POWER TYPE NONLINEARITIES
MATTI LASSAS, TONY LIIMATAINEN, YI-HSUAN LIN, AND MIKKO SALO
Dedicated to the memory of Yaroslav Kurylev
Abstract. We introduce a method for solving Calder´on type inverse problems for semilinear equations with power type nonlinearities. The method is based on higher order linearizations, and it allows one to solve inverse problems for certain nonlinear equations in cases where the solution for a corresponding linear equation is not known. Assuming the knowledge of a nonlinear Dirichlet- to-Neumann map, we determine both a potential and a conformal manifold simultaneously in dimension 2, and a potential on transversally anisotropic manifolds in dimensions n ≥ 3. In the Euclidean case, we show that one can solve the Calder´on problem for certain semilinear equations in a surprisingly simple way without using complex geometrical optics solutions.
Keywords. Inverse boundary value problem, Calder´on problem, semilinear equation, Riemannian manifold, transversally anisotropic.
Contents
1. Introduction 1
2. Preliminaries 8
3. Proof of Theorem 1.1 11
4. Simultaneous recovery on two-dimensional Riemannian surfaces 13 5. Transversally anisotropic manifolds: simplified case 15 6. Transversally anisotropic manifolds: general case 19 Appendix A. Complex geometrical optics solutions 27
Appendix B. Some lemmas 31
References 34
1. Introduction
In this paper we study inverse boundary value problems for nonlinear elliptic equations. A standard example of inverse problems for linear elliptic equations is the problem introduced by Calder´on [Cal80], where the objective is to determine the electrical conductivity of a medium by making voltage and current measurements on its boundary. It is closely related to the problem of determining an unknown potential q in a Schr¨odinger operator ∆ + q from boundary measurements, first solved in [SU87] in dimensions n ≥ 3. There is an extensive theory concerning inverse boundary value problems for linear elliptic equations, and we refer to [Uhl09]
for a survey.
It is also natural to consider analogous inverse problems under nonlinear settings.
Let Ω ⊂ Rn be a bounded domain with C∞ boundary, and consider the reaction- diffusion equation
∂tw − ∆w = a(x, w) in Ω × {t > 0}.
1
Equations of this type arise in the modelling of chemical reactions, population dynamics and pattern formation [Vol14]. Examples include the Fisher, Kolmogorov or logistic diffusion equations with quadratic nonlinearity (i.e. a(x, w) is quadratic in w), the Newell-Whitehead-Segel equation with cubic nonlinearity, and equations in combustion involving polynomial or exponential nonlinearities.
A stationary solution w(x, t) = u(x) satisfies the elliptic equation
∆u + a(x, u) = 0 in Ω.
The Dirichlet problem for this equation is related to maintaining a temperature (or concentration or population) f on the boundary. The boundary measurements for such an equation, provided that it is well-posed for some class of boundary values, may be encoded by a Dirichlet-to-Neumann map (DN map) Λa, which maps the boundary value f to the flux Λa(f ) = ∂νu|∂Ωof the corresponding equilibrium state across the boundary.
In fact, inverse problems for nonlinear elliptic equations have also been widely studied. A standard method, introduced in [Isa93] in the parabolic case, is to show that the first linearization of the nonlinear DN map is actually the DN map of a linear equation, and to use the theory of inverse problems for linear equations. For the semilinear Schr¨odinger equation ∆u + a(x, u) = 0, the problem of recovering the potential a(x, u) was studied in [IS94, Sun10] in dimensions n ≥ 3, and in [IN95, Sun10, IY13] when n = 2. In addition, inverse problems have been studied for quasilinear elliptic equations [Sun96, SU97, KN02, LW07, MU18], the degen- erate elliptic p-Laplace equation [SZ12, BHKS18], and the fractional semilinear Schr¨odinger equation [LL19]. Certain Calder´on type inverse problems for quasilin- ear equations on Riemannian manifolds were recently considered in [LLS19]. We refer to the survey articles [Sun05, Uhl09] for further details on inverse problems for nonlinear elliptic equations.
Inverse problems have also been studied for hyperbolic equations with various nonlinearities. Many of the works mentioned above rely on a solution to a related inverse problem for a linear equation. This is in contrast to the study of inverse problems for nonlinear hyperbolic equations, where it has been realized that the nonlinearity can actually be beneficial in solving inverse problems.
By using the nonlinearity as a tool, some still unsolved inverse problems for hyperbolic linear equations have been solved for their nonlinear counterparts. For the scalar wave equation with a quadratic nonlinearity, Kurylev-Lassas-Uhlmann [KLU18] proved that local measurements determine the global topology, differ- entiable structure and the conformal class of the metric g on a globally hyper- bolic 4-dimensional Lorentzian manifold. The authors of [LUW18] studied in- verse problems for general semilinear wave equations on Lorentzian manifolds, and in [LUW17] they studied analogous problem for the Einstein-Maxwell equations.
For more inverse problems of nonlinear hyperbolic equations, we refer readers to [CLOP19, dHUW18,KLOU14,WZ19] and references there in.
