Last but not least, we present a few results on reconstruction procedures and finite measurement statements for the fractional Calderón problem with drift. More precisely, we show that uniqueness for the fractional Calderón problem with \(C^\infty _c\) drift and potential can be guaranteed from \(n+1\) measurements only.
Higher order approximation
Before addressing the finite measurement results, we recall the higher order Runge approximation property and present the proof of Theorem 1.2 (b). The structure of the proof for the higher order Runge approximation is similar as the arguments presented in Sect. 3. However, since we seek to approximate solutions in high regularity function spaces, by using a duality argument, we need to consider the corresponding Dirichlet problem in Sobolev spaces of negative orders. The argument for this follows along the same lines as the proofs in [20, Section 7], which in the sequel we recall for self-containedness for the fractional Schrödinger equation with drift.
Let us consider the fractional Laplacian \((-\Delta )^s\) with \(s \in (\frac{1}{2},1)\). Let \(\Omega \subset \mathbb {R}^n\) be a bounded domain with a \(C^\infty \)-smooth boundary, \(b\in C^\infty _c(\Omega )^n\), \(c\in C^\infty _c(\Omega )\) with \({{\,\mathrm{supp}\,}}(b),{{\,\mathrm{supp}\,}}(c)\Subset \Omega \) satisfy (1.5). Here we impose the described compact support as well as the high regularity conditions for b and c in order to satisfy the assumptions from the theory which is presented in [24]. In the sequel, we consider the function space
$$\begin{aligned} \mathcal {E}^s(\overline{\Omega }):=e_\Omega d(x)^s C^\infty (\overline{\Omega }), \end{aligned}$$
where \(e_\Omega \) denotes extension by zero from \(\Omega \) to \(\mathbb {R}^n\), and \(d=d(x)\) is a \(C^\infty \) function in \(\overline{\Omega }\) with \(d(x)>0\) for \(x\in \Omega \) and \(d(x)={{\,\mathrm{dist}\,}}(x,\partial \Omega )\) near \(\partial \Omega \).
Further for \(\mu > s- \frac{1}{2}\) we work in the Banach space \(H^{s(\mu )}(\overline{\Omega })\) which is the space which is introduced in [24] as the Banach space tailored for solutions u solving the problem
$$\begin{aligned} r_\Omega ((-\Delta )^s + b\cdot \nabla +c)u \in H^{\mu -2s}(\Omega ) \text { with }u=0 \text { in }\Omega _e, \end{aligned}$$
where \(r_\Omega \) is the restriction map from \(\mathbb {R}^n\) to \(\Omega \) such that \(r_\Omega u=u|_\Omega \).
In order to deduce the desired higher order approximation property, we recall the following result from [24], which was also used in [20, Lemma 7.1] for deducing higher order approximation for the fractional Schrödinger equation.
Proposition 6.1
(Lemma 7.1 in [20]) For \(\mu >s-\frac{1}{2}\) and a smooth bounded domain \(\Omega \subset \mathbb {R}^n\), there exists a Banach space \(H^{s(\mu )}(\overline{\Omega })\) with the following properties:
- (a)
\(H^{s(\mu )}(\overline{\Omega })\subset H^{s-\frac{1}{2}}_{\overline{\Omega }}\) with a continuous inclusion;
- (b)
\(H^{s(\mu )}(\overline{\Omega })=H^{\mu }_{\overline{\Omega }}\) for \(\mu \in (s-\frac{1}{2},s+\frac{1}{2})\);
- (c)
\(r_\Omega ((-\Delta )^{s}+b\cdot \nabla +c)\) is a homeomorphism from \(H^{s(\mu )}(\overline{\Omega })\) onto \(H^{\mu -2s}(\Omega )\);
- (d)
\(H^{\mu }_{\overline{\Omega }}\subset H^{s(\mu )}(\overline{\Omega })\subset H^{\mu }_{loc}(\Omega )\) with continuous inclusions, or the multiplication by any smooth cut-off \(\chi \in C^\infty _c(\Omega )\) is bounded from \(H^{s(\mu )}(\overline{\Omega })\) to \(H^{\mu }(\Omega )\);
- (e)
\(\mathcal {E}^s (\overline{\Omega })=\cap _{\mu >s-\frac{1}{2}}H^{s(\mu )}(\overline{\Omega })\) and the set \(\mathcal {E}^s (\overline{\Omega })\) is dense in \(H^{s(\mu )}(\overline{\Omega })\).
For the proof of this result we refer to [20, 23, 24]. We remark that equipped with the topology induced by \(\{\Vert \cdot \Vert _{H^{s(k)}}\}_{k=1}^\infty \), the space \(\mathcal {E}^s (\overline{\Omega })\) is a Fréchet space.
Building on these properties of the spaces \(H^{s(\mu )}(\overline{\Omega })\), following the argument of [20], we prove a higher order approximation property for solutions to (1.4) in \(\mathcal {E}^s (\overline{\Omega })\). The following result was proved by [20, Lemma 7.2 ] for the case \(b=0\).
Lemma 6.2
Let \(\Omega \subset \mathbb {R}^n\) be a bounded domain with a \(C^\infty \)-smooth boundary, and \(\frac{1}{2}<s <1\). Let \(W\subset \Omega _e\) be a non-empty, open set, and let \(b\in C^\infty _c(\Omega )^n\), \(c\in C^\infty _c(\Omega )\) with \({{\,\mathrm{supp}\,}}(b),{{\,\mathrm{supp}\,}}(c)\Subset \Omega \) be such that (1.5) holds. Let \(P_{b,c}\) be the Poisson operator given by (2.20) and
$$\begin{aligned} \mathcal {D}:=\{e_\Omega (r_\Omega P_{b,c}f): f\in C^\infty _c(W)\}. \end{aligned}$$
Then the set \(\mathcal {D}\) is dense in the Fréchet space \(\mathcal {E}^s (\overline{\Omega })\) with the topology induced by \(\{\Vert \cdot \Vert _{H^{s(k)}}\}_{k=1}^\infty \).
Proof
We follow the argument of [20, Lemma 7.2]. First, notice that for any \(f\in C^\infty _c(W)\), by the definition of the Poisson operator (2.20), one has \(P_{b,c}f=f+v\), where \(v \in \widetilde{H}^s(\Omega )\) satisfies
$$\begin{aligned} r_\Omega ((-\Delta )^s+b\cdot \nabla +c)v\in C^\infty (\overline{\Omega }). \end{aligned}$$
By Proposition 6.1, we have \(v\in \mathcal {E}^s(\overline{\Omega })\), which implies that \(\mathcal {D}\subset \mathcal {E}^s (\overline{\Omega })\). Next, let \(\mathcal {L}\) be a continuous linear functional defined on \(\mathcal {E}^s (\overline{\Omega })\) satisfying
$$\begin{aligned} \mathcal {L}(e_\Omega (r_\Omega P_{b,c}f))=0, \text { for all }f\in C^\infty _c(W). \end{aligned}$$
By the Hahn-Banach theorem, it suffices to show that \(\mathcal {L}\equiv 0\). By using the definition of the topology of the Fréchet space \(\mathcal {E}^s (\overline{\Omega })\), one can find an integer \(\ell \) so that
$$\begin{aligned} |\mathcal {L}(u)|\le C\sum _{m=1}^{\ell } \Vert u\Vert _{H^{s(m)}(\overline{\Omega })}\le C\Vert u\Vert _{H^{s(\ell )}(\overline{\Omega })}, \text { for }u\in \mathcal {E}^{s}(\overline{\Omega }), \end{aligned}$$
for some constant \(C>0\) which is independent of u. By virtue of Proposition 6.1 (e), \(\mathcal {E}^{s}(\overline{\Omega })\) is dense in \(H^{s(\ell )}(\overline{\Omega })\). Thus, \(\mathcal {L}\) has a unique bounded extension \(\widetilde{\mathcal {L}}\in (H^{s(\ell )}(\overline{\Omega }))^{*}\).
