(\delta_{il}\delta_{jm}+\delta_{im}\delta_{jl}),$

where $\lambda,\mu$ are called _Lamé parameters_. Consequently, Hooke’s law
becomes

$\sigma_{ij}=\lambda
tr(\varepsilon)\delta_{ij}+2\mu\varepsilon_{ij}=\left(\lambda+\frac{2\mu}{n}\right)tr(\varepsilon)\delta_{ij}+2\mu\left(\varepsilon_{ij}-\frac{tr(\varepsilon)}{n}\delta_{ij}\right).$

The quantities $k:=\lambda+2\mu/n$ and $\mu$ (this last one sometimes
indicated by $G$) are respectively called _bulk_ and _shear moduli_.
Thermodynamic considerations ensure the positivity of both $k$ and $\mu$.

The corresponding tensors $tr(\varepsilon)\mbox{Id}$ and
$\varepsilon-\frac{tr(\varepsilon)}{n}\mbox{Id}$ are called _dilational_ and
_deviatoric strain tensors_. The first one of them represents the _hydrostatic
compression_ of the body, i.e. a deformation in scale but not in shape, and it
is independent of the coordinate system. On the other hand, the second tensor
represents a _pure shear_ , i.e. a deformation in which the volume is
unchanged, and it is trace-free.

It is also possible to compute the strain tensor given the stress tensor. By
Hooke’s law we have

$tr(\sigma)=(n\lambda+2\mu)tr(\varepsilon)=nktr(\varepsilon),$

and since $k>0$ we can write $\sigma_{ij}=\frac{\lambda
tr(\sigma)}{nK}\delta_{ij}+2\mu\varepsilon_{ij}$. This eventually gives

$\varepsilon_{ij}=\frac{1}{2\mu}\left(\sigma_{ij}-\frac{\lambda
tr(\sigma)}{nk}\delta_{ij}\right)=s_{ijlm}\sigma_{lm},$

where the new tensor
$s_{ijlm}:=\frac{\delta_{il}\delta_{jm}}{2\mu}-\frac{\lambda\delta_{ij}\delta_{lm}}{2n\mu
k}$ is the _isotropic compliance tensor_.

The Poisson effect indicates the physical phenomenon observed in the study of
elastic materials in which a body reacts to a compression (resp. extension)
along one axis with an extension (resp. compression) in the perpendicular
directions. The _Poisson ratio_ $\nu$ is defined as the amount of transverse
extension divided by the amount of axial compression. It is easily computed in
the case of a _homogeneous deformation_ of a thin rod along its axis. If such
axis is oriented in the $e_{n}$ direction and the applied pressure is $p$,
then $\sigma_{ii}=p\delta_{in}$ for all $i\in\\{1,...,n\\}$. Therefore

$\varepsilon_{ii}=\frac{1}{2\mu}\left(\sigma_{ii}-\frac{\lambda
tr(\sigma)}{nk}\right)=\frac{p}{2\mu}\left(\delta_{in}-\frac{\lambda}{nk}\right),\quad\mbox{
for all }i\in\\{1,...,n\\},$

which gives

$\nu:=-\frac{\varepsilon_{11}}{\varepsilon_{nn}}=-\frac{\delta_{1n}-\frac{\lambda}{nk}}{\delta_{nn}-\frac{\lambda}{nk}}=\frac{\lambda}{nk-\lambda}=\frac{k-\frac{2\mu}{n}}{(n-1)k+\frac{2\mu}{n}}=\frac{\frac{k}{\mu}-\frac{2}{n}}{(n-1)\frac{k}{\mu}+\frac{2}{n}}.$

Given the positivity of $k$ and $\mu$, it is always true that
$\nu\in\left(-1,\frac{1}{n-1}\right)$. It is also clear that $\nu$ depends
only on the ratio $k/\mu$ rather than on the two moduli taken separately.

##  The isotropic fractional elasticity operator

In this section we introduce a model for linear fractional elasticity derived
from the classical one, which is related to the model proposed by Tarasov and
Aifantis in [58]. To this end we use the fractional divergence and gradient,
and the result will be a new self-adjoint operator $\mathbf{E}^{s}$. We also
assume throughout the paper the stiffness tensor $C$ to be isotropic and such
that the associated Lamé parameters $L,M$ satisfy

  1. (A1)

there exists $\varepsilon>0$ such that $L,M\in
C^{2s+\varepsilon}(\mathbb{R}^{n})$,

  2. (A2)

there exist $L_{0},M_{0}\in\mathbb{R}$ such that $L-L_{0}=M-M_{0}=0$ in
$\Omega_{e}$, and

  3. (A3)

the functions $M$ and $K:=L+\frac{2M}{n}$ are positive and bounded away from
$0$.

Given that our nonlocal operators act on two-point functions, we need a
preparatory Lemma.

######  Lemma 3.1 (Square root of stiffness tensor).

Let the Lamé parameters $L,M$ of the isotropic stiffness tensor $C$ be as in
assumptions (A1)-(A3). There exists a unique pair of real valued functions
$\lambda,\mu$ verifying (A1)-(A3) and

$C^{1/2}:C^{1/2}=C,$

where the new tensor $C^{1/2}$ is defined as

$C^{1/2}_{ijlm}:=\lambda\delta_{ij}\delta_{lm}+\mu(\delta_{il}\delta_{jm}+\delta_{im}\delta_{jl}).$

Moreover, we have

$\begin{aligned} C^{1/2}_{ij\alpha\beta}(x)C^{1/2}_{\alpha\beta
lm}(y)&=(n\lambda(x)\lambda(y)+2\lambda(x)\mu(y)+2\lambda(y)\mu(x))\delta_{ij}\delta_{lm}\\\
&\quad+2\mu(x)\mu(y)\delta_{il}\delta_{jm}+2\mu(x)\mu(y)\delta_{im}\delta_{jl}\end{aligned},$
(7)

and therefore

$C^{1/2}_{ij\alpha\beta}(x)C^{1/2}_{\alpha\beta
lm}(y)=C^{1/2}_{lm\alpha\beta}(y)C^{1/2}_{\alpha\beta ij}(x).$

###### Proof.

Note that

$\displaystyle C^{1/2}_{ij\alpha\beta}C^{1/2}_{\alpha\beta lm}$
$\displaystyle=(\lambda\delta_{ij}\delta_{\alpha\beta}+\mu\delta_{i\alpha}\delta_{j\beta}+\mu\delta_{i\beta}\delta_{j\alpha})(\lambda\delta_{\alpha\beta}\delta_{lm}+\mu\delta_{\alpha
l}\delta_{\beta m}+\mu\delta_{\alpha m}\delta_{\beta l})$
$\displaystyle=(n\lambda^{2}+4\lambda\mu)\delta_{ij}\delta_{lm}+2\mu^{2}\delta_{il}\delta_{jm}+2\mu^{2}\delta_{im}\delta_{jl}$

and

$C_{ijlm}=L\delta_{ij}\delta_{lm}+M\delta_{il}\delta_{jm}+M\delta_{im}\delta_{jl}.$

We look for $\lambda,\mu$ such that
$C^{1/2}_{ij\alpha\beta}C^{1/2}_{\alpha\beta lm}=C_{ijlm}$. Since $\mu$ must
be positive, we have $\mu:=\sqrt{M/2}$, which also ensures that $\mu$ is
bounded away from $0$. This gives two possible choices for $\lambda$, namely
$\lambda_{\pm}=\frac{1}{n}(\pm\sqrt{2M+nL}-\sqrt{2M})$. However, the required
positivity of the coefficient $k:=\lambda+2\mu/n$ ensures that
$\lambda=\frac{1}{n}(\sqrt{2M+nL}-\sqrt{2M})$. Now $k=\sqrt{K/n}$, and thus it
is bounded away from $0$. This proves that $\lambda,\mu$ satisfy condition
(A3). Given that the square root is a smooth function when considered far from
$0$, by formula (2) we deduce $\mu,k\in C^{2s+\varepsilon}(\mathbb{R}^{n})$,
which in turn implies $\lambda\in C^{2s+\varepsilon}(\mathbb{R}^{n})$ as well
and proves (A1). Since $L,M$ are constant outside of $\Omega$, so must be
$\lambda,\mu$ too, which proves (A2). The last equalities in the statement of
the lemma follow easily from the computations above. ∎

Similarly to what was done in [14], we can define the new fractional
elasticity operator

$\mathbf{E}^{s}u:=(\nabla\cdot)^{s}(C^{1/2}(x):C^{1/2}(y)(\nabla^{s}u+\nabla^{s}u^{T})(x,y)).$
(8)

Observe that this corresponds to taking as fractional Cauchy’s strain tensor

$\varepsilon_{ik}^{s}:=\frac{1}{2}(\partial^{s}_{k}u_{i}+\partial^{s}_{i}u_{k}),\qquad\mbox{i.e.}\quad\varepsilon^{s}:=\frac{1}{2}(\nabla^{s}u+\nabla^{s}u^{T}),$
(9)

and then as a fractional, ”symmetrized” version of Hooke’s law

$\sigma^{s}_{ij}(x,y)=C^{1/2}_{ij\alpha\beta}(x)C^{1/2}_{\alpha\beta
lm}(y)\varepsilon^{s}_{lm}(x,y).$

We next prove the following lemma about $\mathbf{E}^{s}$, which motivates
definition (8):

######  Lemma 3.2 (Properties of $\mathbf{E}^{s}$).

Let $L,M$ satisfy assumptions (A1), (A2), and let $s\in(0,1)$. The operator
$\mathbf{E}^{s}$ is self-adjoint, and it maps $H^{s}(\mathbb{R}^{n})$ to
$H^{-s}(\mathbb{R}^{n})$.

###### Proof.

We start from the proof of the mapping property. By Lemma 3.1 both $\lambda$
and $\mu$ belong to $C^{2s+\varepsilon}({\mathbb{R}}^{n})\subset
L^{\infty}(\mathbb{R}^{n})$, which implies that the two-point functions
$\lambda(x)\lambda(y),\mu(x)\mu(y)$ and $\lambda(x)\mu(y)$ all belong to
$L^{\infty}(\mathbb{R}^{2n})$. The result then follows by the mapping
properties of $\nabla^{s}$ and $(\nabla\cdot)^{s}$ from Section 2.3 and
equation (7).