In this work we introduce a method which uses nonlinearity as a tool that helps in solving inverse problems for certain nonlinear elliptic equations. The method is based on higher order linearizations of the DN map, and essentially amounts to using sources with several parameters and obtaining new linearized equations after differentiating with respect to these parameters. We demonstrate the scope of the method by solving Calder´on type problems for three mathematical models.
The first model is the Calder´on problem for a semilinear Schr¨odinger equation with quadratic nonlinearity,
(1.1) ∆u + qu2= 0 in Ω ⊂ Rn,
where q ∈ C∞(Ω) and n ≥ 2. The solution to a related inverse problem with a(x, u) in place of qu2is known under assumptions like ∂ua(x, u) ≤ 0 [IS94,IN95,Sun10].
Theorem 1.1 proves uniqueness for the nonlinearity qu2, which appears to be a new result. The method applies to more general models, but we begin with the equation (1.1) in order to introduce our approach in the simplest possible setting.
The second new result is Theorem 1.2, where we simultaneously determine the metric, the manifold and the potential up to gauge symmetry from the knowledge of the DN map of a semilinear Schr¨odinger equation on two-dimensional Riemannian surfaces. The analogous result for a linear Schr¨odinger equation is not known in this generality. Here we use nonlinearity to simultaneously determine the topology and the conformal structure of the Riemannian surface, as well as the potential, up to a natural gauge transformation.
The third result, Theorem1.3, is the recovery of the potential q from the knowl- edge of the DN map of a Schr¨odinger operator with nonlinearity of the form qum, m ≥ 3, on transversally anisotropic manifolds in dimensions n ≥ 3. Transversally anisotropic manifolds are product type manifolds which appear in several works related to the anisotropic Calder´on problem. Again, the solution to the analogous inverse problem for a linear equation is not known in this generality. Existing results will be discussed in more detail later in this introduction.
Let us introduce the mathematical setting for this article. We will denote by (M, g) a compact Riemannian manifold with C∞boundary ∂M , where dim(M ) = n, n ≥ 2. For example, one could have M = Ω where Ω is a bounded C∞ domain in Rn, and g could be the Euclidean metric. Let q ∈ C∞(M ). We will consider semilinear elliptic equations of the form
(∆gu + qum= 0 in M,
u = f on ∂M,
(1.2) where
m ∈ N and m ≥ 2.
Here ∆g is the Laplace-Beltrami operator, given in local coordinates by
∆gu = 1 det(g)1/2
n
X
a,b=1
∂
∂xa
det(g)1/2gab∂u
∂xb
,
where g = (gab(x)) and g−1 = (gab(x)).
We will show that the Dirichlet problem (1.2) has a unique small solution u for sufficiently small boundary data f ∈ Cs(∂M ), where s > 2 with s /∈ N. More precisely this means that there is δ > 0 such that whenever kf kCs(∂M )≤ δ , there is a unique solution uf to (1.2) with sufficiently small Cs(M ) norm (see Section 2 for more details on well-posedness). We will call uf the unique small solution.
Here Csis the standard H¨older space for s > 2 with s /∈ N (often written as Ck,α if s = k + α where k ∈ Z and 0 < α < 1), see e.g. [Tay11a, Section 13.8]. Hence, the DN map is defined by using the unique small solution in a following way:
ΛM,g,q: Cs(∂M ) → Cs−1(∂M ), f 7→ ∂νuf|∂M,
where ∂ν denotes the normal derivative on the boundary ∂M . In what follows, we use the notation ΛM,gto denote the DN map when q = 0. When M = Ω ⊂ Rnand g is the identity matrix, we denote the DN map by Λq.
As a warm-up, we begin with a theorem that illustrates our method in a simple setting. This theorem is in Rnfor n ≥ 2, where ∆gis the Euclidean Laplacian and M = Ω with Ω a bounded smooth domain in Rn.
Theorem 1.1 (Global uniqueness for a quadratic nonlinearity). Let n ≥ 2, and let Ω ⊂ Rn be a bounded domain with C∞ boundary ∂Ω. Let q1, q2∈ C∞(Ω). Assume the DN maps Λqj for the equations
(∆u + qju2= 0 in Ω,
u = f on ∂Ω,
(1.3)
for j = 1, 2 satisfy
Λq1(f ) = Λq2(f )
for all f ∈ Cs(∂Ω) with kf kCs(∂M ) < δ, where δ > 0 is any sufficiently small number. Then q1= q2 in Ω.
We will offer a detailed proof of Theorem 1.1 in Section 3, but let us briefly discuss the idea how to prove the theorem by using the method of higher order linearization. The second order linearization of the nonlinear DN map has already been used in the works [Sun96, SU97] related to nonlinear equations with matrix coefficients. First and second order linearizations were also used in [KN02] for a nonlinear conductivity equation (see also [CNV19]). Under certain assumptions on the nonlinearity, by using the second order linearization, they can recover quadratic parts of the nonlinearity (see [KN02, Theorem 1.2 and Theorem 1.3]). In this work, we use similar ideas but obtain interesting new phenomena for related nonlinear inverse problems.