Let us consider the same homeomorphism in Proposition 6.1 (c),
$$\begin{aligned} \mathcal {T}=r_\Omega ((-\Delta )^s+b\cdot \nabla +c):H^{s(\ell )}(\overline{\Omega })\rightarrow H^{\ell -2s}(\Omega ). \end{aligned}$$
The adjoint of \(\mathcal T\) is a bounded map between the dual Banach spaces with
$$\begin{aligned} \mathcal {T}^{*}:(H^{\ell -2s}(\Omega ))^{*}\rightarrow (H^{s(\ell )}(\overline{\Omega }))^*. \end{aligned}$$
Note that the adjoint map \(\mathcal {T}^*\) is also homeomorphism with the inverse \((\mathcal {T}^{-1})^{*}\). Moreover, by Remark 2.1, we have \((H^{\ell -2s}(\Omega ))^*=H^{-\ell +2s}_{\overline{\Omega }}\) such that
$$\begin{aligned} \mathcal {T}^{*}v(w)=(v,\mathcal {T}w)_{H^{-\ell +2s}_{\overline{\Omega }} \times H^{\ell -2s}(\Omega )}, \text { for }v\in H^{-\ell +2s}_{\overline{\Omega }} \text { and }w\in H^{s(\ell )}(\overline{\Omega }). \end{aligned}$$
Let \(v\in H^{-\ell +2s}_{\overline{\Omega }}\) be the unique function satisfying \(\mathcal {T}^{*}v=\widetilde{\mathcal {L}}\) and choose a sequence \(\{v_k\}_{k\in \mathbb {N}}\subset C^\infty _c(\Omega )\) with \(v_k \rightarrow v\) in \(H^{-\ell +2s}(\Omega )\) as \(k\rightarrow \infty \). Now, let \(f\in C^\infty _c(W)\), recalling that \(e_\Omega (r_\Omega P_{b,c}f)=P_{b,c}f-f\), then we have
$$\begin{aligned} 0&=\mathcal {L}(e_\Omega (r_\Omega P_{b,c}f))=\widetilde{\mathcal {L}}(P_{b,c}f-f)=\mathcal {T}^*v(P_{b,c}f-f)\nonumber \\&= (v,\mathcal {T}(P_{b,c}f-f)) = -(v,\mathcal {T}f)=\lim _{k\rightarrow \infty }(v_k,((-\Delta )^s +b\cdot \nabla +c)f)\nonumber \\&=-\lim _{k\rightarrow \infty }((-\Delta )^s v_k -\nabla \cdot (bv_k)+cv_k, f), \end{aligned}$$
(6.1)
where we have utilized that \(\mathcal {T}P_{b,c}f=0\) and \(v_k\in C^\infty _c(\Omega )\). Finally, since \(f\in C^\infty _c(W)\) with \(\overline{W}\cap \overline{\Omega }=\emptyset \), then the last equation of (6.1) reads
$$\begin{aligned} ((-\Delta )^s v,f)=\lim _{k\rightarrow \infty }((-\Delta )^s v_k,f)=0, \text { for }f\in C^\infty _c(W). \end{aligned}$$
Thus, we obtain that \(v\in H^{-\ell +2s}(\mathbb {R}^n)\) satisfies
$$\begin{aligned} v|_W=(-\Delta )^s v|_W=0, \end{aligned}$$
and the strong uniqueness (Proposition 3.1) implies that \(v\equiv 0 \) in \(\mathbb {R}^n\). Therefore, we obtain \(\widetilde{\mathcal {L}}=0\) and hence \(\mathcal {L}=0\), which completes the proof. \(\square \)
Now, we are ready to prove the higher regularity Runge approximation property.
Proof of Theorem1.2(b) As \(\Omega \Subset \Omega _1\) with \(\mathrm {int}(\Omega _1 {\setminus } \overline{\Omega })\ne \emptyset \), it is possible to find a small ball W with \(\overline{W}\subset \Omega _1{\setminus } \overline{\Omega }\). Let \(g\in C^\infty (\overline{\Omega })\) and \(h:=e_{\Omega } d(x)^{s} g \in \mathcal {E}^{s}(\overline{\Omega })\), then Lemma 6.2 shows that one can find a sequence of solutions \(\{u_j\} \subset H^s(\mathbb {R}^n)\) satisfying
$$\begin{aligned} ((-\Delta )^s+b\cdot \nabla +c)u_j=0 \text { in }\Omega \text { with }{{\,\mathrm{supp}\,}}(u_j)\subset \overline{\Omega _1}, \end{aligned}$$
so that \(e_\Omega r_\Omega u_j \in \mathcal {E}^s(\overline{\Omega })\) and
$$\begin{aligned} e_\Omega r_\Omega u_j \rightarrow h \text { in }\mathcal {E}^{s}(\overline{\Omega }) \text { as }j\rightarrow \infty . \end{aligned}$$
The higher order approximation will hold if we can show that
$$\begin{aligned} \mathcal {M}:C^\infty (\Omega ) \rightarrow \mathcal {E}^{s}(\overline{\Omega }) \text { with }\mathcal {M}g=e_\Omega d(x)^s g \end{aligned}$$
is a homeomorphism, as it is then possible to apply \(\mathcal {M}^{-1}=d(x)^{-s}r_\Omega \). This then gives
$$\begin{aligned} d(x)^{-s}r_\Omega u_j \rightarrow g \text { in }C^\infty (\overline{\Omega }). \end{aligned}$$
Note that the map \(\mathcal {M}\) is a bijective linear map between Fréchet spaces and has a closed graph, i.e., if \(g_j \rightarrow g\) in \(C^\infty \) and \(\mathcal {M}g_j\rightarrow h\) in \(\mathcal {E}^s\), then also \(\mathcal {M}g_j \rightarrow \mathcal {M}g\) in \(L^\infty \). Then by the uniqueness of the limit, one obtains that \(\mathcal {M}g=h\) as distributional limits. Hence, \(\mathcal {M}\) is a homeomorphism by the closed graph and the open mapping theorems. This finishes the proof. \(\square \)
Finite measurements reconstruction without openness
In this section, we discuss a first result towards the proof of Theorem 1.4 by using higher order Runge approximation [Theorem 1.2 (b)]. However, before proving the full result of Theorem 1.4, we prove a weaker (but technically considerably easier) result, which still proves finite measurement reconstruction but only asserts that the set of measurement data contains a non-empty open set (a priori this argument does not prove the density of the set of good data). The technically more involved statement on the openness and density of the set of good data will be proved in the subsequent sections.