In order to see the self-adjointness, recall the minor and major symmetries of
the stiffness tensor $C_{ij\alpha\beta}^{1/2}=C_{ji\alpha\beta}^{1/2}$,
$C_{ij\alpha\beta}^{1/2}=C_{\alpha\beta ij}^{1/2}$, as well as the last
equality from Lemma 3.1. Then we compute for $\phi\in H^{s}(\mathbb{R}^{n})$

$\displaystyle\langle\mathbf{E}^{s}u,\phi\rangle$
$\displaystyle=\langle(\nabla\cdot)^{s}(C^{1/2}(x):C^{1/2}(y)(\nabla^{s}u+\nabla^{s}u^{T})),\phi\rangle$
$\displaystyle=\langle
C^{1/2}(x):C^{1/2}(y)(\nabla^{s}u+\nabla^{s}u^{T}),\nabla^{s}\phi\rangle$
$\displaystyle=\int_{\mathbb{R}^{2n}}C_{ij\alpha\beta}^{1/2}(x)C^{1/2}_{\alpha\beta
lm}(y)(\partial^{s}_{m}u_{l}+\partial^{s}_{l}u_{m})\partial^{s}_{j}\phi_{i}\,dxdy$
$\displaystyle=2\int_{\mathbb{R}^{2n}}C_{ij\alpha\beta}^{1/2}(x)C^{1/2}_{\alpha\beta
lm}(y)\partial^{s}_{m}u_{l}\partial^{s}_{j}\phi_{i}\,dxdy$
$\displaystyle=\int_{\mathbb{R}^{2n}}\partial^{s}_{m}u_{l}\,C_{ij\alpha\beta}^{1/2}(x)C^{1/2}_{\alpha\beta
lm}(y)(\partial^{s}_{j}\phi_{i}+\partial^{s}_{i}\phi_{j})\,dxdy$
$\displaystyle=\langle\nabla^{s}u,C^{1/2}(x):C^{1/2}(y)(\nabla^{s}\phi+\nabla^{s}\phi^{T})\rangle$
$\displaystyle=\langle
u,(\nabla\cdot)^{s}(C^{1/2}(x):C^{1/2}(y)(\nabla^{s}\phi+\nabla^{s}\phi^{T}))\rangle=\langle
u,\mathbf{E}^{s}\phi\rangle.$

∎

######  Remark 3.3.

For $n=1$, $\mathbf{E}^{s}$ reduces to the fractional conductivity operator.
In fact in that case $C$ is a positive scalar function
$\gamma:\mathbb{R}\rightarrow\mathbb{R}$, and $\nabla^{s}u$ can be written as

$\nabla^{s}u(x,y)=\nabla^{s}u^{T}(x,y)=\frac{\mathcal{C}_{1,s}^{1/2}}{\sqrt{2}}\frac{u(y)-u(x)}{|x-y|^{1/2+s}}\,\mbox{sgn}(x-y)$

We also define the fractional potential energy $U^{s}$ as

$\displaystyle U^{s}(u)$
$\displaystyle:=\frac{1}{2}\int_{\mathbb{R}^{2n}}C^{1/2}_{ij\alpha\beta}(x)C^{1/2}_{\alpha\beta
lm}(y)\varepsilon^{s}_{ij}(x,y)\varepsilon_{lm}^{s}(x,y)\,dydx,$

and prove the following Lemma:

######  Lemma 3.4 (Positive definiteness of $U^{s}$).

Let $K\subset\mathbb{R}^{n}$ be a compact set. There exist two constants
$c,C>0$ such that the inequality $c\|u\|_{H^{s}}\leq U^{s}(u)\leq
C\|u\|^{2}_{H^{s}}$ holds for all $u\in H^{s}_{K}(\mathbb{R}^{n})$. As a
consequence, $U^{s}(u)=0$ if and only if $u=0$.

###### Proof.

We first claim that

$C^{1/2}_{ij\alpha\beta}(x)C^{1/2}_{\alpha\beta
lm}(y)v_{i}w_{j}v_{l}w_{m}\gtrsim v_{i}w_{j}v_{i}w_{j}=|v|^{2}|w|^{2}$ (10)

holds for all vectors $v=v(x,y)$, $w=w(x,y)$ in $\mathbb{R}^{n}$. In fact, if
(10) holds, then the definition of the fractional strain tensor (9), the
symmetries of $C^{1/2}$, and the definition of $\nabla^{s}u$ imply that

$\displaystyle U^{s}(u)$
$\displaystyle=2\int_{\mathbb{R}^{2n}}C^{1/2}_{ij\alpha\beta}(x)C^{1/2}_{\alpha\beta
lm}(y)(\nabla^{s}u)_{ij}(\nabla^{s}u)_{lm}dxdy$
$\displaystyle\gtrsim\langle\mathbb{\nabla}^{s}u,\nabla^{s}u\rangle_{L^{2}(\mathbb{R}^{2n})}=\langle(-\Delta)^{s}u,u\rangle_{L^{2}(\mathbb{R}^{n})}=\|(-\Delta)^{s/2}u\|_{L^{2}(\mathbb{R}^{n})}^{2}.$

Using the fractional Poincaré inequality from Theorem 2.7 for $u\in
H^{s}_{K}(\mathbb{R}^{n})$, we finally get

$\|u\|^{2}_{H^{s}(\mathbb{R}^{n})}\leq\|u\|^{2}_{L^{2}(\mathbb{R}^{n})}+\|(-\Delta)^{s/2}u\|^{2}_{L^{2}(\mathbb{R}^{n})}\lesssim\|(-\Delta)^{s/2}u\|^{2}_{L^{2}(\mathbb{R}^{n})}\lesssim
U^{s}(u).$

Thus for the lower bound it suffices to show (10). Using Lemma 3.1 we see that
the left hand side of (10) can be written as

$\left(nk(x)k(y)+\frac{2(n-2)}{n}\mu(x)\mu(y)\right)(v\cdot
w)^{2}+2\mu(x)\mu(y)|v|^{2}|w|^{2},$

where we recall our previous definition $k:=\lambda+2\mu/n$. Since
$k(x),\mu(x)$ are known to be larger than a positive constant for all
$x\in\mathbb{R}^{n}$ and also $n\geq 2$, the lower bound is proved. The
reverse inequality $U^{s}(u)\lesssim\|u\|_{H^{s}}^{2}$ follows immediately
from the boundedness of the coefficients $L,M$ (see also formula (15)). This
completes the proof of the lemma. ∎

While the definition of the fractional elasticity operator $\mathbf{E}^{s}$ is
very useful in order to prove its self-adjointness, when it comes to studying
other properties it is more convenient to rewrite it in a different way.

######  Lemma 3.5 (Reduction lemma).

Let $s\in(0,1)$ and $u\in H^{s}(\mathbb{R}^{n})$. Then in weak sense we have

$\displaystyle\begin{split}(n/2+s)\mathbf{E}^{s}u&=(2n+4s+n^{\prime})\mu(-\Delta)^{s}(u\mu)+nk(-\Delta)^{s}(uk)\\\
&\quad-2n^{\prime}s\mu(-\Delta)^{s-1}\nabla\nabla\cdot(u\mu)-2nsk(-\Delta)^{s-1}\nabla\nabla\cdot(uk)\\\
&\quad+2su\cdot\left\\{n^{\prime}\mu(-\Delta)^{s-1}\nabla^{2}\mu+nk(-\Delta)^{s-1}\nabla^{2}k\right\\}\\\
&\quad-u\left\\{(2n+4s+n^{\prime})\mu(-\Delta)^{s}\mu+nk(-\Delta)^{s}k\right\\},\end{split}$
(11)

where $n^{\prime}:=2(n-2)/n$.

###### Proof.

Step 1. By the computations in Lemmas 3.2 and 3.1, we see that for all test
functions $\phi\in H^{s}(\mathbb{R}^{n})$

$\displaystyle\frac{1}{2}\langle\mathbf{E}^{s}u,\phi\rangle$
$\displaystyle=\langle C_{ij\alpha\beta}^{1/2}(x)C_{\alpha\beta
lm}^{1/2}(y)\partial^{s}_{l}u_{m},\partial^{s}_{i}\phi_{j}\rangle$
$\displaystyle=\langle(n\lambda(x)\lambda(y)+2\lambda(x)\mu(y)+2\lambda(y)\mu(x))\partial^{s}_{j}u_{j},\partial^{s}_{i}\phi_{i}\rangle+$
$\displaystyle\quad+2\langle\mu(x)\mu(y)\partial^{s}_{j}u_{i},\partial^{s}_{j}\phi_{i}\rangle+2\langle\mu(x)\mu(y)\partial^{s}_{i}u_{j},\partial^{s}_{j}\phi_{i}\rangle$
$\displaystyle=\langle(n\lambda(x)\lambda(y)+2\lambda(x)\mu(y)+2\lambda(y)\mu(x)+2\mu(x)\mu(y))\partial^{s}_{i}u_{j},\partial^{s}_{j}\phi_{i}\rangle+$
$\displaystyle\quad+2\langle\mu(x)\mu(y)\partial^{s}_{j}u_{i},\partial^{s}_{j}\phi_{i}\rangle.$

For the last equality we used the relation
$\partial^{s}_{j}u_{j}\,\partial^{s}_{i}\phi_{i}=\partial^{s}_{i}u_{j}\,\partial^{s}_{j}\phi_{i}$,
which follows directly from the definition of the fractional gradient. Using
the definitions $k:=\lambda+2\mu/n$ and $n^{\prime}:=2(n-2)/n$, we can write

$\displaystyle\langle\mathbf{E}^{s}u,\phi\rangle$ $\displaystyle=\langle
2(nk(x)k(y)+n^{\prime}\mu(x)\mu(y))(\nabla^{s}u)^{T}+4\mu(x)\mu(y)\nabla^{s}u,\nabla^{s}\phi\rangle$
$\displaystyle=:\langle
a(x,y)\nabla^{s}u+b(x,y)(\nabla^{s}u)^{T},\nabla^{s}\phi\rangle.$

By the definition of the fractional gradient we then have

$\displaystyle\langle\mathbf{E}^{s}u,\phi\rangle$ $\displaystyle=\langle
a(x,y)(u(y)-u(x))\otimes\zeta+b(x,y)\zeta\otimes(u(y)-u(x)),(\phi(y)-\phi(x))\otimes\zeta\rangle$
$\displaystyle=\langle
a(x,y)|\zeta|^{2}(u(y)-u(x))+b(x,y)(\zeta\otimes(u(y)-u(x)))\cdot\zeta,\phi(y)-\phi(x)\rangle$
$\displaystyle=\langle a(x,y)|\zeta|^{2}(u(y)-u(x)),\phi(y)-\phi(x)\rangle$
$\displaystyle\quad+\langle
b(x,y)(\zeta\otimes\zeta)\cdot(u(y)-u(x)),\phi(y)-\phi(x)\rangle,$

where at the second and third steps we used Lemma 2.4. Since

$\frac{x-y}{|x-y|^{m+2}}=\frac{\nabla_{y}(|x-y|^{-m})}{m}=-\frac{\nabla_{x}(|x-y|^{-m})}{m}$

holds for all $m\neq 0$, and also the identity $\nabla\psi\otimes
v=\nabla(\psi v)-\psi\nabla v$ holds for scalar $\psi$ and vector $v$ (see
again Lemma 2.4), we have

$\displaystyle\frac{(x-y)\otimes(x-y)}{|x-y|^{n+2s+2}}$
$\displaystyle=\frac{\nabla_{y}(|x-y|^{-(n+2s)})}{n+2s}\otimes(x-y)$
$\displaystyle=\frac{1}{n+2s}\left(\frac{Id}{|x-y|^{n+2s}}+\nabla_{y}\left(\frac{x-y}{|x-y|^{n+2s}}\right)\right)$
$\displaystyle=\frac{1}{n+2s}\left(\frac{Id}{|x-y|^{n+2s}}-\frac{\nabla_{y}\nabla_{x}(|x-y|^{-(n+2s-2)})}{n+2s-2}\right),$

which implies

$(\zeta\otimes\zeta)\cdot(u(y)-u(x))=\frac{|\zeta|^{2}(u(y)-u(x))}{n+2s}\\\
-\frac{C_{n,s}\nabla_{y}\nabla_{x}(|x-y|^{-(n+2s-2)})}{2(n+2s)(n+2s-2)}\cdot(u(y)-u(x)).$

Using $C_{n,s}=2s(n+2s-2)C_{n,s-1}$ we get

$\displaystyle\langle\mathbf{E}^{s}u,\phi\rangle$
$\displaystyle=\langle\left(a(x,y)+\frac{b(x,y)}{n+2s}\right)|\zeta|^{2}(u(y)-u(x)),\phi(y)-\phi(x)\rangle$
$\displaystyle\quad-\frac{s\,C_{n,s-1}}{n+2s}\langle
b(x,y)\nabla_{y}\nabla_{x}(|x-y|^{-(n+2s-2)})\cdot(u(y)-u(x)),\phi(y)-\phi(x)\rangle$
$\displaystyle=:\frac{I_{1}-sI_{2}}{n/2+s}.$

