The higher order fractional Calderón problem for linear local operators: uniqueness

We denote by $M(H^{s-|\alpha|}\rightarrow H^{-s})$ the space of all bounded Sobolev multipliers between the Sobolev spaces $H^{s-|\alpha|}({\mathbb{R}}^{n})$ and $H^{-s}({\mathbb{R}}^{n})$. We denote by $M_{0}(H^{s-|\alpha|}\rightarrow H^{-s})\subset M(H^{s-|\alpha|}\rightarrow H^{-s})$ the space of bounded Sobolev multipliers that can be approximated by smooth compactly supported functions in the multiplier norm of $M(H^{s-|\alpha|}\rightarrow H^{-s})$. We also write $H^{r,\infty}(\Omega)$ for the local Bessel potential space with bounded derivatives. See section 2 for more detailed definitions.

Our first theorem is a generalization of [44, Theorem 1.1] which considered the case $m=0$ with $s\in(0,1)$. It also generalizes [12, Theorem 1.5] which considered the higher order case $s\in{\mathbb{R}}^{+}\setminus{\mathbb{Z}}$ when $m=0$.

In theorem 1.1 one can pick the lower order coefficients ($\left\lvert\alpha\right\rvert<s$) from $L^{p}(\Omega)$ for high enough $p$ (especially from $L^{\infty}(\Omega)$) and higher order coefficients ($s<\left\lvert\alpha\right\rvert<2s$) from the closure of $C^{\infty}_{c}(\Omega)$ in $H^{r,\infty}(\Omega)$ for certain values of $r\in{\mathbb{R}}$. In the following propositions, which are proved in Section 2, we give more examples of Sobolev spaces which belong to the space of multipliers $M_{0}(H^{s-\left\lvert\alpha\right\rvert}\rightarrow H^{-s})$:

Note that the assumptions in theorem 1.1 satisfy the conditions of proposition 1.2 since then $r=\left\lvert\alpha\right\rvert-s$ and $t=s$. The next proposition gives examples of spaces of lower order coefficients ($\left\lvert\alpha\right\rvert\leq s$):

As mentioned above, we put $t=s>0$ in theorem 1.1 and the condition in proposition 1.3 is satisfied. Note that under the assumption $\left\lvert\alpha\right\rvert\leq s$ we have $M_{0}(H^{0}\rightarrow H^{-s})\subset M_{0}(H^{s-\left\lvert\alpha\right\rvert}\rightarrow H^{-s})$. Hence we can choose the lower order coefficients from a less regular space in theorem 1.1 (compare to proposition 1.2). We also note that when $\left\lvert\alpha\right\rvert=0$ the space of multipliers $M_{0}(H^{s}\rightarrow H^{-s})$ coincides with the one studied in [44].

It follows from lemma 2.5 that the space of multipliers is trivial for higher order operators, i.e. $M(H^{s-|\alpha|}\rightarrow H^{-s})=\{0\}$ when $s-\left\lvert\alpha\right\rvert<-s$. It would be possible to state theorem 1.1 for higher order PDOs, but that forces $a_{\alpha}=0$ for all $\left\lvert\alpha\right\rvert>2s$. For this reason we only consider PDOs whose order is $m<2s$. See lemma 2.5 and the related remarks for more details.

Our second theorem generalizes [8, Theorem 1.1] and [18, Theorem 1.1], where similar results are proved when $m=0,1$ and $s\in(0,1)$. It also generalizes [12, Theorem 1.5] where the case $m=0$ and $s\in{\mathbb{R}}^{+}\setminus{\mathbb{Z}}$ was studied.