Proof of Theorem1.4without the density result We show that for any drift \(b \in C^{\infty }_c(\Omega )^n\) and any potential \(c\in C^{\infty }_c(\Omega )\) with \({{\,\mathrm{supp}\,}}(b),{{\,\mathrm{supp}\,}}(c)\Subset \Omega \), there exist exterior Dirichlet data \(f_1,\dots ,f_{n+1}\) such that the b and c can be uniquely reconstructed from the knowledge of \(f_1,\dots ,f_{n+1}\) and \(\Lambda _{b,c}(f_1),\dots , \Lambda _{b,c}(f_{n+1})\).
By Runge approximation in \(C^{k}\) spaces [see Theorem 1.2 (b)], we have that for any \(g\in C^{\infty }(\overline{\Omega })\) there exists a sequence of solutions \(\{u_j\}_{j \in \mathbb {N}}\) to the fractional Schrödinger equation (1.4) with drift (and compactly supported coefficients) such that for any \(k \in \mathbb {N}\)
$$\begin{aligned} \Vert g - d^{-s} u_j \Vert _{C^k(\Omega )} \rightarrow 0 \text { as }j\rightarrow \infty , \end{aligned}$$
where \(d(x)= {{\,\mathrm{dist}\,}}(x,\partial \Omega )\) if \(x\in \Omega \) is sufficiently close to the boundary of \(\Omega \) and d(x) is extended to a positive function smoothly into the interior of \(\Omega \). Next, we choose \(n+1\) smooth functions \(g_1,\dots ,g_{n+1}\) defined in \(\Omega \) with the property that
$$\begin{aligned} h(g_1,g_2,\ldots ,g_{n+1})(x) := \det \begin{pmatrix} \partial _1 g_1 &{} \ldots &{} \partial _n g_1 &{} g_1 \\ \vdots &{} \vdots &{} \vdots &{} \vdots \\ \partial _1 g_{n} &{} \ldots &{} \partial _n g_{n} &{} g_{n}\\ \partial _1 g_{n+1} &{} \ldots &{} \partial _n g_{n+1} &{} g_{n+1} \end{pmatrix} (x)\ne 0. \end{aligned}$$
(6.2)
An example for this would be the functions \(g_j= x_j\) for \(j\in \{1,\dots ,n\}\) and \(g_{n+1}=1\). We further set \(\widetilde{g}_l= d^{-s} (\chi g_l)\), where \(\chi \in C_c^\infty (\Omega )\) and \(\chi = 1\) on K, for \(l\in \{1,\dots ,n+1\}\) and apply Theorem 1.2 (b). As a consequence, for each \(l\in \{1,\dots ,n+1\}\), and in any compact subset \(K \Subset \Omega \) there exists a sequence of solutions \(\{u_{j,K}^l\}_{j\in \mathbb {N}}\) such that for any \(k\in \mathbb {N}\)
$$\begin{aligned} \Vert d^{-s}( g_l-u_{j,K}^l)\Vert _{C^k(K)} \rightarrow 0 \text{ as } j \rightarrow \infty . \end{aligned}$$
As \(K \subset \Omega \) and as \(d(x)>0\) in K, we then also have
$$\begin{aligned} \Vert g_l-u_{j,K}^l\Vert _{C^k(K)} \rightarrow 0 \text{ as } j \rightarrow \infty . \end{aligned}$$
Hence, choosing \(j\ge j_0\) large enough and K such that \({{\,\mathrm{supp}\,}}(b)\cup {{\,\mathrm{supp}\,}}(c)\Subset K\), we obtain that
$$\begin{aligned} h( u_{j,K}^1, \ldots , u_{j,K}^{n+1})(x)= \det \begin{pmatrix} \partial _1 u_{j,K}^1 &{} \ldots &{} \partial _n u_{j,K}^1 &{} u_{j,K}^1 \\ \vdots &{} \vdots &{} \vdots &{} \vdots \\ \partial _1 u_{j,K}^n &{} \ldots &{} \partial _n u_{j,K}^n &{} u_{j,K}^n\\ \partial _1 u_{j,K}^{n+1} &{} \ldots &{} \partial _n u_{j,K}^{n+1} &{} u_{j,K}^{n+1} \end{pmatrix} (x)\ne 0. \end{aligned}$$
(6.3)
As a consequence, for these values of \(u_{j,K}^l\) the (linear) system for \(b_1,\dots ,b_n\) and c
$$\begin{aligned} \begin{pmatrix} \partial _1 u_{j,K}^1 &{} \ldots &{} \partial _n u_{j,K}^1 &{} u_{j,K}^1 \\ \vdots &{} \vdots &{} \vdots &{} \vdots \\ \partial _1 u_{j,K}^n &{} \ldots &{} \partial _n u_{j,K}^n &{} u_{j,K}^n\\ \partial _1 u_{j,K}^{n+1} &{} \ldots &{} \partial _n u_{j,K}^{n+1} &{} u_{j,K}^{n+1} \end{pmatrix} \begin{pmatrix} b_1 \\ \vdots \\ b_n\\ c \end{pmatrix} = -\begin{pmatrix} (-\Delta )^s u_{j,K}^1\\ \vdots \\ (-\Delta )^s u_{j,K}^n \\ (-\Delta )^s u_{j,K}^{n+1} \end{pmatrix} \end{aligned}$$
(6.4)
is solvable (since the \((n+1)\times (n+1)\) matrix in the left hand side of (6.4) is invertible). Thus, from the knowledge of \(u_{j,K}^l\), \(l\in \{1,\dots ,n+1\}\) for some \(j\ge j_0\), it is possible to uniquely recover the drift and the potential simultaneously.
As by the global (nonlocal) unique continuation arguments in [19] (c.f. also Proposition 3.1 from above) it is possible to recover \(u_{j,K}^{l}\) given the measurements \(f_{j,K}^l\) and \(\Lambda _{b,c}(f_{j,K}^l)\) we infer the finite measurement recovery statement.
Finally, in order to infer the openness of the set of possible exterior data, we note that for any \(\epsilon >0\) there exists \(\delta >0\) such that for any \(f=(f^1,\dots , f^{n+1})\in C_c^{\infty }(W)^{n+1}\) with
$$\begin{aligned} \Vert f-f_{j,K}\Vert _{C^{k}_{c}(W)} < \delta , \ k \in \mathbb {N}, \end{aligned}$$
we have by boundedness of the mapping \(C_c^k(W)^{n+1} \ni f \mapsto u \in C^{k}(K)\)
$$\begin{aligned} \Vert u-u_{j,K}\Vert _{C^k(K)} < \epsilon . \end{aligned}$$
Here \(f_{j,K}:=(f_{j,K}^1,\dots ,f_{j,K}^{n+1})\in C_c^{\infty }(W)^{n+1}\) are the exterior data from above, \(u=(u^1,\dots ,u^{n+1})\) are the solutions to (1.4) corresponding to the data f and \(u_{j,K}:=(u_{j,K}^1,\dots ,u_{j,K}^{n+1})\) are the solutions to (1.4) corresponding to the data \(f_{j,K}=(f_{j,K}^1,\dots ,f_{j,K}^{n+1})\). In particular, assuming that
$$\begin{aligned} \det \begin{pmatrix} \partial _1 u_{j,K}^1 &{} \ldots &{} \partial _n u_{j,K}^1 &{} u_{j,K}^1 \\ \vdots &{} \vdots &{} \vdots &{} \vdots \\ \partial _1 u_{j,K}^n &{} \ldots &{} \partial _n u_{j,K}^n &{} u_{j,K}^n\\ \partial _1 u_{j,K}^{n+1} &{} \ldots &{} \partial _n u_{j,K}^{n+1} &{} u_{j,K}^{n+1} \end{pmatrix} \ge c>0 \text{ in } K, \end{aligned}$$
the triangle inequality implies that if \(\epsilon >0\) (and correspondingly \(\delta \)) is chosen sufficiently small, also
$$\begin{aligned} \det \begin{pmatrix} \partial _1 u^1 &{} \ldots &{} \partial _n u^1 &{} u^1 \\ \vdots &{} \vdots &{} \vdots &{} \vdots \\ \partial _1 u^n &{} \ldots &{} \partial _n u^n &{} u^n\\ \partial _1 u^{n+1} &{} \ldots &{} \partial _n u^{n+1} &{} u^{n+1} \end{pmatrix} >0 \text{ in } K. \end{aligned}$$
This concludes the argument. \(\square \)
Remark 6.3
We conclude this section by some comments on the assumptions of the theorem:
- (a)
The compact support condition for the functions \(b_j\), \(c_j\), \(j=1,2\), is assumed here in order to be able to apply the theory of Grubb [24].