Step 2. For the first term $I_{1}$ we compute

$(n+2s)a(x,y)+b(x,y)=2(2n+4s+n^{\prime})\mu(x)\mu(y)+2nk(x)k(y),$

and then obtain

$\displaystyle I_{1}$
$\displaystyle=(2n+4s+n^{\prime})\langle\mu(x)\mu(y)\nabla^{s}u,\nabla^{s}\phi\rangle+n\langle
k(x)k(y)\nabla^{s}u,\nabla^{s}\phi\rangle$
$\displaystyle=(2n+4s+n^{\prime})\langle\mathbf{C}^{s}_{\mu^{2}}u,\phi\rangle+n\langle\mathbf{C}^{s}_{k^{2}}u,\phi\rangle,$

where $\mathbf{C}^{s}_{\mu^{2}},\mathbf{C}^{s}_{k^{2}}$ are fractional
conductivity operators, as studied in [14]. Since by assumptions (A1)-(A3)
$\mu^{2},k^{2}$ are conductivities in the sense of [14], Theorem 3.1 from this
paper can be applied. This leads to

$\begin{split}I_{1}&=(2n+4s+n^{\prime})\langle\mu(-\Delta)^{s}(u\mu)-\mu
u(-\Delta)^{s}({\mu-\mu_{0}}),\phi\rangle\\\ &\quad+n\langle
k(-\Delta)^{s}(uk)-ku(-\Delta)^{s}({k-k_{0}}),\phi\rangle.\end{split}$ (12)

For the second term $I_{2}$ we want to integrate by parts twice. For the sake
of simplicity, we will show our computations only for the term with $k$ coming
from $b$ (the term with $\mu$ is treated in the same way). For $w:=ku$ we have
the integral

$\displaystyle\int_{{\mathbb{R}}^{2n}}$
$\displaystyle\nabla_{y}\nabla_{x}(|x-y|^{-(n+2s-2)}):[(\phi(y)-\phi(x))\otimes(k(x)w(y)-k(y)w(x))]dydx$
$\displaystyle=-\int_{{\mathbb{R}}^{2n}}\nabla_{x}(|x-y|^{-(n+2s-2)})\cdot\left\\{(\nabla\cdot\phi)(y)(k(x)w(y)-k(y)w(x))\right.$
$\displaystyle\qquad\qquad+\left.(\phi(y)-\phi(x))\cdot(k(x)\nabla w(y)-\nabla
k(y)\otimes w(x))\right\\}dydx$
$\displaystyle=\int_{{\mathbb{R}}^{2n}}\frac{1}{|x-y|^{n+2s-2}}(\nabla\cdot)_{x}\left\\{(\nabla\cdot\phi)(y)(k(x)w(y)-k(y)w(x))\right.$
$\displaystyle\qquad\qquad+\left.(\phi(y)-\phi(x))\cdot(k(x)\nabla w(y)-\nabla
k(y)\otimes w(x))\right\\}dydx$
$\displaystyle=\int_{{\mathbb{R}}^{2n}}\frac{1}{|x-y|^{n+2s-2}}\left\\{(\nabla
k(x)\cdot w(y)-k(y)(\nabla\cdot w)(x))(\nabla\cdot\phi)(y)\right.$
$\displaystyle\qquad\qquad+(w(x)\otimes\nabla k(y)-k(x)(\nabla
w(y))^{T}):\nabla\phi(x)$ $\displaystyle\qquad\qquad\left.+(\nabla
w(y)\cdot\nabla k(x)-\nabla k(y)(\nabla\cdot
w)(x))\cdot(\phi(y)-\phi(x))\right\\}dydx$
$\displaystyle=\int_{{\mathbb{R}}^{2n}}\frac{1}{|x-y|^{n+2s-2}}\left\\{(\nabla
k(x)\cdot w(y)-k(y)(\nabla\cdot w)(x))(\nabla\cdot\phi)(y)\right.$
$\displaystyle\qquad\qquad+(w(y)\otimes\nabla k(x)-k(y)(\nabla
w(x))^{T}):\nabla\phi(y)$ $\displaystyle\qquad\qquad\left.+[\nabla
w(y)\cdot\nabla k(x)-\nabla k(y)(\nabla\cdot w)(x)\right.$
$\displaystyle\qquad\qquad\quad\left.-\nabla w(x)\cdot\nabla k(y)+\nabla
k(x)(\nabla\cdot w)(y)]\cdot\phi(y)\right\\}dydx,$

where at the last step we exchanged the $x,y$ variables in the last two terms.
Observe that the $n+2s-2<n$ exponent ensures that all the above integrals are
well-defined. If we define the new operator
$R:=\frac{(-\Delta)^{s-1}}{C_{n,s-1}}$, we can rewrite the last line as

$\displaystyle\int_{{\mathbb{R}}^{n}}$ $\displaystyle\\{R\nabla k\cdot
w-kR(\nabla\cdot w)\\}(\nabla\cdot\phi)+\\{w\otimes R\nabla k-kR(\nabla
w)^{T}\\}:\nabla\phi$ $\displaystyle+\\{\nabla w\cdot R\nabla k-\nabla
kR(\nabla\cdot w)-R\nabla w\cdot\nabla k+R\nabla k(\nabla\cdot
w)\\}\cdot\phi\,dy.$

We integrate by parts in the first two terms one more time. Since $R$ commutes
with the derivatives, we get many cancellations, and eventually

$\displaystyle\int_{{\mathbb{R}}^{n}}-\nabla\\{R\nabla k\cdot w-kR(\nabla\cdot
w)\\}\cdot\phi-\nabla\cdot\\{w\otimes R\nabla k-kR(\nabla w)^{T}\\}\cdot\phi$
$\displaystyle\qquad+\\{\nabla w\cdot R\nabla k-\nabla kR(\nabla\cdot
w)-R\nabla w\cdot\nabla k+R\nabla k(\nabla\cdot w)\\}\cdot\phi\,dy$
$\displaystyle=2\langle kR\nabla\nabla\cdot w-w\cdot R\nabla^{2}k,\phi\rangle$
$\displaystyle=2\langle kR\nabla\nabla\cdot(ku)-ku\cdot
R\nabla^{2}k,\phi\rangle.$

Coming back to $I_{2}$, we have obtained

$\begin{split}I_{2}&=2n\langle
k(-\Delta)^{s-1}\nabla\nabla\cdot(ku)-ku\cdot(-\Delta)^{s-1}\nabla^{2}k,\phi\rangle\\\
&\quad+2n^{\prime}\langle\mu(-\Delta)^{s-1}\nabla\nabla\cdot(\mu u)-\mu
u\cdot(-\Delta)^{s-1}\nabla^{2}\mu,\phi\rangle,\end{split}$ (13)

and the wanted formula finally follows as a combination of (12) and (13);
however, we still need to make sure that all the involved terms make sense in
$H^{-s}({\mathbb{R}}^{n})$. Recall that by assumptions (A1), (A2) there exist
$\mu_{0},k_{0}>0$ such that $\tilde{\mu}:=\mu-\mu_{0}\in
C^{2s+\varepsilon}_{c}(\Omega)$ and $\tilde{k}:=k-k_{0}\in
C^{2s+\varepsilon}_{c}(\Omega)$. For each term we can use a decomposition of
the kind

$\mu(-\Delta)^{s}(u\mu)=\tilde{\mu}(-\Delta)^{s}(u\tilde{\mu})+\mu_{0}\tilde{\mu}(-\Delta)^{s}u+\mu_{0}(-\Delta)^{s}(u\tilde{\mu})+\mu_{0}^{2}(-\Delta)^{s}u,$

and thus it suffices to study the terms in (12) and (13) with $\mu,k$
substituted by $\tilde{\mu},\tilde{k}$.

Given that $\tilde{\mu},\tilde{k}\in C^{2s+\epsilon}({\mathbb{R}}^{n})$, they
both map $H^{s}({\mathbb{R}}^{n})$ to itself by [60], Section 3.3.2. Since the
space of multipliers on $H^{s}(\mathbb{R}^{n})$ coincides with that of the
multipliers on $H^{-s}({\mathbb{R}}^{n})$ (see [49, 11]), we have
$\mu(-\Delta)^{s}(u\mu)\in H^{-s}(\mathbb{R}^{n})$. This same reasoning shows
that the first two terms on the right hand sides of (12) and (13) all make
sense in $H^{-s}(\mathbb{R}^{n})$.

For the remaining parts of (12) and (13) we can proceed as follows, taking as
an example the term $u\cdot\tilde{\mu}(-\Delta)^{s-1}\nabla^{2}\tilde{\mu}$.
It suffices to show $\tilde{\mu}(-\Delta)^{s-1}\nabla^{2}\tilde{\mu}\in
L^{\infty}(\mathbb{R}^{n})$, since this is a set of multipliers on
$L^{2}(\mathbb{R}^{n})$. Moreover, given that $\tilde{\mu}\in
C^{2s+\varepsilon}_{c}(\Omega)\subset L^{\infty}({\mathbb{R}}^{n})$, it is
enough to show that the operator $(-\Delta)^{s-1}\nabla^{2}$ maps
$C^{2s+\varepsilon}_{c}(\Omega)$ to $L^{\infty}(\mathbb{R}^{n})$. However, by
formula (5) we have
$(-\Delta)^{s-1}\nabla^{2}:C^{2s+\varepsilon}({\mathbb{R}}^{n})\rightarrow
C^{\varepsilon}({\mathbb{R}}^{n})$, so that the wanted mapping property
follows from $C^{\varepsilon}({\mathbb{R}}^{n})\subset
L^{\infty}({\mathbb{R}}^{n})$. This concludes the proof. ∎

######  Remark 3.6.

The above reduction can similarly be performed if $u\in
H^{s}({\mathbb{R}}^{n})$ is matrix-valued instead than vector-valued. In this
case the fractional elasticity operator $\mathbf{E}^{s}$ is weakly defined as

$\langle\mathbf{E}^{s}u,\phi\rangle=2\langle
C_{ij\alpha\beta}^{1/2}(x)C_{\alpha\beta
lm}^{1/2}(y)\partial^{s}_{l}u_{mp},\partial^{s}_{i}\phi_{jp}\rangle,$

for all matrix-valued test functions $\phi\in H^{s}({\mathbb{R}}^{n})$. The
first step of the proof, which only deals with the vector $\zeta$, and formula
(12) are unchanged in the matrix case, apart from the additional component
indicated by the index $p$. For the term $I_{2}$ in this case we compute the
integral

$\displaystyle\int_{{\mathbb{R}}^{2n}}\partial_{y,i}\partial_{x,j}$
$\displaystyle(|x-y|^{-(n+2s-2)})[(\phi_{il}(y)-\phi_{il}(x))(k(x)w_{jl}(y)-k(y)w_{jl}(x))]dydx$
$\displaystyle=2\int_{{\mathbb{R}}^{n}}\phi:\left(kR\nabla\nabla\cdot
w-R\nabla^{2}k\cdot w\right)dy$

following the same integration by parts technique shown in the second step of
the proof of Lemma 3.5. If in particular there exists a scalar function $r$
such that $u=rId$, and thus $w$ commutes with all matrices, we obtain that
(11) holds. This observation will be used in the last steps of the proof of
the main theorem.