Our first theorem is formulated for general bounded open sets and the second theorem for Lipschitz domains. The difference arises in the proof of the well-posedness of the forward problem. We note that theorem 1.4 holds for coefficients $a_{\alpha}$ which are smooth up to the boundary ($a_{\alpha}=g|_{\Omega}$ where $g\in C^{\infty}({\mathbb{R}}^{n})$). The conditions (4) imply that one can choose $a_{\alpha}\in L^{\infty}(\Omega)$ for every $\alpha$ such that $|\alpha|<s$. The case $|\alpha|=s$ never happens, as $s$ is assumed not to be an integer. If $|\alpha|>s$, we have $a_{\alpha}\in H^{|\alpha|-s,\infty}(\Omega)$ when $|\alpha|-s\not\in\{1/2,3/2,...\}$. Thus the conditions (4) coincide with [8, 18] when $m=0,1$ and $s\in(0,1)$.

Our article is roughly divided into two parts. The first part of the article (theorem 1.1 and section 3) generalizes the study of the uniqueness problem for singular potentials in [44] and the second part (theorem 1.4 and section 4) generalizes the uniqueness problem for bounded first order perturbations in [8].

The approach to prove theorems 1.1 and 1.4 is the following. First one shows that the forward problem is well-posed and the corresponding bilinear forms are bounded. This leads to the boundedness of the exterior DN maps and an Alessandrini identity. By a unique continuation property of the higher order fractional Laplacian one obtains a Runge approximation property for equation (2). Using the Runge approximation and the Alessandrini identity for suitable test functions one proves the uniqueness of the inverse problem.

Equation (2) and theorems 1.1 and 1.4 are related to the Calderón problem for the fractional Schrödinger equation first introduced in [18]. There one tries to uniquely recover the potential $q$ in $\Omega$ by doing measurements in the exterior $\Omega_{e}$. This is a nonlocal (fractional) counterpart of the classical Calderón problem arising in electrical impedance tomography, where one obtains information about the electrical properties of some bounded domain by doing voltage and current measurements on the boundary [49, 50]. In [44] the study of the fractional Calderón problem is extended for “rough” potentials $q$, i.e. potentials which are in general bounded Sobolev multipliers. First order perturbations were studied in [8] assuming that the fractional part dominates the equation, i.e. $s\in(1/2,1)$, and that the perturbations have bounded fractional derivatives. A higher order version ($s\in{\mathbb{R}}^{+}\setminus{\mathbb{Z}}$) of the fractional Calderón problem was introduced and studied in [12]. These three articles [8, 12, 44] motivate the study of higher order (rough) perturbations to the fractional Laplacian $(-\Delta)^{s}$ in equation (2). The natural restriction for the order of  $P(x,D)$ in theorems  1.1 and 1.4 is then $2s>m$, so that the fractional part governs the equation (2).

The fractional Calderón problem for $s\in(0,1)$ has been studied in many settings. We refer to the survey  [46] for a more detailed treatment. In the work [44] stability was proved for singular potentials, and in [43] the related exponential instability was shown. The fractional Calderón problem has also been solved under single measurement [17]. The perturbed equation is related to the fractional magnetic Schrödinger equation which is studied in [10, 30, 31, 32]. See also [5] for a fractional Schrödinger equation with a lower order nonlocal perturbation. Other variants of the fractional Calderón problem include semilinear fractional (magnetic) Schrödinger equation  [24, 25, 30, 32], fractional heat equation [26, 45] and fractional conductivity equation [11] (see also [7, 16] for equations arising from a nonlocal Schrödinger-type elliptic operator). In the recent work [12], the first three authors of this article studied higher order versions ($s\in{\mathbb{R}}^{+}\setminus{\mathbb{Z}}$) of the fractional Calderón problem and proved uniqueness for the Calderón problem for the fractional magnetic Schrödinger equation (up to a gauge). This article continues these studies by showing uniqueness for the fractional Schrödinger equation with higher order perturbations and gives positive answer to a question posed in  [12, Question 7.5].

Equations involving fractional Laplacians like (2) have applications in mathematics and natural sciences. Fractional Laplacians appear in the study of anomalous and nonlocal diffusion, and these diffusion phenomena can be used in many areas such as continuum mechanics, graph theory and ecology just to mention a few [2, 6, 15, 34, 41]. Another place where the fractional counterpart of the classical Laplacian naturally shows up is the formulation of fractional quantum mechanics [27, 28, 29]. See [42] and references therein for possible applications of higher order fractional Laplacians. For more applications of fractional mathematical models, see  [6, 37] and the references therein.