- (b)
In order to obtain our result we in principle do not need the full strength of the \(C^k\), \(k\in \mathbb {N}\), approximation result from [20]. It would for instance be sufficient to use a \(C^1\) approximation result for which only lower regularity on the coefficients is needed. Since the theory of Grubb is however formulated in the smooth set-up, we do not optimize the regularity dependences here.
- (c)
We again emphasize that although the variant of Theorem 1.4 which is proved in this section is interesting from a theoretical point of view, a word of caution is needed as follows: In contrast to the single measurement result from [19], the exterior data \(f_1,\dots ,f_{n+1}\) are not arbitrary. In general the specific choice of these data depends on \(b_1, b_2\), \(c_1,c_2\) and hence the explicit choice of the functions \(f_1,\dots , f_{n+1}\) is not known in general. This has a similar background (see the following proof) as many results on hybrid inverse problems, where it is important that constraints are satisfied, see [1, 2]. The result from this section will be improved considerably in the argument leading to the proof of Theorem 1.4.
- (d)
In addition to the previous point, there are examples of matrices with entries satisfying elliptic equations, for which the determinant vanishes on an open set, see [10]. This indicates that the zero set of the determinant (6.3) can indeed be large.
Variations and extensions of the reconstruction result
We discuss a slight variation of the reconstruction result from Sect. 6.2 by relaxing the condition that the fields b, c are compactly supported in \(\Omega \). Recall that a \(C^\infty \)-smooth function f is vanishing to infinite order at a point \(x_0\) provided that \(\partial ^\alpha f(x_0)=0\) holds for any multi-index \(\alpha =(\alpha _1,\ldots ,\alpha _n) \in (\mathbb {N}\cup \{0\})^n\).
Proposition 6.4
Let \(\Omega \subset \mathbb {R}^n\) be a bounded domain with a \(C^\infty \)-smooth boundary. Let \(W \subset \Omega _e\) be a non-empty open, smooth domain containing an open neighbourhood of \(\partial \Omega \). Let \(\frac{1}{2}<s<1\) and assume that \(b_j\in C^{\infty }(\overline{\Omega })^n\), \(c_j\in C^{\infty }(\overline{\Omega })\) satisfy (1.5) and
$$\begin{aligned} b_1-b_2, \ c_1-c_2 \text{ vanish } \text{ to } \text{ infinite } \text{ order } \text{ on } \partial \Omega . \end{aligned}$$
Then, there exist \(n+1\) exterior Dirichlet data \(f_1,\dots ,f_{n+1}\in C^{\infty }_c(W)\) such that if
$$\begin{aligned} \Lambda _{b_1,c_1}(f_l) = \Lambda _{b_2,c_2}(f_l) \text{ for } l \in \{1,\dots ,n+1\}, \end{aligned}$$
then \(b_1 = b_2\) and \(c_1=c_2\) in \(\Omega \). Moreover, for any \(k\in \mathbb {N}\) the set of exterior data \(f_1,\dots ,f_{n+1}\), which satisfies this property forms an open subset in \(C^{k}_{ c}(W)\).
This follows from an auxiliary result, which states that under geometric restrictions on the set where we measure the Dirichlet data, we may enlarge our domain and that the DN map on the larger domain is determined by the DN map on the smaller set. This is well-known in the study of local inverse problems – see e.g. [43, Lemma 4.2].
In what follows, we denote by \(r_X \widetilde{H}^s(Y)\) the set of all restrictions \(f|_{X}\) of functions \(f \in \widetilde{H}^s(Y)\) for open \(X, Y \subset \mathbb {R}^n\).
Lemma 6.5
Assume that \(\Omega \) and \(\Omega '\) with \(\Omega \subset \Omega '\) are bounded domains with Lipschitz boundaries and \(W_1, W_2 \subset \mathbb {R}^n\) are two non-empty open sets, such that \(\Omega '{\setminus } \Omega \Subset W_1\) (so, in particular, \(W_1 \cap \Omega '_e\) is non-empty). Then let \(b_1, b_2 \in W^{1-s, \infty }(\Omega ')\) and \(c_1, c_2 \in L^\infty (\Omega ')\), and \(\frac{1}{2}< s < 1\), and assume \(c_1 = c_2\) and \(b_1 = b_2\) in \(\Omega ' {\setminus } \Omega \).
Additionally, assume that the coefficients satisfy the eigenvalue condition (1.5) both on \(\Omega '\) and on \(\Omega \). Assume the equality of DN maps \(\Lambda _{b_j,c_j}\) with respect to \(\Omega \)
$$\begin{aligned} \Lambda _{b_1, c_1}f|_{W_2} = \Lambda _{b_2, c_2}f|_{W_2} \text { for all } f \in r_{W_1 \cap \Omega _e}\widetilde{H}^s(W_1), \end{aligned}$$
Then we have the equality of DN maps \(\Lambda '_{b_j,c_j}\) with respect to \(\Omega '\):
$$\begin{aligned} \Lambda '_{b_1, c_1}f|_{W_2} = \Lambda '_{b_2, c_2}f|_{W_2} \text { for all }f \in r_{W_1 \cap \Omega '_e}\widetilde{H}^s(W_1). \end{aligned}$$
Proof
Assume \(u_1' \in H^s(\mathbb {R}^n)\) solves the Dirichlet problem for \(f \in \widetilde{H}^s(W_1 \cap \Omega '_e)\)
$$\begin{aligned} ((-\Delta )^s + b_1 \cdot \nabla + c_1)u_1'&= 0 \, \text { in } \, \Omega ',\\ u_1'|_{\Omega '_e}&= f|_{\Omega '_e}. \end{aligned}$$
Then solve the analogous Dirichlet problem with respect to the smaller domain \(\Omega \):
$$\begin{aligned} ((-\Delta )^s + b_2 \cdot \nabla + c_2)u_2&= 0 \, \text { in } \, \Omega ,\\ u_2|_{\Omega _e}&= u_1'|_{\Omega _e}. \end{aligned}$$
By using the computation from Remark 2.12 for \(\phi \in C_c^\infty (\Omega _e \cap W_2)\) (we need the support condition here) and the hypothesis of equality of the DN maps, we get:
$$\begin{aligned}&\Lambda _{b_1, c_1}f|_{W_2 \cap \Omega _e} = \Lambda _{b_2, c_2}f|_{W_2 \cap \Omega _e}, \\&\quad \implies B_1(u_1', \phi ) = B_2(u_2, \phi ),\\&\quad \implies \int _{\mathbb {R}^n} \phi (-\Delta )^s u_1' dx= \int _{\mathbb {R}^n} \phi (-\Delta )^s u_2 dx, \\&\quad \implies (-\Delta )^s u_1' \equiv (-\Delta )^s u_2 \text { on } W_2 \cap \Omega _e. \end{aligned}$$
Here, for convenience, we have used the abbreviation \(B_{j}(\cdot , \cdot ) = B_{b_{j},c_j}(\cdot , \cdot )\) for \(j\in \{1,2\}\). Furthermore, note that here we also use that \(u_1'|_{\Omega _e} \in r_{\Omega _e \cap W_1} \widetilde{H}^s(W_1)\), since \(\Omega ' {\setminus } \Omega \Subset W_1\). Therefore, we have the following relations:
$$\begin{aligned} (-\Delta )^s(u_1' - u_2) = 0 \text { on } W_2 \cap \Omega _e \quad \text {and} \quad u_1' - u_2 = 0 \text { on } W_2 \cap \Omega _e . \end{aligned}$$
(6.5)
By the strong uniqueness Proposition from 3.1, we conclude \(u_1' \equiv u_2\) on the whole of \(\mathbb {R}^n\).