##  Well-posedness and the DN map

We begin this section by defining our problem of interest. Let $s\in(0,1)$,
and assume $\Omega\subset\mathbb{R}^{n}$ is a bounded open set. In the _direct
problem for the isotropic fractional elasticity equation_ we are given an
exterior value $f\in H^{s}(\mathbb{R}^{n})$, and we want to find a weak
solution $u$ to

$\begin{array}[]{rll}\mathbf{E}^{s}u&=0&\quad\mbox{ in }\Omega\\\
u&=f&\quad\mbox{ in }\Omega_{e}\end{array}.$ (14)

Here the condition $u=f$ in $\Omega_{e}$ should be intended in the sense that
$u-f\in\widetilde{H}^{s}(\Omega)$. In order to define what we mean by a weak
solution, we introduce the following bilinear form. Using the definition of
the operator, we write for $u,v\in C^{\infty}_{c}(\mathbb{R}^{n})$

$\displaystyle B_{L,M}(u,v)$ $\displaystyle:=\langle
C^{1/2}(x):C^{1/2}(y)\nabla^{s}u,\nabla^{s}v\rangle.$

It is immediately seen that $B_{L,M}$ is symmetric. Boundedness in
$H^{s}(\mathbb{R}^{n})\times H^{s}(\mathbb{R}^{n})$ follows easily thanks to
the assumption $L,M\in L^{\infty}(\mathbb{R}^{n})$, which implies
$\lambda,\mu\in L^{\infty}(\mathbb{R}^{n})$ as well:

$\displaystyle\begin{split}|B_{L,M}(u,v)|&\leq\|C^{1/2}(x):C^{1/2}(y)\nabla^{s}u\|_{L^{2}(\mathbb{R}^{2n})}\|\nabla^{s}v\|_{L^{2}(\mathbb{R}^{2n})}\\\
&\lesssim\|\nabla^{s}u\|_{L^{2}(\mathbb{R}^{2n})}\|\nabla^{s}v\|_{L^{2}(\mathbb{R}^{2n})}\leq\|u\|_{H^{s}(\mathbb{R}^{n})}\|v\|_{H^{s}(\mathbb{R}^{n})}.\end{split}$
(15)

With this, we can extend the definition of $B_{L,M}$ to act on
$H^{s}({\mathbb{R}}^{n})\times H^{s}({\mathbb{R}}^{n})$ by density. We can now
say that

$u\in H^{s}(\mathbb{R}^{n})\mbox{ is a weak solution to \eqref{original-
problem} if and only if }B_{L,M}(u,\phi)=0\mbox{ for all
}\phi\in\widetilde{H}^{s}(\Omega),$

and $u-f\in\widetilde{H}^{s}(\Omega)$. More generally, we say that $u\in
H^{s}(\mathbb{R}^{n})$ is a weak solution to the inhomogeneous problem

$\begin{array}[]{rll}\mathbf{E}^{s}u&=F&\quad\mbox{ in }\Omega\\\
u&=f&\quad\mbox{ in }\Omega_{e}\end{array},$ (16)

where $F\in H^{-s}(\Omega)$, if and only if $B_{L,M}(u,\phi)=F(\phi)$ holds
for all $\phi\in\widetilde{H}^{s}(\Omega)$, and
$u-f\in\widetilde{H}^{s}(\Omega)$.

For our problem (16) we have the following well-posedness result:

######  Proposition 4.1 (Well-posedness).

Let $s\in(0,1)$ and assume $\Omega\subset\mathbb{R}^{n}$ is a bounded open
set. For any $f\in H^{s}(\mathbb{R}^{n})$ and $F\in H^{-s}(\Omega)$ there
exists a unique $u\in H^{s}(\mathbb{R}^{n})$ such that
$u-f\in\widetilde{H}^{s}(\Omega)$ and

$B_{L,M}(u,\phi)=F(\phi)\quad\mbox{for
all}\quad\phi\in\widetilde{H}^{s}(\Omega).$

Moreover, the following estimate holds:

$\|u\|_{H^{s}(\mathbb{R}^{n})}\leq
C\left(\|f\|_{H^{s}(\mathbb{R}^{n})}+\|F\|_{H^{-s}(\Omega)}\right).$

###### Proof.

By letting $\tilde{u}:=u-f$, we can reduce the above problem to the one of
finding a unique $\tilde{u}\in\widetilde{H}^{s}(\Omega)$ such that
$B_{L,M}(\tilde{u},\phi)=\tilde{F}(\phi)$, where
$\tilde{F}:=F-B_{L,M}(f,\cdot)$ belongs to $(\widetilde{H}^{s}(\Omega))^{*}$
because of the boundedness estimate (15) for the bilinear form:

$|\tilde{F}(\phi)|\leq|F(\phi)|+|B_{L,M}(f,\phi)|\leq(\left\lVert
F\right\rVert_{H^{-s}(\Omega)}+c\|f\|_{H^{s}(\mathbb{R}^{n})})\|\phi\|_{H^{s}(\mathbb{R}^{n})}.$

Observe that $B_{L,M}(\cdot,\cdot)$ gives an equivalent inner product on
$\widetilde{H}^{s}(\Omega)$, because by Lemma 3.4 the fractional potential
energy $U^{s}(v)$ always verifies $\|v\|^{2}_{H^{s}}\lesssim U^{s}(v)$ , and
also it vanishes if and only if $v=0$. In fact,

$\displaystyle B_{L,M}(v,v)$
$\displaystyle=\langle\nabla^{s}v,C^{1/2}(x):C^{1/2}(y)\nabla^{s}v\rangle$
$\displaystyle=\int_{\mathbb{R}^{2n}}C^{1/2}_{ij\alpha\beta}(x)C^{1/2}_{\alpha\beta
lm}(y)\varepsilon^{s}_{ij}(x,y)\varepsilon_{lm}^{s}(x,y)\,dydx=2\,U^{s}(v).$

The Riesz representation theorem now ensures the existence of a bounded linear
operator $G:{H}^{-s}(\Omega)\rightarrow\widetilde{H}^{s}(\Omega)$ associating
each functional in ${H}^{-s}(\Omega)$ to its unique representative in the
inner product given by $B_{L,M}(\cdot,\cdot)$ on $\widetilde{H}^{s}(\Omega)$.
Thus the wanted (unique) solution $\tilde{u}\in\widetilde{H}^{s}(\Omega)$ can
be defined as $\tilde{u}:=G\tilde{F}$, and it verifies

$B_{L,M}(\tilde{u},\phi)=\tilde{F}(\phi)\quad\mbox{for
all}\quad\phi\in\widetilde{H}^{s}(\Omega).$

The boundedness of $G$ and the definition of $\tilde{u}$ eventually give the
estimate

$\displaystyle\|u\|_{H^{s}(\mathbb{R}^{n})}$
$\displaystyle\leq\|f\|_{H^{s}(\mathbb{R}^{n})}+\|\tilde{u}\|_{H^{s}(\Omega)}=\|f\|_{H^{s}(\mathbb{R}^{n})}+\|G\tilde{F}\|_{H^{s}(\Omega)}$
$\displaystyle\leq\|f\|_{H^{s}(\mathbb{R}^{n})}+C\|\tilde{F}\|_{H^{-s}(\Omega)}\leq
C\left(\|f\|_{H^{s}(\mathbb{R}^{n})}+\|F\|_{H^{-s}(\Omega)}\right).$

∎

In light of Proposition 4.1, we can define a Poisson operator $P_{L,M}$ of
$H^{s}(\mathbb{R}^{n})$ into itself: if $f\in H^{s}(\mathbb{R}^{n})$ is any
exterior datum, then $P_{L,M}f$ is by definition the unique solution to the
homogeneous problem (14). It of course follows from Proposition 4.1 that
$P_{L,M}$ is a bounded operator.

######  Remark 4.2.

Let $f,g$ be exterior values, let $u_{f}:=P_{L,M}f$ and $u_{g}:=P_{L,M}g$ be
the unique solutions corresponding to them, and let $e_{f},e_{g}$ be any
extensions. Then

$\displaystyle B_{L,M}(u_{f},e_{g})$ $\displaystyle=\langle
C^{1/2}(x):C^{1/2}(y)\nabla^{s}u_{f},\nabla^{s}e_{g}\rangle=\langle\mathbf{E}^{s}u_{f},e_{g}\rangle=\langle\mathbf{E}^{s}u_{f},e_{g}\rangle_{\Omega_{e}}$
$\displaystyle=\langle\mathbf{E}^{s}u_{f},u_{g}\rangle_{\Omega_{e}}=\langle\mathbf{E}^{s}u_{f},u_{g}\rangle=\langle
u_{f},\mathbf{E}^{s}u_{g}\rangle,$

where we used the properties of $(\nabla\cdot)^{s}$, the fact that
$\mathbf{E}^{s}u_{f}=0$ in $\Omega$ and the self-adjointness of
$\mathbf{E}^{s}$. Now following the same computations backwards gives
$B_{L,M}(u_{f},e_{g})=B_{L,M}(u_{g},e_{f})$.

With the well-posedness of the direct problem, we can now define the DN map.
Consider first the abstract trace space
$X:=H^{s}({\mathbb{R}}^{n})/\widetilde{H}^{s}(\Omega)$. It is such that two
functions $f_{1},f_{2}\in H^{s}({\mathbb{R}}^{n})$ belong to the same
equivalence class if and only if they agree in $\Omega_{e}$, in the sense that
$f_{1}-f_{2}\in\widetilde{H}^{s}(\Omega)$. If $\Omega$ happens to be
Lipschitz, then it has been proved in [27, p.463] that $X=H^{s}(\Omega_{e})$.

######  Lemma 4.3 (DN map).

Let $s\in(0,1)$, and assume that $\Omega\subset\mathbb{R}^{n}$ is a bounded
open set. There exists a continuous, self-adjoint linear map

$\Lambda_{L,M}:X\rightarrow X^{*}\qquad\mbox{defined
by}\qquad\langle\Lambda_{L,M}[f],[g]\rangle:=B_{L,M}(P_{L,M}f,g),$

where $f,g\in H^{s}(\mathbb{R}^{n})$.

###### Proof.

The proof is quite standard, and it follows the arguments presented in [27,
11, 15]. The DN map $\Lambda_{L,M}$ is well-defined because of the well-
posedness of the direct problem: in fact, since
$[f]=f+\widetilde{H}^{s}(\Omega)$, we have $P_{L,M}f^{\prime}=P_{L,M}f$ for
all $f^{\prime}\in[f]$. Moreover,
$B_{L,M}(P_{L,M}f,g^{\prime})=B_{L,M}(P_{L,M}f,g)$ for all $g^{\prime}\in[g]$
by the definition of the Poisson operator. The boundedness of the bilinear
form and the well-posedness estimate give the continuity of $\Lambda_{L,M}$.
Finally, the self-adjointness of $\Lambda_{L,M}$ follows from Remark 4.2:

$\langle\Lambda_{L,M}[f],[g]\rangle=B_{L,M}(P_{L,M}f,e_{g})=B_{L,M}(P_{L,M}g,e_{f})=\langle\Lambda_{L,M}[g],[f]\rangle.$

∎

##  The fractional Liouville reduction

In this Section we show an equivalence between our original problem (14) and a
Schrödinger-like problem in which the nonlocal part does not depend on the
coefficients. In analogy to the classical transformation from the conductivity
equation to Schrödinger’s, we call this procedure _fractional Liouville
reduction_ (see also [14]). The reason of such transformation will be made
clear in Remark 7.3.

Recall that every vector valued $u\in H^{s}(\mathbb{R}^{n})$ admits a
_Helmholtz decomposition_ , i.e., it can be written as $u=\nabla\phi+F$, where
$\phi$ is the Newtonian potential of $\nabla\cdot u$ and $\nabla\cdot F=0$.
See e.g. [18] and references therein.