In section 2 we introduce the notation and give preliminaries on Sobolev spaces and fractional Laplacians. We also define the spaces of rough coefficients (Sobolev multipliers) and discuss some of the basic properties. In section 3 we prove theorem 1.1 in detail. Finally, in section 4 we prove theorem 1.4 but as the proofs of both theorems are very similar we do not repeat all identical steps and we keep our focus in the differences of the proofs.

G.C. was partially supported by the European Research Council under Horizon 2020 (ERC CoG 770924). K.M. and J.R. were supported by Academy of Finland (Centre of Excellence in Inverse Modelling and Imaging, grant numbers 284715 and 309963). G.U. was partly supported by NSF, a Walker Family Endowed Professorship at UW and a Si Yuan Professorship at IAS, HKUST. G.C. would like to thank Angkana Rüland for helpful discussions and her hospitality during his visit to Max Planck Institute for Mathematics in the Sciences. J.R. and G.U. wish to thank Maarten V. de Hoop, Rice University and Simons Foundation for providing the support to participate 2020 MATH + X Symposium on Inverse Problems and Deep Learning, Mitigating Natural Hazards, where this project was initiated.

The (inhomogeneous) fractional $L^{2}$-based Sobolev space of order $r\in{\mathbb{R}}$ is defined to be

equipped with the norm

Here $\hat{u}=\mathcal{F}(u)$ is the Fourier transform of a tempered distribution $u\in\mathscr{S}^{\prime}({\mathbb{R}}^{n})$, $\mathcal{F}^{-1}$ is the inverse Fourier transform and $\langle x\rangle=(1+\left\lvert x\right\rvert^{2})^{1/2}$. We define the fractional Laplacian of order $s\in{\mathbb{R}}^{+}\setminus{\mathbb{Z}}$ as $(-\Delta)^{s}\varphi=\mathcal{F}^{-1}(\left\lvert\cdot\right\rvert^{2s}\hat{\varphi})$ where $\varphi\in\mathscr{S}({\mathbb{R}}^{n})$ is a Schwartz function. Then $(-\Delta)^{s}$ extends to a bounded operator $(-\Delta)^{s}\colon H^{r}({\mathbb{R}}^{n})\rightarrow H^{r-2s}({\mathbb{R}}^{n})$ for all $r\in{\mathbb{R}}$ by density of $\mathscr{S}({\mathbb{R}}^{n})$ in $H^{r}({\mathbb{R}}^{n})$ [18, Lemma 2.1] (see also [12, Section 2.2]). It would be possible to define the fractional Laplacian in many other ways, according to the intended application (check e.g. [13, 23, 33]). In particular, our global definition of the fractional Laplacian using Fourier transform will be different from the spectral definition used in [21] (see for example [13, 47]).

Let $\Omega\subset{\mathbb{R}}^{n}$ be an open set and $F\subset{\mathbb{R}}^{n}$ a closed set. We define the following Sobolev spaces

It trivially follows that $\widetilde{H}^{r}(\Omega)\subset H_{0}^{r}(\Omega)$ and $\widetilde{H}^{r}(\Omega)\subset H^{r}_{\overline{\Omega}}({\mathbb{R}}^{n})$. Further, we have $(\widetilde{H}^{r}(\Omega))^{*}=H^{-r}(\Omega)$ and $(H^{r}(\Omega))^{*}=\widetilde{H}^{-r}(\Omega)$ for any open set $\Omega$ and $r\in{\mathbb{R}}$ [9, Theorem 3.3]. If $\Omega$ is in addition a Lipschitz domain, then we have $\widetilde{H}^{r}(\Omega)=H_{\overline{\Omega}}^{r}({\mathbb{R}}^{n})$ for all $r\in{\mathbb{R}}$ and $H_{0}^{r}(\Omega)=H^{r}_{\overline{\Omega}}({\mathbb{R}}^{n})$ when $r\geq 0$ such that $r\notin\{\frac{1}{2},\frac{3}{2},\frac{5}{2}\dotso\}$ [36, Theorems 3.29 and 3.33].