Now we proceed to compare the DN maps on the domain \(\Omega '\). We have the following chain of equalities for \(w \in C_c^\infty (W_2 \cup \Omega )\) and \(u_1', u_2\) as above:
$$\begin{aligned} B_2'(u_2, w)&= \int _{\mathbb {R}^n} (-\Delta )^{\frac{s}{2}} u_2 \cdot (-\Delta )^{\frac{s}{2}} w dx + \int _{\Omega '} b_2 \cdot \nabla u_2 w dx + \int _{\Omega '} c_2 u_2 w dx\\&= B_2(u_2, w) + \int _{\Omega ' {\setminus } \Omega } b_2 \cdot \nabla u_2 w dx + \int _{\Omega ' {\setminus } \Omega } c_2 u_2 w\\&= B_1(u_1', w) + \int _{\Omega ' {\setminus } \Omega } b_1 \cdot \nabla u_1' w dx + \int _{\Omega ' {\setminus } \Omega } c_1 u_1' w\\&= B_1'(u_1', w). \end{aligned}$$
Here we split the integrals over \(\Omega \) and \(\Omega ' {\setminus } \Omega \), used the notation \(B'_1, B_2'\) for the bilinear form associated to the equation on \(\Omega '\), the definition of the DN map (see Definition 2.11), the fact that \(u_2 = u_1'\), and \(b_1 = b_2\) and \(c_1 = c_2\) on \(\Omega ' {\setminus } \Omega \).
We conclude that \(u_2\) solves \((-\Delta )^{s} u_2 + b_2 \cdot \nabla u_2 + c_2 u_2 = 0\) in \(\Omega '\), with \(u_2|_{\Omega '_e} = f|_{\Omega '_e}\). Also, we conclude that for all w as above
$$\begin{aligned}(\Lambda '_{b_2, c_2}f, w) = B_2'(u_2, w) = B_1'(u_1', w) = (\Lambda '_{b_1, c_1} f, w).\end{aligned}$$
Here we use \(\Lambda '\) notation for the DN map on \(\Omega '\). The main claim follows by observing that we may pick arbitrary \(w \in C_c^\infty (W_2)\). Note that we need to assume that the operators satisfy condition (1.5) on the bigger set \(\Omega '\) to assume well-definedness of the DN maps. \(\square \)
Sketch of the argument for Proposition6.4 The proof of Proposition 6.4 is similar to Section 6.2, so we only sketch the arguments here. With the auxiliary result of Lemma 6.5 at hand, we deduce the claim of Proposition 6.4 by extending \(\Omega \) to \(\Omega '\) suitably such that the geometric conditions on the domains are satisfied. As our operator is not self-adjoint, in the extended domain \(\Omega '\) we might possibly work with Cauchy data, as we could catch a finite number of Dirichlet eigenvalues (c.f. Remark 6.6 below on how to avoid this in some cases). However, using arguments as in [41], it is possible to deduce analogous results. \(\square \)
Remark 6.6
(Domain monotonicity) We remark that when the drift term \(b=0\), then it is possible to avoid dealing with Cauchy data by using perturbation of domain arguments. Indeed, for the fractional Laplacian (and self-adjoint fractional Schrödinger operators), it is possible to characterize the Dirichlet spectrum through min-max formulations [17] (see also [11] for similar settings for the classical Laplacian). Relying on the weak unique continuation property, it is possible to show the monotonicity of eigenvalues
$$\begin{aligned} \lambda _k(\Omega _1) > \lambda _k(\Omega _2)\text { for }\Omega _1 \subsetneq \Omega _2\text { and }k\in \mathbb {N}, \end{aligned}$$
where \(\lambda _k\) is the k-th Dirichlet eigenvalue of the fractional Laplacian. Thus, perturbing the domain suitably and only considering a finite number of eigenvalues in the case of self-adjoint fractional Schrödinger operators, one might work with the DN map instead of having to resort to Cauchy data.
Proof of Theorem 1.4 and generic properties of determinants via singularity theory
In this section, we prove the full result of Theorem 1.4, i.e. we show that the set of exterior data from which we can choose \(n + 1\) measurements in order to recover the coefficients on a compact set K as in previous sections is open and dense. This significantly improves the result from Sect. 6.2 in that the data still depend on the unknown potentials b, c, but in a precise sense they form a large set, i.e. given (random) exterior data, we know that an arbitrarily small perturbation of them will render them admissible in our reconstruction scheme.
The main idea of our argument is to relax the condition that for admissible exterior data \(f_1,\dots ,f_{n+1}\in C_c^{\infty }(W)\) we require
$$\begin{aligned} h(P_{b,c}(f_1),\dots ,P_{b,c}(f_{n+1})) \ne 0 \text{ in } K \subset \Omega , \end{aligned}$$
where h was the function from (6.2). Instead, we consider data \(f_1,\dots ,f_{n+1}\in C_c^{\infty }(W)\) such that \(h(P_{b,c}(f_1),\dots ,P_{b,c}(f_{n+1}))\) is only allowed to vanish to a finite (dimension-dependent) order. Then, by known results from [3, Lemma 3], it follows that the set
$$\begin{aligned} \left\{ x\in K; \ h(P_{b,c}(f_1),\dots ,P_{b,c}(f_{n+1}))(x)=0\right\} \end{aligned}$$
is of measure zero in \(K \Subset \Omega \).Footnote 1 As a consequence, due to the continuity of b, c, it is then possible to reconstruct both coefficients (see Lemma 6.8). Simultaneously, the set of exterior data \(f_1,\dots ,f_{n+1}\in C_c^{\infty }(W)\) for which \(h(P_{b,c}(f_1),\dots ,P_{b,c}(f_{n+1}))\) vanishes only of finite (dimension-dependent) order immediately by definition is open in \(C_c^{\infty }(W)^{n+1}\). The density of such data will be obtained via small perturbations, relying on ideas of Whitney’s work [53], which had been developed in the context of singularity theory. Technically, this is the most involved part of our arguments.