This allows us to define an operator $(\cdot)^{\prime}:u\mapsto u^{\prime}$
for all $u\in H^{s}(\mathbb{R}^{n})\times H^{s}(\mathbb{R}^{n})$ such that, if
the Helmholtz decomposition of $u$ is

$u=\begin{pmatrix}u_{1}\\\
u_{2}\end{pmatrix}=\begin{pmatrix}\nabla\phi_{1}+F_{1}\\\
\nabla\phi_{2}+F_{2}\end{pmatrix},\quad\mbox{ with }\nabla\cdot
F_{1}=\nabla\cdot F_{2}=0,$

then $u^{\prime}\in H^{s}(\mathbb{R}^{n})\times H^{s}(\mathbb{R}^{n})$ is

$u^{\prime}:=\begin{pmatrix}(2n+4s+n^{\prime}+2n^{\prime}s)\nabla\phi_{1}+(2n+4s+n^{\prime})F_{1}\\\
n(1+2s)\nabla\phi_{2}+nF_{2}\end{pmatrix}.$

Recall that $n^{\prime}=2(n-2)/n$. The operator $(\cdot)^{\prime}$ is bounded,
with

$\begin{split}\|u^{\prime}\|_{H^{s}\times
H^{s}}&\lesssim\sum_{j=1}^{2}(\|\nabla\phi_{j}\|_{H^{s}}+\|F_{j}\|_{H^{s}})\leq\|u\|_{H^{s}\times
H^{s}}.\end{split}$ (17)

Moreover, if we define the differential operator $D$ acting as

$D\begin{pmatrix}u_{1}\\\ u_{2}\end{pmatrix}:=\begin{pmatrix}D_{1}u_{1}\\\
D_{2}u_{2}\end{pmatrix}:=-\begin{pmatrix}d_{1}\Delta
u_{1}+d_{2}\nabla\nabla\cdot u_{1}\\\ d_{3}\Delta u_{2}+d_{4}\nabla\nabla\cdot
u_{2}\end{pmatrix}:=-\begin{pmatrix}(2n+4s+n^{\prime})\Delta
u_{1}+2n^{\prime}s\nabla\nabla\cdot u_{1}\\\ n\Delta
u_{2}+2ns\nabla\nabla\cdot u_{2}\end{pmatrix},$

we can compute

$\displaystyle-Du$
$\displaystyle=\begin{pmatrix}\left((2n+4s+n^{\prime})\Delta+2n^{\prime}s\nabla\nabla\cdot\right)(\nabla\phi_{1}+F_{1})\\\
\left(n\Delta+2ns\nabla\nabla\cdot\right)(\nabla\phi_{2}+F_{2})\end{pmatrix}$
$\displaystyle=\begin{pmatrix}(2n+4s+n^{\prime}+2n^{\prime}s)\Delta\nabla\phi_{1}+(2n+4s+n^{\prime})\Delta
F_{1}\\\ n(1+2s)\Delta\nabla\phi_{2}+n\Delta F_{2}\end{pmatrix}$
$\displaystyle=\Delta\begin{pmatrix}(2n+4s+n^{\prime}+2n^{\prime}s)\nabla\phi_{1}+(2n+4s+n^{\prime})F_{1}\\\
n(1+2s)\nabla\phi_{2}+nF_{2}\end{pmatrix}=\Delta u^{\prime},$

which shows that $Du=-\Delta u^{\prime}$. In particular,
$(-\Delta)^{s-1}Du=(-\Delta)^{s}u^{\prime}$ whenever $u\in
H^{s}(\mathbb{R}^{n})\times H^{s}(\mathbb{R}^{n})$.

######  Proposition 5.1 (Fractional Liouville reduction).

Let $L,M\in C^{2s+\varepsilon}(\mathbb{R}^{n})$ satisfy assumptions (A1)-(A3),
and assume $f\in C^{\infty}_{c}(W)$ with $W\subset\Omega_{e}$ open and
bounded. Define

$\Gamma(x):=\left(\mu(x),k(x)\right),\qquad
Q(x):=\frac{\Gamma(x)}{|\Gamma(x)|^{2}}\cdot(-\Delta)^{s-1}D(\Gamma(x)\otimes
Id).$

If $u\in H^{s}(\mathbb{R}^{n})$ solves the original problem

$\begin{array}[]{rll}\mathbf{E}^{s}u&=F&\quad\mbox{ in }\Omega\\\
u&=f&\quad\mbox{ in }\Omega_{e}\end{array}$ (18)

in the weak sense for some $F\in L^{2}(\Omega)$, then $w:=\Gamma\otimes u$
solves the transformed problem

$\begin{array}[]{rll}(-\Delta)^{s-1}Dw-w\cdot Q&=G&\quad\mbox{ in }\Omega\\\
w&=\Gamma\otimes f&\quad\mbox{ in }\Omega_{e}\end{array}$ (19)

in the weak sense in $\Gamma\otimes H^{s}$, where
$G:=\Gamma\otimes\frac{(n/2+s)F}{|\Gamma|^{2}}$. Conversely, if
$w\in\Gamma\otimes H^{s}(\mathbb{R}^{n})$ solves (19) in the weak sense in
$\Gamma\otimes H^{s}$ for some $G\in\Gamma\otimes L^{2}(\Omega)$, then
$u:=\frac{\Gamma\cdot w}{|\Gamma|^{2}}$ solves (18) in the weak sense, where
$F=\frac{\Gamma\cdot G}{n/2+s}$.

Observe that saying that $w\in\Gamma\otimes H^{s}(\mathbb{R}^{n})$ is a
solution in the weak sense in $\Gamma\otimes H^{s}$ means that

$B_{Q}(w,\Gamma\otimes\phi)=\langle G,\Gamma\otimes\phi\rangle,\qquad\mbox{
for all }\quad\phi\in\widetilde{H}^{s}(\Omega).$

This can be equivalently written as

$\Gamma\cdot((-\Delta)^{s-1}Dw-w\cdot Q)=\Gamma\cdot G\text{ in $\Omega$}.$

###### Proof.

Because of the assumptions on $L,M$ we have $\Gamma\in
C^{2s+\varepsilon}(\mathbb{R}^{n})\times C^{2s+\varepsilon}(\mathbb{R}^{n})$,
with $\Gamma(x)=\left(\mu_{0},k_{0}\right)=:\gamma$ for all $x\in\Omega_{e}$.
Therefore

$Q(x)=\frac{\Gamma(x)}{|\Gamma(x)|^{2}}\cdot(-\Delta)^{s-1}D((\Gamma(x)-\gamma)\otimes
Id),$

where $\Gamma(x)-\gamma=:\tilde{\Gamma}\in
C^{2s+\varepsilon}_{c}(\Omega)\times C^{2s+\varepsilon}_{c}(\Omega)$. By
formula (5), we have $(-\Delta)^{s-1}D((\Gamma(x)-\gamma)\otimes Id)\in
C^{\varepsilon}(\mathbb{R}^{n})\subset L^{\infty}(\mathbb{R}^{n})$. Since
$\frac{\Gamma(x)}{|\Gamma(x)|^{2}}$ is also in $L^{\infty}(\mathbb{R}^{n})$ by
property (2), we conclude that $Q\in L^{\infty}({\mathbb{R}}^{n})$.

Let now $u\in H^{s}(\mathbb{R}^{n})$. In terms of the new symbols introduced
in the statement of the Proposition, after a straightforward computation the
equality from Lemma 3.5 can equivalently be written as

$\Gamma\cdot\left((-\Delta)^{s-1}D(\Gamma\otimes u)-(\Gamma\otimes u)\cdot
Q\right)=(n/2+s)\mathbf{E}^{s}u.$ (20)

Observe that problem (19) is well-posed in the Hilbert space $\Gamma\otimes
H^{s}(\mathbb{R}^{n})$. In fact, in $(\Gamma\otimes
H^{s}(\mathbb{R}^{n}))\times(\Gamma\otimes H^{s}(\mathbb{R}^{n}))$ the
bilinear form

$B_{Q}(v_{1},v_{2}):=\langle(-\Delta)^{s/2}v_{1}^{\prime},(-\Delta)^{s/2}v_{2}\rangle-\langle
v_{1}\cdot Q,v_{2}\rangle$

is clearly bounded, and it is also coercive by equation (20):

$\displaystyle B_{Q}(\Gamma\otimes u,\Gamma\otimes u)$
$\displaystyle=\langle(-\Delta)^{s-1}D(\Gamma\otimes u)-(\Gamma\otimes u)\cdot
Q,\Gamma\otimes u\rangle$
$\displaystyle=\langle\Gamma\cdot\left((-\Delta)^{s-1}D(\Gamma\otimes
u)-(\Gamma\otimes u)\cdot Q\right),u\rangle$
$\displaystyle=(n/2+s)\langle\mathbf{E}^{s}u,u\rangle$
$\displaystyle=(2n+4s)U^{s}(u)\geq 0.$

Thus (19) has a unique solution of the kind $w=\Gamma\otimes v$ for some $v\in
H^{s}(\mathbb{R}^{n})$, provided that $G\in L^{2}(\Omega)$. Of course one must
have $v=f$ in $\Omega_{e}$.

If now $u\in H^{s}({\mathbb{R}}^{n})$ solves (18) and
$G:=\Gamma\otimes\frac{(n/2+s)F}{|\Gamma|^{2}}$, then by (20)
$w:=\Gamma\otimes u$ solves (19). Conversely, if $w=\Gamma\otimes v$ solves
(19) in $\Gamma\otimes H^{s}$ and $F=\frac{\Gamma\cdot G}{n/2+s}$, then by
(20) $v$ solves (18). Therefore, $u=\frac{\Gamma\cdot
w}{|\Gamma|^{2}}=\frac{\Gamma\cdot(\Gamma\otimes v)}{|\Gamma|^{2}}=v$ must
also solve (18). ∎

We also define the adjoint bilinear form

$B^{*}_{Q}(u,v):=\langle(-\Delta)^{s/2}u,(-\Delta)^{s/2}v^{\prime}\rangle-\langle
v\cdot Q,u\rangle,$

which of course shares the same boundedness inequality as $B_{Q}$ and can
similarly be extended to act on
$(\Gamma\otimes\widetilde{H}^{s}(\Omega))\times(\Gamma\otimes\widetilde{H}^{s}(\Omega))$.
It is also clear that we have $B_{Q}(u,v)=B^{*}_{Q}(v,u)$. Given the well-
posedness in $\Gamma\otimes H^{s}$ of problem (19), we can define the Poisson
operator $P_{Q}$ associating to the exterior datum $\Gamma\otimes f$ the
unique solution $w=\Gamma\otimes v$ to the problem (19) with $G=0$. We can
also define the DN map $\Lambda_{Q}$ in a similar fashion as in our Lemma 4.3:

$\displaystyle\langle\Lambda_{Q}[\Gamma\otimes f_{1}]$
$\displaystyle,[\Gamma\otimes f_{2}]\rangle:=B_{Q}(P_{Q}(\Gamma\otimes
f_{1}),\Gamma\otimes f_{2})$
$\displaystyle=\langle(-\Delta)^{s/2}(P_{Q}(\Gamma\otimes
f_{1}))^{\prime},(-\Delta)^{s/2}(\Gamma\otimes f_{2})\rangle-\langle
P_{Q}(\Gamma\otimes f_{1})\cdot Q,\Gamma\otimes f_{2}\rangle$
$\displaystyle=\langle\Gamma\cdot\left((-\Delta)^{s-1}DP_{Q}(\Gamma\otimes
f_{1})-P_{Q}(\Gamma\otimes f_{1})\cdot Q\right),f_{2}\rangle,$

and similarly for $\Lambda_{Q}^{*}$. With the usual computation (see e.g. the
analogous result in [11]), we get
$\langle\Lambda_{Q}[g_{1}],[g_{2}]\rangle=\langle[g_{1}],\Lambda_{Q}^{*}[g_{2}]\rangle$,
which motivates the choice of symbols.