More generally, let $1\leq p\leq\infty$ and $r\in{\mathbb{R}}$. We define the Bessel potential space

equipped with the norm

We also write $\mathcal{F}^{-1}(\langle\cdot\rangle^{r}\hat{u})=:J^{r}u$ where the Fourier multiplier $J=(\mathrm{Id}-\Delta)^{1/2}$ is called the Bessel potential. We have the continuous inclusions $H^{r,p}({\mathbb{R}}^{n})\hookrightarrow H^{t,p}({\mathbb{R}}^{n})$ whenever $r\geq t$ [4, Theorem 6.2.3]. By the Mikhlin multiplier theorem one can show that $(-\Delta)^{s}\colon H^{r,p}({\mathbb{R}}^{n})\rightarrow H^{r-2s,p}({\mathbb{R}}^{n})$ is continuous whenever $s\geq 0$ and $1<p<\infty$ (see [18, Remark 2.2] and [1, Theorem 7.2]). The local version of the space $H^{r,p}({\mathbb{R}}^{n})$ is defined as earlier by the restrictions

where $\Omega\subset{\mathbb{R}}^{n}$ is any open set. This space is equipped with the quotient norm

We have the continuous inclusions $H^{r,p}(\Omega)\hookrightarrow H^{t,p}(\Omega)$ whenever $r\geq t$ by the definition of the quotient norm.

We also define the spaces

where $F\subset{\mathbb{R}}^{n}$ is a closed set. Note that $\widetilde{H}^{r,p}(\Omega)\subset H_{0}^{r,p}(\Omega)$ since the restriction map $|_{\Omega}\colon H^{r,p}({\mathbb{R}}^{n})\rightarrow H^{r,p}(\Omega)$ is by definition continuous. One can also see that $\widetilde{H}^{r,p}(\Omega)\subset H^{r,p}_{\overline{\Omega}}({\mathbb{R}}^{n}).$ If $\Omega$ is a bounded $C^{\infty}$-domain and $1<p<\infty$, then we have [48, Theorem 1 in section 4.3.2]

Some authors (especially in [8, 44]) use the notation $W^{r,p}(\Omega)$ for Bessel potential spaces. We have decided to use the notation $H^{r,p}(\Omega)$ so that these spaces are not confused with the Sobolev-Slobodeckij spaces which are in general different from the Bessel potential spaces [14, Remark 3.5].

The equation (2) we study is nonlocal. Instead of putting boundary conditions we impose exterior values for the equation. This can be done by saying that $u=f$ in $\Omega_{e}$ if $u-f\in\widetilde{H}^{s}(\Omega)$. Motivated by this we define the (abstract) trace space $X=H^{r}({\mathbb{R}}^{n})/\widetilde{H}^{r}(\Omega)$, i.e. functions in $X$ are the same (have the same trace) if they agree in $\Omega_{e}$. If $\Omega$ is a Lipschitz domain, then we have $X=H^{r}(\Omega_{e})$ and $X^{*}=H^{-r}_{\overline{\Omega}_{e}}({\mathbb{R}}^{n})$ [18, p.463].

The fractional Laplacian admits two important properties which we need in our proofs. The first one is unique continuation property (UCP) which is used in proving the Runge approximation property.

Lemma 2.1 is proved in [12] for $s>1$ by reducing the problem to the UCP result for $s\in(0,1)$ in [18]. Note that such property is not true for local operators like the classical Laplacian $(-\Delta)$. The second property we need is the Poincaré inequality, which is used in showing that the forward problem for the perturbed fractional Schrödinger equation is well-posed.

Many different proofs for lemma 2.2 are given in [12]. We note that in the literature the fractional Poincaré inequality is typically considered only when $s\in(0,1)$.