Let us introduce the set of our admissible exterior conditions.
Definition 6.7
Let b, c satisfy the conditions in Theorem 1.4. Let \(n\in \mathbb {N}\) and \(k(n)= \Big \lceil {\sqrt{n + 1}}\Big \rceil \in \mathbb {N}\) (i.e. k(n) is the smallest positive integer greater or equal to \(\sqrt{n + 1}\)). Let \(\Omega \subset \mathbb {R}^n\) be as in Theorem 1.4 and \(K\Subset \Omega \) be a compact set as in Sect. 6.2. Then, we define the set \(\mathcal {F} \subset C_c^{\infty }(W)^{n + 1}\) to be the set of exterior data, such that \((f_1, \cdots , f_{n + 1}) \in {{\,\mathrm{\mathcal {F}}\,}}\), if \(h(P_{b,c} f_1, \cdots , P_{b, c} f_{n + 1})\) has at most order of vanishing \(k(n)-2\) at each point of \(K \Subset \Omega \).
We claim that the set \({{\,\mathrm{\mathcal {F}}\,}}\) yields the desired set of exterior data for the proof of Theorem 1.4. To this end, we first show that given exterior data \((f_1, \cdots , f_{n + 1})\in {{\,\mathrm{\mathcal {F}}\,}}\), it is possible to recover b and c:
Lemma 6.8
Assume that the conditions of Theorem 1.4 hold. Let \((f_1, \cdots , f_{n + 1})\in {{\,\mathrm{\mathcal {F}}\,}}\). Then it is possible reconstruct b, c from the exterior measurements of \(f_l\) and \(\Lambda _{b,c}(f_l)\), for \(l \in \{1,\cdots ,n+1\}\).
Proof
We first recall that the zero set of a smooth function not vanishing to infinite order is contained in a countable union of codimension one submanifolds (see e.g. [3, Lemma 3]). In particular, by definition of the set \({{\,\mathrm{\mathcal {F}}\,}}\), we thus infer that \(h(P_{b,c}(f_1),\dots ,P_{b,c}(f_{n+1}))\) vanishes only on a set of Lebesgue measure zero. Let us denote this measure zero set by \(B \subset K\). Therefore, the argument in (6.4) goes through on the set \(K {\setminus } B\). But b, c satisfy \({{\,\mathrm{supp}\,}}(b)\cup {{\,\mathrm{supp}\,}}(c)\Subset K\) and are smooth, so have unique continuous extensions to K, which can be determined from \(b|_{K{\setminus } B}\) and \(c|_{K {\setminus } B}\). This concludes the argument for the reconstruction of b, c from \(f_l\) and \(\Lambda _{b,c}(f_l)\) for \(l\in \{1,\cdots ,n+1\}\). \(\square \)
Hence, it remains to prove the openness and density of the set \({{\,\mathrm{\mathcal {F}}\,}}\subset C^\infty _0(W)^{n+1}\). While the openness is a direct consequence of the definition of the set \({{\,\mathrm{\mathcal {F}}\,}}\), the density of the set \({{\,\mathrm{\mathcal {F}}\,}}\) requires careful arguments. This will be the content of the remaining subsections.
Generic properties of determinants via singularity theory
In order to deduce the density of the set \({{\,\mathrm{\mathcal {F}}\,}}\), we seek to argue by perturbation: The main idea is that for an m-tuple of functions \(f = (f_1, \cdots , f_m)\) on \(\mathbb {R}^n\), and some differential relation \(P(x,D)(f)(x)=0\) on \(\mathbb {R}^n\), we may generate a parametric family \(f_{\alpha }\), in such a way that on a compact set, near any \(\alpha _0\) there is an \(\alpha \) arbitrarily close, such that \(P(x,D)(f_\alpha ) \ne 0\) on K. Here \(\alpha \in \mathbb {R}^N\) for some (large) \(N \in \mathbb {N}\) and \(f_\alpha \) is given by adding a polynomial of degree r (related to N) to each of the entries \(f_i\) of f, with coefficients given by reading off indices of \(\alpha \) in a suitable order.
A famous example, due to Morse, of this fact is just a \(C^2\) function \(f: \mathbb {R}^n \rightarrow \mathbb {R}\). Then by looking at \(f_\alpha (x) = f(x) + \langle {\alpha , x}\rangle \), where \(\langle {\cdot , \cdot }\rangle \) is the inner product and \(\alpha \in \mathbb {R}^n\), by Sard’s theorem the set of \(\alpha \) for which \(f_\alpha \) has a degenerate critical point is of measure zero. These ideas were generalised by Whitney [53] and others in the area of singularity theory to study generic maps \(\mathbb {R}^n \rightarrow \mathbb {R}^m\).
In our case, the idea is that by adding generic polynomials of degree \(k(n)\in \mathbb {N}\) with small coefficients to \(P_{b, c} f_1, \cdots , P_{b, c} f_{n + 1}\), we may obtain perturbations such that the determinant function h from (6.2) only vanishes of order at most \(k(n)-2\). Here \(k(n)\in \mathbb {N}\) is the constant from Definition 6.7. Since the perturbation is just by polynomials of order k(n), i.e. by a linear combination of one of
$$\begin{aligned} N_0 = \sum \limits _{j=0}^{k(n)} {n + j - 1 \atopwithdelims ()j} = {n +k(n) \atopwithdelims ()k(n)} \end{aligned}$$
(6.6)
linearly independent polynomials of the form \(1, x_i, x_i x_j, ...\), by Runge approximation we may approximate these by \(P_{b, c} (f_{i,m})\) for \(i\in \{1 , \cdots , N_0\}\) arbitrarily close, for some suitable exterior data \(\left\{ f_{i,m}\right\} \)’s. By adding a linear combination of \(f_{i,m}\) with coefficients \(\alpha \) to the exterior data, one can obtain an arbitrary close measurement for which the zero set of h is “good”, i.e. is just given by a stratification of smooth hypersurfaces (see Propositions 6.10 and 6.14 below). In the sequel, we present the details of this argument.
Preliminaries
We need to import some (old) technology from [53, Parts A and B], which allows us to modify functions in a favorable way by simply applying dimension-counting arguments. We consider a mapping \(f = (f_1, \cdots , f_m): \mathbb {R}^n \rightarrow \mathbb {R}^m\). We consider derivatives of order up to \(r \in \mathbb {N}\) of f and a map \(\bar{f}: \mathbb {R}^n \rightarrow \mathbb {R}^N\) given by arranging the partial derivatives of f in some fixed order. Then there is a “bad” set \(S \subset \mathbb {R}^N\) that we would like to avoid. In general, the space S is stratified, i.e. there is a splitting \(S = \cup _{i \le \mu } S_i\), where \(S_i\) are smooth manifolds of dimension \(\dim S_i\) for \(i\in \{1,\cdots ,\mu \}\). More precisely, we say S is a manifold collection of defect\(\delta \), if \(\cup _{i \le j} S_i\) is closed for all \(j\in \{1,\cdots ,\mu \}\) and \({{\,\mathrm{codim}\,}}S_i \le \delta \) for all i.