##  The Alessandrini identity

The most important instruments needed for proving our main theorem are the so
called Alessandrini identity and Runge approximation property, which we study
in this and the next section. Let us start from a simple Lemma relating the DN
maps of the original and transformed problems:

######  Lemma 6.1 (Relation between the DN maps).

Let $f_{j}\in C^{\infty}_{c}(W_{j})$ for $j=1,2$, where
$W_{1},W_{2}\subset\Omega_{e}$ are open, bounded and disjoint. Then the
following equation holds:

$(n/2+s)\langle\Lambda_{L,M}[f_{1}],[f_{2}]\rangle=\langle\Lambda_{Q}[\Gamma\otimes
f_{1}],[\Gamma\otimes f_{2}]\rangle.$ (21)

###### Proof.

Let $u_{1}$ be the unique solution to problem (18) corresponding to the
exterior value $f_{1}$, and let $w_{1}$ be the unique solution to (19)
corresponding to $u_{1}$ via the fractional Liouville reduction. In light of
formula (20), we can compute

$\displaystyle(n/2+s)\langle\Lambda_{L,M}[f_{1}],$
$\displaystyle[f_{2}]\rangle=(n/2+s)\langle\mathbf{E}^{s}u_{1},f_{2}\rangle=\langle\Gamma\cdot\left((-\Delta)^{s-1}Dw_{1}-w_{1}\cdot
Q\right),f_{2}\rangle$ $\displaystyle=\langle(-\Delta)^{s-1}Dw_{1}-w_{1}\cdot
Q,\Gamma\otimes f_{2}\rangle=\langle\Lambda_{Q}[\Gamma\otimes
f_{1}],[\Gamma\otimes f_{2}]\rangle.$

Therefore, complete knowledge of the DN map $\Lambda_{L,M}$ is equivalent to
knowledge of the DN map $\Lambda_{Q}$ on functions $g_{j}$ of the kind
$\Gamma\otimes f_{j}$. ∎

Next, we state and prove the Alessandrini identity:

######  Lemma 6.2 (Alessandrini identity).

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded open set and $s\in(0,1)$. Let
$L_{j},M_{j}$ for $j=1,2$ be two sets of Lamé parameters satisfying
assumptions (A1)-(A3), corresponding to $\Gamma_{j},Q_{j}$ for $j=1,2$ through
the fractional Liouville reduction. Assume
$\Gamma_{1}(x)=\Gamma_{2}(x)=:\gamma$ for all $x\in\Omega_{e}$ and that the
relative Poisson ratios of $(L_{1},M_{1})$ and $(L_{2},M_{2})$ agree in
$\mathbb{R}^{n}$, i.e. $(L_{1},M_{1})\sim(L_{2},M_{2})$. Then the following
integral identity holds for all $f_{1},f_{2}\in C^{\infty}_{c}(\Omega_{e})$

$(n/2+s)\langle(\Lambda_{L_{1},M_{1}}-\Lambda_{L_{2},M_{2}})[f_{1}],[f_{2}]\rangle=\langle
u_{1}\cdot(Q_{1}-Q_{2}),u_{2}^{*}\rangle,$

where $u_{1}:=P_{Q_{1}}(\gamma\otimes f_{1})$ and
$u_{2}^{*}:=P_{Q_{2}}^{*}(\gamma\otimes f_{2})$.

######  Remark 6.3.

Let $\nu$ be the Poisson ratio corresponding to the Lamé parameters $M,K$. By
the definition of $\nu$ and Lemma 3.1, we see that there is a one-to-one
correspondence between $\nu$ and the ratio $r:=\mu/k$ of the Lamé parameters
$\mu,k$ of the square root of the stiffness tensor. Thus
$(L_{1},M_{1})\sim(L_{2},M_{2})$ if and only if $\mu_{1}/k_{1}=\mu_{2}/k_{2}$,
that is, if and only if $\Gamma_{1}=\rho\Gamma_{2}$ for some fixed function
$\rho$. We say that in this case $\Gamma_{1}$ and $\Gamma_{2}$ are themselves
in gauge, and we indicate this by $\Gamma_{1}\sim\Gamma_{2}$.

###### Proof of Lemma 6.2.

The proof is a computation following from Lemma 6.1:

$\displaystyle(n/2+s)\langle(\Lambda_{L_{1},M_{1}}$
$\displaystyle-\Lambda_{L_{2},M_{2}})[f_{1}],[f_{2}]\rangle$
$\displaystyle=(n/2+s)(\langle\Lambda_{L_{1},M_{1}}[f_{1}],[f_{2}]\rangle-\langle\Lambda_{L_{2},M_{2}}[f_{1}],[f_{2}]\rangle)$
$\displaystyle=\langle\Lambda_{Q_{1}}[\Gamma_{1}\otimes
f_{1}],[\Gamma_{1}\otimes
f_{2}]\rangle-\langle\Lambda_{Q_{2}}[\Gamma_{2}\otimes
f_{1}],[\Gamma_{2}\otimes f_{2}]\rangle$
$\displaystyle=\langle\Lambda_{Q_{1}}[\gamma\otimes f_{1}],[\gamma\otimes
f_{2}]\rangle-\langle[\gamma\otimes f_{1}],\Lambda_{Q_{2}}^{*}[\gamma\otimes
f_{2}]\rangle$ $\displaystyle=B_{Q_{1}}(P_{Q_{1}}(\gamma\otimes
f_{1}),{\Gamma_{2}}\otimes f_{2})-B_{Q_{2}}^{*}(P_{Q_{2}}^{*}(\gamma\otimes
f_{2}),{\Gamma_{1}}\otimes f_{1})$
$\displaystyle=B_{Q_{1}}(u_{1},u_{2}^{*})-B_{Q_{2}}^{*}(u_{2}^{*},u_{1})$
$\displaystyle=\langle u_{1}\cdot(Q_{1}-Q_{2}),u_{2}^{*}\rangle.$

Here we have used the fact that $\Gamma_{1}\sim\Gamma_{2}$ in order to deduce
that $B_{Q_{1}}(u_{1},{\Gamma_{2}}\otimes f_{2})=B_{Q_{1}}(u_{1},u_{2}^{*})$,
and similarly for the other term. In fact, this will be true as soon as

$B_{Q_{1}}(u_{1},\Gamma_{2}\otimes v_{2})=0$

for all $v_{2}\in\widetilde{H}^{s}(\Omega)$, which is granted by the fact that
$u_{1}$ is a weak solution in $\Gamma_{1}\otimes H^{s}$and
$\Gamma_{2}=\rho\Gamma_{1}$. ∎

Given that the right hand side of the Alessandrini identity from Lemma 6.2
only contains the difference of the transformed potentials $Q_{1}$ and
$Q_{2}$, we can at most hope to recover $Q$ from the complete knowledge of the
DN map $\Lambda_{L,M}$. This suggests that we may encounter a gauge invariance
for our inverse problem: if many different couples of Lamé parameters $(L,M)$
give rise to the same transformed potential $Q$, they will remain
indistinguishable. Thus we are now left with two tasks: to find appropriate
solutions to use in the Alessandrini identity which will let us recover
information about $Q$, and to study the relative gauge. These problems will be
considered in the coming sections.

##  Runge approximation property and proof of the main Theorem

Because of the particular exterior values associated to the solutions
appearing in our Alessandrini identity, we do not need to prove a full Runge
approximation property in the sense of [27] or [11]. We rather need only the
following result:

######  Lemma 7.1 (Runge approximation property).

Let $\Omega,W\subset\mathbb{R}^{n}$ be bounded open sets such that
$W\subset\Omega_{e}$, and assume $s\in(0,1)$. Define

$\mathcal{R}:=\left\\{\frac{\Gamma\cdot P^{*}_{Q}(\Gamma\otimes
f)}{|\Gamma|^{2}}-f:f\in
C^{\infty}_{c}(W)\right\\}\subset\widetilde{H}^{s}(\Omega).$

Then the set $\\{w|_{\Omega}:w\in\mathcal{R}\\}$ is dense in $L^{2}(\Omega)$.
The same result holds when we substitute $P_{Q}$ to $P_{Q}^{*}$ in the
definition of $\mathcal{R}$.

###### Proof.

By the Hahn-Banach theorem, it is enough to show that any $F\in L^{2}(\Omega)$
such that $\langle F,v\rangle=0$ for all $v\in\mathcal{R}$ must vanish
identically. Fix any $F\in L^{2}(\Omega)$ with such property, and consider the
problem

$\begin{array}[]{rll}(-\Delta)^{s-1}Dw-w\cdot Q&=-\frac{\Gamma\otimes
F}{|\Gamma|^{2}}&\quad\mbox{ in }\Omega\\\ w&=0&\quad\mbox{ in
}\Omega_{e}\end{array}.$

It has a unique weak solution in $\Gamma\otimes\widetilde{H}^{s}(\Omega)$ of
the form $w=\Gamma\otimes\phi$ by Proposition 5.1. Then for any $f\in
C^{\infty}_{c}(W)$ we have

$\displaystyle 0$ $\displaystyle=\langle F,\frac{\Gamma\cdot
P^{*}_{Q}(\Gamma\otimes f)}{|\Gamma|^{2}}-f\rangle$ $\displaystyle=\langle
F,\Gamma\cdot\frac{P^{*}_{Q}(\Gamma\otimes f)-\Gamma\otimes
f}{|\Gamma|^{2}}\rangle=\langle\frac{\Gamma\otimes
F}{|\Gamma|^{2}},P^{*}_{Q}(\Gamma\otimes f)-\Gamma\otimes f\rangle$
$\displaystyle=\langle(-\Delta)^{s-1}Dw-w\cdot Q,\Gamma\otimes
f\rangle-\langle(-\Delta)^{s-1}Dw-w\cdot Q,P_{Q}^{*}(\Gamma\otimes f)\rangle.$

The second term on the right hand side is

$B_{Q}(w,P_{Q}^{*}(\Gamma\otimes f))=B_{Q}^{*}(P_{Q}^{*}(\Gamma\otimes
f),w)=0,$

because of the fact that $w\in\Gamma\otimes\widetilde{H}^{s}(\Omega)$ and the
definition of weak solution to the adjoint problem. Thus we are left with

$0=\langle(-\Delta)^{s-1}Dw-w\cdot Q,\Gamma\otimes f\rangle\\\
=\langle(-\Delta)^{s-1}Dw-w\cdot Q,\gamma\otimes
f\rangle=\langle(-\Delta)^{s-1}Dw,\gamma\otimes f\rangle$

because of the assumption that the supports of $f,w$ are disjoint. Eventually

$0=\langle(-\Delta)^{s-1}Dw,\gamma\otimes
f\rangle=\langle(-\Delta)^{s}w^{\prime},\gamma\otimes
f\rangle=\langle(-\Delta)^{s}(\gamma\cdot w^{\prime}),f\rangle,$

which by the arbitrariety of $f$ implies $(-\Delta)^{s}(\gamma\cdot
w^{\prime})=0$ in $W$. The UCP for the fractional Laplacian and the exterior
datum of $w$ now imply $\gamma\cdot w^{\prime}\equiv 0$. Thus in $\Omega$ by
Lemma 2.4

$\displaystyle\frac{(\gamma\cdot\Gamma)F}{|\Gamma|^{2}}$
$\displaystyle=\gamma\cdot\frac{\Gamma\otimes
F}{|\Gamma|^{2}}=\gamma\cdot(-(-\Delta)^{s-1}Dw+w\cdot Q)$
$\displaystyle=-(-\Delta)^{s}(\gamma\cdot w^{\prime})+\gamma\cdot(w\cdot
Q)=\gamma\cdot(w\cdot Q)$ $\displaystyle=\gamma\cdot((\Gamma\otimes\phi)\cdot
Q)=(\gamma\cdot\Gamma)(\phi\cdot Q),$

and by the positivity of $\gamma\cdot\Gamma$ we get $\phi\cdot
Q=F/|\Gamma|^{2}$ in $\Omega$. Therefore in $\Omega$

$(-\Delta)^{s}w^{\prime}=w\cdot Q{-}\frac{\Gamma\otimes
F}{|\Gamma|^{2}}=\Gamma\otimes\left(\phi\cdot
Q{-}\frac{F}{|\Gamma|^{2}}\right)=0,$

which means that $w^{\prime}$ solves

$\begin{array}[]{rll}(-\Delta)^{s}w^{\prime}&=0&\quad\mbox{ in }\Omega\\\
w^{\prime}&=0&\quad\mbox{ in }\Omega_{e}\end{array}.$

By the well-posedness of the direct problem for the fractional Laplacian, we
deduce $w^{\prime}\equiv 0$, which entails $w\equiv 0$ and eventually $F\equiv
0$. ∎

With this result at hand, we can prove our main Theorem.