Finally, we recall the fractional Leibniz rule, also known as the Kato-Ponce inequality. It is used to show the boundedness of the bilinear forms associated to the perturbed fractional Schrödinger equation in the case when the coefficients of the PDO have bounded fractional derivatives.

The proof of lemma 2.3 can be found in [20] (see also [19, 22]).

Following [35, Ch. 3], we introduce the space of multipliers $M(H^{r}\rightarrow H^{t})$ between pairs of Sobolev spaces. Here we are assuming that $r,t\in\mathbb{R}$. The coefficients of $P(x,D)$ in theorem 1.1 will be picked from such spaces of multipliers.

If $f\in\mathcal{D}^{\prime}({\mathbb{R}}^{n})$ is a distribution, we say that $f\in M(H^{r}\rightarrow H^{t})$ whenever the norm

is finite. Here $uv$ indicates the pointwise product of functions, while $\left\langle\cdot,\cdot\right\rangle$ is the duality pairing. If the distribution $f$ happens to be a function, the duality pairing can be defined as

By $M_{0}(H^{r}\rightarrow H^{t})$ we indicate the closure of $C^{\infty}_{c}(\mathbb{R}^{n})$ in $M(H^{r}\rightarrow H^{t})\subset\mathcal{D}^{\prime}(\mathbb{R}^{n})$. If $f\in M(H^{r}\rightarrow H^{t})$ and $u,v\in C_{c}^{\infty}(\mathbb{R}^{n})$ are both non-vanishing, we have the multiplier inequality

By the density of $C_{c}^{\infty}({\mathbb{R}}^{n})\times C_{c}^{\infty}({\mathbb{R}}^{n})$ in $H^{r}({\mathbb{R}}^{n})\times H^{-t}({\mathbb{R}}^{n})$ with respect to the product norm $\left\lVert(u,v)\right\rVert=\max\{\left\lVert u\right\rVert_{H^{r}({\mathbb{R}}^{n})},\left\lVert v\right\rVert_{H^{-t}({\mathbb{R}}^{n})}\}$ and estimate (22), there is a unique continuous extension of $(u,v)\mapsto\langle f,uv\rangle$ acting on $(u,v)\in H^{r}({\mathbb{R}}^{n})\times H^{-t}({\mathbb{R}}^{n})$. More precisely, each $f\in M(H^{r}\rightarrow H^{t})$ gives rise to a linear multiplication map $m_{f}:H^{r}({\mathbb{R}}^{n})\rightarrow H^{t}({\mathbb{R}}^{n})$ defined by

where $(u_{i},v_{i})\in C_{c}^{\infty}({\mathbb{R}}^{n})\times C_{c}^{\infty}({\mathbb{R}}^{n})$ is any Cauchy sequence in $H^{r}({\mathbb{R}}^{n})\times H^{-t}({\mathbb{R}}^{n})$ converging to $(u,v)$. The existence of the limit is granted by completeness and formula (22), which ensures that $\langle f,u_{i}v_{i}\rangle$ is also a Cauchy sequence. In fact, we have

where $\left\lVert u_{m}\right\rVert_{H^{r}({\mathbb{R}}^{n})}+\left\lVert v_{n}\right\rVert_{H^{-t}({\mathbb{R}}^{n})}$ is bounded by a constant independent of $m$ and $n$. The independence of the limit on the particular sequence $(u_{i},v_{i})$ can be showed by a similar estimate.

We can analogously define the unique adjoint multiplication map $m_{f}^{*}:H^{-t}({\mathbb{R}}^{n})\to H^{-r}({\mathbb{R}}^{n})$ such that

Since one sees that the adjoint of $m_{f}$ is $m_{f}^{*}$, the chosen notation is justified. For convenience, in the rest of the paper we will just write $fu$ for both $m_{f}(u)$ and $m_{f}^{*}(u)$.

In the next lemma we state some elementary properties of the spaces of multipliers. Other interesting properties may be found in [35].