We alter f by adding to it a polynomial of degree \(\le r\) whose coefficients form a set \(\alpha \) of very small numbers. If \(f_{\alpha }\) is the resulting mapping \(\mathbb {R}^n \rightarrow \mathbb {R}^N\), we may prove that for compact subsets \(K \subset \mathbb {R}^n\) and \(T \subset S\) there exists \(\alpha \) arbitrarily small such that \(f_{\alpha }(K) \cap T = \emptyset \). If we fix K, by taking \(\bar{f}(K) \subset B_L \subset \mathbb {R}^N\) for some large L, we prove that for any K, there is an \(\alpha \) such that \(f_{\alpha }(K) \cap S = \emptyset \).
Let \(N \in \mathbb {N}\) be the number given as above. Assume that we have a smooth map for \((p, \alpha ) \in \Omega \times R_1 \subset \mathbb {R}^n \times \mathbb {R}^N\)
$$\begin{aligned}F(p, \alpha ) = f_\alpha (p) = f_p^*(\alpha )\end{aligned}$$
into \(\mathbb {R}^N\). The family \(f_{\alpha }\) is called an N-parameter family of mappings of \(\Omega \) into \(\mathbb {R}^N\) if the matrix \(\nabla _\alpha f_p^*(\alpha )\) has full rank for all \(p \in \Omega \).
Next, we say a subset Q of a Euclidean space is of finite\(\mu \)-extent if there is a number A with the following property. For any integer \(\kappa \), there are sets \(Q_1, \cdots , Q_a\) such that
$$\begin{aligned} Q = Q_1 \cup \cdots \cup Q_a, \quad {{\,\mathrm{diam}\,}}(Q_j) < 1/2^\kappa \ (\text {for all } j),\quad a \le 2^{\mu \kappa }A. \end{aligned}$$
(6.7)
Lemma 6.9
(Lemma 9a in [53]) Let the \(f_{\alpha }\) form an N-parameter family of mappings of \(\Omega \subset \mathbb {R}^n\) into \(\mathbb {R}^N\) and let \(\Omega _1\subset \Omega \) and Q be compact subsets of \(\mathbb {R}^n\) and of \(\mathbb {R}^N\) of finite \(\omega \)-extent and of finite q-extent, respectively, and suppose \(\omega +q < N\). Then for any \(\alpha _0\in \mathbb {R}^N\) and for any \(\epsilon >0\) there exists \(\alpha \) with \(|\alpha -\alpha _0|<\epsilon \) such that
$$\begin{aligned} f_\alpha (\Omega _1) \cap Q = 0. \end{aligned}$$
By [53, Section 10], the family of functions \(f_\alpha \) given by adding polynomials of order \(\le r\) is an N-parameter family of mapping of \(\mathbb {R}^n\) into \(\mathbb {R}^N\). We call \(\delta := N - q\) the defect of S in \(\mathbb {R}^N\), so the condition in Lemma 6.9 can be restated as simply \(\delta > \omega \). It can be easily seen that the conclusion of the above lemma holds if Q is a manifold collection of defect \(\delta > \omega \).
Density argument
Let \(\mathcal {F} \subset C_c^{\infty }(W)^{n + 1}\) be the set, which contains exterior measurements from Definition 6.7. By Lemma 6.8, we may reconstruct the drift b and potential c for such exterior measurements.
Notice that \(\mathcal {F}\) is an open set, since the set of all functions g with finite order of vanishing at each point is open and the operator \(P_{b, c}\) is continuous in given topologies. In the sequel, we seek to prove that \(\mathcal {F}\) is also dense.
We first illustrate the argument for the case \(n=1\) in which case it is rather transparent. We will then present the proof for the general case below.
Proposition 6.10
Assume \(n = 1\). Then \(\mathcal {F} \subset C_c^{\infty }(W)^2\) is open and dense.
Proof
Take \(f = (f_1, f_2) \in C_c^{\infty }(W)^2\) to be any exterior data and consider \(g_i := P_{b, c}f_i \in C^\infty (K)\) for \(i = 1, 2\). Write \(g = (g_1, g_2)\). We will construct an approximation of \((f_1, f_2)\) in the topology of \(C_c^{\infty }(W)^2\) lying in \(\mathcal {F}\). In one dimension, we have
$$\begin{aligned} \begin{aligned} h(g_1, g_2) = \begin{pmatrix} g_1' &{} g_1\\ g_2' &{} g_2 \end{pmatrix} = g_1' g_2 - g_1 g_2'. \end{aligned} \end{aligned}$$
(6.8)
In other words, h is the Wronskian of \(g_1\) and \(g_2\). Now, in this case, we have
$$\begin{aligned} \bar{g}(x) = \big (g_1(x), g_2(x), g_1'(x), g_2'(x), g_1''(x), g_2''(x)\big ) \end{aligned}$$
and \(N = 2N_0= 6\) [where \(N_0\) was defined in (6.6)]. We consider perturbations of the form
$$\begin{aligned} g_\alpha (x) = g(x) + (\alpha _1 + \alpha _1' x + \alpha _1'' x^2)e_1 + (\alpha _2 + \alpha _2' x + \alpha _2'' x^2)e_2 \in \mathbb {R}^2, \end{aligned}$$
where \(e_1, e_2\) are the canonical coordinate vectors in \(\mathbb {R}^2\).
We compute the defect of the bad set given by \(h=0\) and \(\nabla h = 0\), i.e.
$$\begin{aligned} S := \{J_1 = \alpha _1' \alpha _2 - \alpha _1 \alpha _2' = 0\} \cap \{J_2 = \alpha _1'' \alpha _2 - \alpha _1 \alpha _2'' =0 \} \subset \mathbb {R}^6. \end{aligned}$$
We need to show that the defect \(\delta = 6 - q\), where q is the extent of S, is bigger that \(n = 1\) to apply Lemma 6.9. To this end, we compute the gradients
$$\begin{aligned} \nabla \bar{J}_1 = (-\alpha _2', \alpha _1', \alpha _2, -\alpha _1, 0, 0),\\ \nabla \bar{J}_2 = (-\alpha _2'', \alpha _1'', 0, 0, \alpha _2, -\alpha _1). \end{aligned}$$
These are clearly linearly independent for \(\alpha _1 \ne 0\) or \(\alpha _2 \ne 0\) or \(\det \begin{pmatrix} \alpha _1' &{} \alpha _2'\\ \alpha _1'' &{} \alpha _2'' \end{pmatrix} \ne 0\), so S is a manifold collection of defect \(\delta = 2\). So we apply Lemma 6.9 to obtain arbitrarily small values of \(\alpha = (\alpha _1, \alpha _1', \alpha _1'', \alpha _2, \alpha _2', \alpha _2'')\) such that \(g_\alpha \) satisfies the property that \(h(g_\alpha )\) has an empty critical zero set on \(K \subset \overline{\Omega }\).