###### Proof of Theorem 1.2.

Step 1. Given that the known data can always be restricted, we can without
loss of generality assume that the sets $W_{1}$ and $W_{2}$ are disjoint. Let
$f_{j}\in C^{\infty}_{c}(W_{j})$ for $j=1,2$, and define
$u_{1}:=P_{Q_{1}}(\gamma\otimes f_{1})$, $u_{2}:=P_{Q_{2}}^{*}(\gamma\otimes
f_{2})$. By the definition of the Poisson operators we have
$u_{1}=\Gamma_{1}\otimes v_{1}$, $u_{2}^{*}=\Gamma_{2}\otimes v_{2}$ for some
$v_{1},v_{2}\in H^{s}({\mathbb{R}}^{n})$. Thus by the Alessandrini identity
from Lemma 6.2 and Lemma 2.4 it holds that

$\begin{split}0&=(n/2+s)\langle(\Lambda_{L_{1},M_{1}}-\Lambda_{L_{2},M_{2}})[f_{1}],[f_{2}]\rangle\\\
&=\langle u_{1}\cdot(Q_{1}-Q_{2}),u_{2}^{*}\rangle\\\
&=\langle(\Gamma_{1}\otimes v_{1})\cdot(Q_{1}-Q_{2}),\Gamma_{2}\otimes
v_{2}\rangle\\\ &=\langle
v_{1}\cdot(Q_{1}-Q_{2}),(\Gamma_{1}\cdot\Gamma_{2})v_{2}\rangle.\end{split}$
(22)

Let now $g_{1},g_{2}$ be any functions belonging to $C^{\infty}_{c}(\Omega)$.
Using the Runge approximation property from Lemma 7.1, we can find two
sequences $\\{f_{j,i}\\}_{i}\subset C^{\infty}_{c}(W_{j})$, $j=1,2$, such that

$v_{1,i}:=\frac{\Gamma_{1}\cdot P_{Q_{1}}(\gamma\otimes
f_{1,i})}{|\Gamma_{1}|^{2}}=f_{1,i}+g_{1}+r_{1,i},\qquad\mbox{with}\quad\|r_{1,i}\|_{L^{2}(\Omega)}\leq
1/i,$

and

$v_{2,i}:=\frac{\Gamma_{2}\cdot P_{Q_{2}}^{*}(\gamma\otimes
f_{2,i})}{|\Gamma_{2}|^{2}}=f_{2,i}+g_{2}+r_{2,i},\qquad\mbox{with}\quad\|r_{2,i}\|_{L^{2}(\Omega)}\leq
1/i.$

Substituting $v_{1,i},v_{2,i}$ into (22) gives

$\displaystyle 0$ $\displaystyle=\langle
v_{1}\cdot(Q_{1}-Q_{2}),(\Gamma_{1}\cdot\Gamma_{2})v_{2}\rangle$
$\displaystyle=\langle(f_{1,i}+g_{1}+r_{1,i})\cdot(Q_{1}-Q_{2}),(\Gamma_{1}\cdot\Gamma_{2})(f_{2,i}+g_{2}+r_{2,i})\rangle$
$\displaystyle=\langle(g_{1}+r_{1,i})\cdot(Q_{1}-Q_{2}),(\Gamma_{1}\cdot\Gamma_{2})(g_{2}+r_{2,i})\rangle$

by the support assumptions. Moreover, the terms containing the errors
$r_{j,i}$ vanish as $i\rightarrow\infty$, since for example

$|\langle
r_{1,i}\cdot(Q_{1}-Q_{2}),(\Gamma_{1}\cdot\Gamma_{2})r_{2,i}\rangle|\leq\|r_{1,i}\cdot(Q_{1}-Q_{2})\|_{L^{2}(\Omega)}\|(\Gamma_{1}\cdot\Gamma_{2})r_{2,i}\|_{L^{2}(\Omega)}\leq
C/i^{2},$

and similarly for the other ones. Therefore, we are left with

$\displaystyle 0$ $\displaystyle=\langle
g_{1}\cdot(Q_{1}-Q_{2}),(\Gamma_{1}\cdot\Gamma_{2})g_{2}\rangle.$

Since $\Gamma_{1}\cdot\Gamma_{2}=\mu_{1}\mu_{2}+k_{1}k_{2}>0$, by the
arbitrariety of $g_{1},g_{2}\in C^{\infty}_{c}(\Omega)$ we obtain
$Q_{1}=Q_{2}$ in $\Omega$.

Step 2. We are left with the task of proving that $Q_{1}=Q_{2}$ in $\Omega$
implies $\Gamma_{1}=\Gamma_{2}$. By Remark 6.3, the assumption
$(L_{1},M_{1})\sim(L_{2},M_{2})$ already implies that $\Gamma_{2}=r\Gamma_{1}$
for some fixed but unknown function $r$. Thus $\Gamma_{2}$ solves

$\displaystyle\Gamma_{2}\cdot(-\Delta)^{s-1}D(\Gamma_{2}\otimes
Id)=|\Gamma_{2}|^{2}Q_{1}$ $\displaystyle\quad\mbox{ in }\Omega,$
$\displaystyle\Gamma_{2}=\gamma$ $\displaystyle\quad\mbox{ in }\Omega_{e},$

which implies that $r$ solves

$\displaystyle\Gamma_{1}\cdot\left((-\Delta)^{s-1}D(\Gamma_{1}\otimes(rId))-(\Gamma_{1}\otimes(rId))\cdot
Q_{1}\right)=0$ $\displaystyle\quad\mbox{ in }\Omega,$ $\displaystyle r=1$
$\displaystyle\quad\mbox{ in }\Omega_{e}.$

By Remark 3.6, equation (11) holds when $u=rId$. Observe that the right hand
side of equation (11) can be rewritten as the left hand side of the above
equation in $\Omega$ (see Proposition 5.1). This lets us deduce that
$\mathbf{E}^{s}_{L_{1},M_{1}}(rId)=0$ holds in $\Omega$. Because the direct
problem for the fractional elasticity equation can be showed to be well-posed
in the matrix case as well by the same strategy used in Proposition 4.1, we
conclude that it must necessarily be $r\equiv 1$. ∎

######  Remark 7.2.

In order to show that without previous knowledge of the Poisson ratio $\nu$
one may indeed incur in a gauge invariance, we analyze the simple case $n=1$.

Given that it has only one element, the square root of the stiffness tensor is
just $C_{1111}^{1/2}=\lambda+2\mu=k$. The fractional elasticity operator
$\mathbf{E}^{s}$ becomes the conductivity operator $\mathbf{C}^{s}_{k^{2}}$ in
one dimension, and the fractional Liouville reduction given in this work for
the former coincides with the reduction shown in [14] for the latter. Since
the operator has evidently lost all information about $\mu$, it will certainly
be impossible to recover such function, which proves that at least this gauge
invariance is present in the problem of recovering both Lamé parameters
$(\mu,k)$. However, [14] also shows that the conductivity (in our current case
$k^{2}$) can be recovered without gauge from exterior data in the form of the
DN map. This proves that in one dimension the gauge is in fact limited to the
shear modulus $\mu$.

######  Remark 7.3.

At this point, we can clarify why did we need a fractional Liouville reduction
in the first place. In principle, it would have been possible after the
definition of $\Lambda_{L,M}$ to naïvely prove an Alessandrini identity
directly for such DN map. We would have obtained

$\langle(\Lambda_{L_{1},M_{1}}-\Lambda_{L_{2},M_{2}})[f_{1}],[f_{2}]\rangle=\langle(\mathbf{E}_{1}^{s}-\mathbf{E}_{2}^{s})u_{1},u_{2}^{*}\rangle,$
(23)

with clear meaning of the involved symbols. However, the operator on the right
hand side of (23) is _nonlocal_ , as opposed to the _local_ multiplication
operator we obtained on the right hand side of the Alessandrini identity from
Lemma 6.2. This is no minor distinction, as it affects the success of our
plan. In fact, when proving the main theorem we would have obtained

$0=\langle(\Lambda_{L_{1},M_{1}}-\Lambda_{L_{2},M_{2}})[f_{1,k}],[f_{2,k}]\rangle=\langle(\mathbf{E}_{1}^{s}-\mathbf{E}_{2}^{s})(f_{1,k}+v_{1}+r_{1,k}),f_{2,k}+v_{2}+r_{2,k}\rangle$

for some fixed $v_{1},v_{2}\in C^{\infty}_{c}(\Omega)$ and $f_{j,k}\in
C^{\infty}_{c}(W_{j})$, $j=1,2$. Given the nonlocality of
$\mathbf{E}_{1}^{s}-\mathbf{E}_{2}^{s}$, here we could not dismiss the
exterior data $f_{j,k}$ by support assumptions. This is quite dangerous, as it
is known that as the Runge approximation improves a corresponding lost of
control is expected for the exterior data (see e.g. [54]). Thus we are
prevented from obtaining the wanted result.

Observe that in most previous works ([27], [9], [11] come to mind) the
operator on the right hand side of the Alessandrini identity was local as
well. In [13] a similar, albeit simpler fractional Liouville reduction was
needed for the same reason as in our current work. Finally, the recent paper
[15] considers an Alessandrini identity with a nonlocal operator on the right
hand side, but this is assumed to be such that a fine control can still be
established over the exterior data.

## References

  * AA [11] Harm Askes and Elias C. Aifantis.  Gradient elasticity in statics and dynamics: an overview of formulations, length scale identification procedures, finite element implementations and new results.  International Journal of Solids and Structures, 48(13):1962–1990, 2011.
  * AKNT [02] Habib Ammari, Hyeonbae Kang, Gen Nakamura, and Kazumi Tanuma.  Complete asymptotic expansions of solutions of the system of elastostatics in the presence of an inclusion of small diameter and detection of an inclusion.  J. Elasticity, 67(2):97–129 (2003), 2002.
  * Bar [76] Victor Barcilon.  Inverse problem for a vibrating beam.  Z. Angew. Math. Phys., 27(3):347–358, 1976.
  * BC [94] P. Ballard and A. Constantinescu.  On the inversion of subsurface residual stresses from surface stress measurements.  J. Mech. Phys. Solids, 42(11):1767–1787, 1994.
  * BC [05] M. Bonnet and A. Constantinescu.  Inverse problems in elasticity.  Inverse Problems, 21:R1–R50, 2005.
  * BG [04] Paul E. Barbone and Nachiket H. Gokhale.  Elastic modulus imaging: on the uniqueness and nonuniqueness of the elastography inverse problem in two dimensions.  Inverse Problems, 20(1):283–296, 2004.
  * BGU [21] S. Bhattacharyya, T. Ghosh, and G. Uhlmann.  Inverse problems for the fractional-Laplacian with lower order non-local perturbations.  Trans. Amer. Math. Soc., 374(5):3053–3075, 2021.
  * CCS [11] A. Carpinteri, P. Cornetti, and A. Sapora.  A fractional calculus approach to nonlocal elasticity.  The European Physical Journal Special Topics, 193(1):193–204, 2011\.
  * CLR [20] Mihajlo Cekić, Yi-Hsuan Lin, and Angkana Rüland.  The Calderón problem for the fractional Schrödinger equation with drift.  Calc. Var. Partial Differential Equations, 59(3):Paper No. 91, 46, 2020.
  * CMR [21] Giovanni Covi, Keijo Mönkkönen, and Jesse Railo.  Unique continuation property and Poincaré inequality for higher order fractional Laplacians with applications in inverse problems.  Inverse Probl. Imaging, 15(4):641–681, 2021.
  * CMRU [22] Giovanni Covi, Keijo Mönkkönen, Jesse Railo, and Gunther Uhlmann.  The higher order fractional Calderón problem for linear local operators: uniqueness.  Adv. Math., 399:Paper No. 108246, 29, 2022.
  * Con [95] A. Constantinescu.  On the identification of elastic moduli from displacement-force boundary measurements.  Inverse Problems Eng., 1:293–315, 1995.
  * [13] Giovanni Covi.  An inverse problem for the fractional Schrödinger equation in a magnetic field.  Inverse Problems, 36(4):045004, 24, 2020.
  * [14] Giovanni Covi.  Inverse problems for a fractional conductivity equation.  Nonlinear Anal., 193:111418, 18, 2020.
  * Cov [21] Giovanni Covi.  Uniqueness for the fractional Calderón problem with quasilocal perturbations.  Preprint, arXiv:2110.11063, to appear in SIAM Journal on Mathematical Analysis (SIMA), 2021.
  * DGLZ [12] Qiang Du, Max Gunzburger, R. B. Lehoucq, and Kun Zhou.  Analysis and approximation of nonlocal diffusion problems with volume constraints.  SIAM Rev., 54(4):667–696, 2012.
  * DGLZ [13] Qiang Du, Max Gunzburger, R. B. Lehoucq, and Kun Zhou.  A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws.  Math. Models Methods Appl. Sci., 23(3):493–540, 2013.
  * DP [09] Erwan Deriaz and Valérie Perrier.  Orthogonal Helmholtz decomposition in arbitrary dimension using divergence-free and curl-free wavelets.  Appl. Comput. Harmon. Anal., 26(2):249–269, 2009.
  * ER [02] G. Eskin and J. Ralston.  On the inverse boundary value problem for linear isotropic elasticity.  Inverse Problems, 18(3):907–921, 2002.
  * ER [04] Gregory Eskin and James Ralston.  On the inverse boundary value problem for linear isotropic elasticity and Cauchy-Riemann systems.  In Inverse problems and spectral theory, volume 348 of Contemp. Math., pages 53–69. Amer. Math. Soc., Providence, RI, 2004.
  * Eri [72] A. Cemal Eringen.  Linear theory of nonlocal elasticity and dispersion of plane waves.  International Journal of Engineering Science, 10(5):425–435, 1972\.
  * Eri [02] A. Cemal Eringen.  Nonlocal continuum field theories.  Springer-Verlag, New York, 2002.
  * FZ [20] Giuseppe Failla and Massimiliano Zingales.  Advanced materials modelling via fractional calculus: Challenges and perspectives, 2020.
  * GM [89] Z. Gao and T. Mura.  On the inversion of residual stresses from surface measurements.  ASME J. Appl. Mech., 56:508–13, 1989.
  * GP [03] Giuseppe Geymonat and Stéphane Pagano.  Identification of mechanical properties by displacement field measurement: a variational approach.  Meccanica, 38(5):535–545, 2003.  Dedicated to Piero Villaggio on the occasion of his 70th birthday.
  * GRSU [20] Tuhin Ghosh, Angkana Rüland, Mikko Salo, and Gunther Uhlmann.  Uniqueness and reconstruction for the fractional Calderón problem with a single measurement.  J. Funct. Anal., 279(1):108505, 42, 2020.
  * GSU [20] Tuhin Ghosh, Mikko Salo, and Gunther Uhlmann.  The Calderón problem for the fractional Schrödinger equation.  Anal. PDE, 13(2):455–475, 2020.
  * Hör [03] Lars Hörmander.  The analysis of linear partial differential operators. I.  Classics in Mathematics. Springer-Verlag, Berlin, 2003.  Distribution theory and Fourier analysis, Reprint of the second (1990) edition [Springer, Berlin; MR1065993 (91m:35001a)].
  * Ike [90] Masaru Ikehata.  Inversion formulas for the linearized problem for an inverse boundary value problem in elastic prospection.  SIAM J. Appl. Math., 50(6):1635–1644, 1990.
  * Ike [93] Masaru Ikehata.  An inverse problem for the plate in the Love-Kirchhoff theory.  SIAM J. Appl. Math., 53(4):942–970, 1993.
  * Ike [95] Masaru Ikehata.  The linearization of the Dirichlet to Neumann map in anisotropic plate theory.  Inverse Problems, 11(1):165–181, 1995.
  * IY [15] O. Yu. Imanuvilov and M. Yamamoto.  Global uniqueness in inverse boundary value problems for the Navier-Stokes equations and Lamé system in two dimensions.  Inverse Problems, 31(3):035004, 46, 2015.
  * JR [15] Bangti Jin and William Rundell.  A tutorial on inverse problems for anomalous diffusion processes.  Inverse Problems, 31(3):035003, 40, 2015.
  * KST [06] Anatoly A. Kilbas, Hari M. Srivastava, and Juan J. Trujillo.  Theory and applications of fractional differential equations, volume 204 of North-Holland Mathematics Studies.  Elsevier Science B.V., Amsterdam, 2006.
  * Kwa [17] Mateusz Kwaśnicki.  Ten equivalent definitions of the fractional Laplace operator.  Fract. Calc. Appl. Anal., 20(1):7–51, 2017.
  * LBZA [97] Vladimir Lyakhovsky, Yehuda Ben-Zion, and Amotz Agnon.  Distributed damage, faulting, and friction.  Journal of Geophysical Research: Solid Earth, 102(B12):27635–27649, 1997.
  * [37] Li Li.  The Calderón problem for the fractional magnetic operator.  Inverse Problems, 36(7):075003, 14, 2020.
  * [38] Li Li.  A semilinear inverse problem for the fractional magnetic Laplacian.  Preprint, arXiv:2005.06714, 2020.
  * [39] Li Li.  Determining the magnetic potential in the fractional magnetic Calderón problem.  Comm. Partial Differential Equations, 46(6):1017–1026, 2021.
  * [40] Li Li.  On inverse problems arising in fractional elasticity.  Preprint, arXiv:2109.03387, 2021.
  * LL [59] L. D. Landau and E. M. Lifshitz.  Theory of elasticity.  Course of Theoretical Physics, Vol. 7. Pergamon Press, London-Paris-Frankfurt; Addison-Wesley Publishing Company, Inc., Reading, Mass., 1959.  Translated by J. B. Sykes and W. H. Reid.
  * LL [19] Ru-Yu Lai and Yi-Hsuan Lin.  Global uniqueness for the fractional semilinear Schrödinger equation.  Proc. Amer. Math. Soc., 147(3):1189–1199, 2019.
  * LLR [20] Ru-Yu Lai, Yi-Hsuan Lin, and Angkana Rüland.  The Calderón problem for a space-time fractional parabolic equation.  SIAM J. Math. Anal., 52(3):2655–2688, 2020.
  * LO [22] Ru-Yu Lai and Laurel Ohm.  Inverse problems for the fractional Laplace equation with lower order nonlinear perturbations.  Inverse Probl. Imaging, 16(2):305–323, 2022.
  * LW [03] Ching-Lung Lin and Jenn-Nan Wang.  Uniqueness in inverse problems for an elasticity system with residual stress by a single measurement.  Inverse Problems, 19(4):807–820, 2003.
  * McL [00] William McLean.  Strongly elliptic systems and boundary integral equations.  Cambridge University Press, Cambridge, 2000.
  * Min [63] Raymond David Mindlin.  Microstructure in linear elasticity.  Technical report, Columbia Univ., New York, Dept. of Civil Engineering and Engineering Mechanics, 1963.
  * Min [65] Raymond David Mindlin.  Second gradient of strain and surface-tension in linear elasticity.  International Journal of Solids and Structures, 1(4):417–438, 1965\.
  * MS [09] Vladimir G. Maz’ya and Tatyana O. Shaposhnikova.  Theory of Sobolev multipliers, volume 337 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences].  Springer-Verlag, Berlin, 2009.  With applications to differential and integral operators.
  * NU [94] Gen Nakamura and Gunther Uhlmann.  Global uniqueness for an inverse boundary problem arising in elasticity.  Invent. Math., 118(3):457–474, 1994.
  * NU [03] Gen Nakamura and Gunther Uhlmann.  Erratum: “Global uniqueness for an inverse boundary value problem arising in elasticity” [Invent. Math. 118 (1994), no. 3, 457–474; MR1296354 (95i:35313)].  Invent. Math., 152(1):205–207, 2003.
  * OS [07] Zaid M. Odibat and Nabil T. Shawagfeh.  Generalized Taylor’s formula.  Appl. Math. Comput., 186(1):286–293, 2007.
  * Rob [97] R. L. Robertson.  Boundary identifiability of residual stress via the Dirichlet to Neumann map.  Inverse Problems, 13(4):1107–1119, 1997.
  * [54] Angkana Rüland and Mikko Salo.  The fractional Calderón problem: low regularity and stability.  Nonlinear Anal., 193:111529, 56, 2020.
  * [55] Angkana Rüland and Mikko Salo.  Quantitative approximation properties for the fractional heat equation.  Math. Control Relat. Fields, 10(1):1–26, 2020.
  * Rül [18] Angkana Rüland.  Unique continuation, Runge approximation and the fractional Calderón problem.  Journées équations aux dérivées partielles, Exp. No. 8:10p, 2018.
  * Sal [17] Mikko Salo.  The fractional Calderón problem.  Journées équations aux dérivées partielles, Exp. No. 7:8p, 2017\.
  * TA [19] Vasily E. Tarasov and Elias C. Aifantis.  On fractional and fractal formulations of gradient linear and nonlinear elasticity.  Acta Mech., 230(6):2043–2070, 2019.
  * Tay [11] Michael E. Taylor.  Partial differential equations III. Nonlinear equations, volume 117 of Applied Mathematical Sciences.  Springer, New York, second edition, 2011.
  * Tri [83] Hans Triebel.  Theory of function spaces, volume 78 of Monographs in Mathematics.  Birkhäuser Verlag, Basel, 1983.
  * ZO [20] Dušan Zorica and Ljubica Oparnica.  Energy dissipation for hereditary and energy conservation for non-local fractional wave equations.  Philos. Trans. Roy. Soc. A, 378(2172):20190295, 24, 2020.

  * Giovanni Covi - Institut fur Angewandte Mathematik, Ruprecht-Karls-Universität Heidelberg, Germany<EMAIL_ADDRESS>
  * Maarten de Hoop - Department of Computational and Applied Mathematics, Rice University, Houston, TX, USA<EMAIL_ADDRESS>
  * Mikko Salo - Department of Mathematics and Statistics, University of Jyväskylä, Finland<EMAIL_ADDRESS>