Next, by Runge approximation (see Lemma 6.2), there exists \(f_{0, m}, f_{1, m}, f_{2, m} \in C_c^{\infty }(W)\) with \(P_{b, c} f_{i, m} \rightarrow x^i\) in \(C^\infty (K)\) for \(i = 0,1, 2\) and as \(m \rightarrow \infty \). Therefore, we define
$$\begin{aligned} f_{\alpha , m} = f + (\alpha _1 f_{0, m} + \alpha _1' f_{1, m} + \alpha _1'' f_{2, m}) e_1 + (\alpha _2 f_{0, m} + \alpha _2' f_{1, m} + \alpha _2'' f_{2, m}) e_2 \in \mathbb {R}^2. \end{aligned}$$
Now fix m large enough, so \(P_{b, c} f_{i, m}\) is close to \(x^i\), such that the perturbation by elements \(f_{i, m}\) of f makes a 6-parameter family of mappings \(\Omega ' \rightarrow \mathbb {R}^6\), for \(\alpha \in B_1(0) \subset \mathbb {R}^6\) in the unit ball (say), where \(K \subset \Omega ' \Subset \Omega \), by compactness. Then we again apply Lemma 6.9 and get that \(g_{\alpha ,m} := P_{b, c} f_{\alpha , m} \rightarrow g\) in \(C^\infty (K)\) on a sequence of \(\alpha \) converging to zero. By construction \(f_{\alpha , m} \in \mathcal {F}\), so this finishes the proof. \(\square \)
Remark 6.11
To prove the desired genericity property, we need to approximate \(P_{b, c} f\) by either polynomials or other nice functions (analytic, generic etc.), by use of a linear approximation operator \(T_m f\), but such that
$$\begin{aligned} T_m P_{b, c} f = P_{b, c} T_m f \quad \text {and} \quad \lim _{m \rightarrow \infty } T_m f = f. \end{aligned}$$
This is the reason why the usual approximation operators, such is the Bernstein polynomials operator, or the general Weierstrass approximation theorem approach are not good for this purpose. The above approximation argument however proves the existence of such \(T_m\), which is obtained by adding a finite linear combination of suitable functions with coefficients going to zero as \(m \rightarrow \infty \).
Remark 6.12
There is an alternative proof by hand of the above statement for \(n = 1\), not using the Whitney machinery, but only the genericity of Morse functions.
We seek to extend the previous argument to dimension \(n=2\) and higher. Unfortunately, in this context, it does not suffice to consider the critical zero set of h, i.e. the set \(h=0\) and \(\nabla h=0\). The computations below show that we have to include higher order derivatives of h to obtain genericity in the sense of the previous proposition.
Let \(N = (n + 1) \times {n + k(n) \atopwithdelims ()k(n)}\) be the number of polynomials of degree \(\le k(n)\) with which we perturb (we multiply by \(n + 1\) as this is our number of functions). Then we consider the map \(\bar{g}:\mathbb {R}^n \rightarrow \mathbb {R}^N\) (see previous subsection) of evaluating the derivatives of order \(\le k(n)\) at each point. We define the bad set to be \(S = S_{k(n)} \subset \mathbb {R}^N\) consisting of points given by the condition that h and its derivatives up to order \(k(n) - 1\) vanish. For a given function \(g = (g_1, \cdots , g_{n + 1})\), we denote this set by \(Z_0(g) = h^{-1}(0)\), \(Z_1(g) = Z_0 \cap (\nabla h)^{-1} (0)\), and inductively we define \(Z_j(g) = Z_{j-1}(g) \cap (\nabla ^j h)^{-1}(0)\).
Similarly as in one dimension, we then also have the following lemma, which we prove in the “Appendix”:
Lemma 6.13
(Determinant genericity) The bad set \(S_{k(n)} \subset \mathbb {R}^N\) is a manifold collection of defect \(n + 1\). In particular, there are arbitrarily small perturbations \(g_\alpha \) with \(\alpha \in \mathbb {R}^N\), such that \(Z_{k(n) - 1}(g_{\alpha }) = \emptyset \) on an arbitrary compact set.
As a corollary, we deduce the main result.
Proposition 6.14
For any \(n \in \mathbb {N}\), the set \(\mathcal {F} \subset C_c^{\infty }(W)^{n + 1}\) is open and dense.
Proof
The proof is an immediate corollary of Lemma 6.13 and the method of proof of Proposition 6.10. Indeed, openness again follows from the definition of the set \({{\,\mathrm{\mathcal {F}}\,}}\). In order to infer the density of \({{\,\mathrm{\mathcal {F}}\,}}\), we argue along the lines of Proposition 6.10: Let \(f=(f_1,\dots ,f_{n+1})\in C_c^{\infty }(W)^{n+1}\) be arbitrary but fixed. By Runge approximation (see Theorem 1.2 (b)), for each \(\beta \in \mathbb {N}^{n}\) with \(|\beta |\le \lceil {\sqrt{n + 1}}\Big \rceil \) there exists \(f_{\beta ,m} \in C_c^{\infty }(W)\) such that
$$\begin{aligned} P_{b,c}(f_{\beta ,m}) \rightarrow x^{\beta } \text{ in } C^{\infty }(K). \end{aligned}$$
As the set of polynomials up to degree \(|\beta |\le \Big \lceil {\sqrt{n + 1}\Big \rceil }\) forms a \(\nu \)-parameter family with \(\nu = N_0\) and \(N_0\) as in (6.6), for \(m\ge m_0\) sufficiently large, also the set \(P_{b, c} (f_{\beta ,m})\) with \(|\beta | \le \Big \lceil {\sqrt{n + 1}}\Big \rceil \) forms a \(N_0\)-parameter family. As a result, Lemma 6.9 can be applied to
$$\begin{aligned} P_{b,c}(f_{\alpha }):= P_{b,c}(f) + \sum \limits _{j=1}^{ n + 1} \sum \limits _{\beta \in \mathbb {N}^n, \ |\beta | \le \Big \lceil {\sqrt{n + 1}}\Big \rceil } \alpha _{\beta j} P_{b,c}(f_{\beta , m}) e_j, \end{aligned}$$
where \(\{e_1, \dots , e_{n+1}\}\) denotes the canonical basis of \(\mathbb {R}^{n + 1}\), \(\alpha _{\beta j} \in \mathbb {R}\) and \(x^{\beta }:= \prod \nolimits _{\ell =1}^{n} x_{\ell }^{\beta _{\ell }}\). Thus, for any \(\epsilon >0\) there exists \(\alpha _{\epsilon } \in \mathbb {R}^{N_0 (n +1)}\) with \(|\alpha _{\epsilon }|\le \epsilon \) such that \(Z_{ k(n) - 1}(P_{b,c}(f_{\alpha _{\epsilon }})) = \emptyset \) on \(K \subset \Omega \). By construction, we have \(f_{\alpha _{\epsilon }} \in \mathcal {{{\,\mathrm{\mathcal {F}}\,}}}\). This concludes the density proof. \(\square \)
Combining Lemma 6.8 and Propositions 6.10, 6.14 then implies the result of Theorem 1.4.
Remark 6.15
We remark that Theorem 1.4 together with the results from [19] also yields a constructive reconstruction algorithm for the fractional Calderón problem with drift.
Remark 6.16
Last but not least, we point out that similar openness and density results can also be obtained for the Jacobian by arguing along the same lines as in the “Appendix”. More specifically, genericity results in Lemma 6.13 can be shown to hold with the same critical index \(k(n) - 1\), if instead of the determinant in Eq. (6.2) we consider the Jacobian determinant of n functions, which might be of independent interest.