For the equation (1.3) with quadratic nonlinearity, the first linearization of the nonlinear DN map Λq, linearized at the zero boundary value, is just the DN map for the standard Laplace equation:
(DΛq)0: Cs(∂Ω) → Cs−1(∂Ω), f 7→ ∂νvf|∂Ω,
where vf is the unique solution of ∆vf = 0 in Ω with vf|∂Ω = f . Thus the first linearization does not carry any information about the unknown potential q.
However, for a quadratic nonlinearity the second linearization (D2Λq)0, which is a symmetric bilinear map from Cs(∂Ω) × Cs(∂Ω) to Cs−1(∂Ω), turns out to be very useful: it is characterized by the identity (see (2.9))
Z
∂Ω
(D2Λq)0(f1, f2)f3dS = −2 Z
Ω
qvf1vf2vf3dx
where vfj is the harmonic function with boundary value fj. See formula (1.4) bel . Thus we have the implications
Λq1(f ) = Λq2(f ) for small f
=⇒ (D2Λq1)0= (D2Λq2)0
=⇒
Z
Ω
(q1− q2)v1v2v3dx = 0 for any functions v1, v2, v3∈ Cs(Ω) that are harmonic in Ω.
The last statement is very close to the linearized Calder´on problem for a linear Schr¨odinger equation (the difference is that here one has the product of three har- monic functions, instead of two). Choosing v1and v2to be harmonic exponentials as in the work of Calder´on [Cal80], and choosing v3 ≡ 1, shows that the Fourier transform of q1− q2vanishes and hence q1= q2. Thus, somewhat strikingly, we can solve a Calder´on type inverse problem for the nonlinear equation ∆u + qu2= 0 in a much simpler way than for the linear equation ∆u+qu = 0 (the latter requires com- plex geometrical optics solutions as in [SU87]). The method also provides extremely simple reconstruction of the potential q, see Corollary 3.1.
We also mention that the second order linearization can be described as (D2Λq)0(f1, f2) = ∂1∂2u1f1+2f2|1=2=0 on ∂Ω.
(1.4)
That is, one considers boundary data
f = 1f1+ 2f2∈ Cs(∂Ω),
where 1, 2 are sufficiently small parameters, and takes the mixed derivative
∂
∂1
∂
∂2
1=2=0
of the equation (1.3). This idea is similar to the recent works on inverse problems for nonlinear hyperbolic equations mentioned above, and it yields the equations (1.5) ∆wj = −2qjvf1vf2,
for j = 1, 2, where wj = ∂∂
1
∂
∂2
1=2=0uj and vfj are harmonic functions, i.e.
solutions to the linearized equation ∆v = 0. Taking the mixed derivative of the DN maps yields (see Section2)
∂νw1= ∂νw2on ∂Ω.
Subtracting the equations (1.5) for j = 1, 2 and integrating the resulting equation against the harmonic function vf3 yields the desired formula
Z
Ω
(q1− q2)vf1vf2vf3dx = 0 which was mentioned in the discussion above.
We move on to describe our next result. By using higher order linearizations we prove the following simultaneous recovery on a two-dimensional Riemannian surface.
Theorem 1.2 (Simultaneous recovery of metric and potential). Let (M1, g1) and (M2, g2) be two compact connected manifolds with mutual C∞ smooth boundaries
∂M1= ∂M2=: ∂M and dim(M1) = dim(M2) = 2. Let m ≥ 2, and let ΛMj,gj,qj be the DN maps of
∆gju + qjum= 0 in Mj
(1.6)
for j = 1, 2. Let s > 2 with s /∈ N and assume that ΛM1,g1,q1f = ΛM2,g2,q2f on ∂M,
for any f ∈ Cs(∂M ) with kf kCs(∂M )≤ δ, where δ > 0 is sufficiently small. Then:
(1) There exists a conformal diffeomorphism J : M1 → M2 and a positive smooth function σ on M1 such that for x ∈ M1 we have
(σJ∗g2)(x) = g1(x), with J |∂M = Id and σ|∂M = 1.
(2) Moreover, one can also recover the potential up to a natural gauge invari- ance in the sense that
σq1= q2◦ J in M1.
We see that the conformal factor σ (and also the diffeomorphism J ) couples to the potential. This is due to the gauge symmetry of the inverse problem:
ΛM1,σJ∗g,σ−1J∗q= ΛJ (M1),g,q
where J is a conformal diffeomorphism and σ is a positive smooth function satisfy the boundary conditions J |∂M = Id and σ|∂M = 1. For the linear equation ∆gu + qu = 0, an analogous result has been proved when M is a domain in R2 with
a Riemannian metric [IUY12], when M is a manifold and the potentials are zero [LU01], and when the manifold M is a priori known [GT11]. The recovery of properties of both the manifold and potential is stated as an open question in [GT13], where further references to two-dimensional results are given. The proof of Theorem1.2uses the first linearization of the DN map to recover the metric and the manifold up to a conformal transformation. Then the second linearization is used to recover the potential on a single fixed manifold (up to the gauge symmetry).
The final new result in this article is to consider inverse problems for the semi- linear Schr¨odinger equation on transversally anisotropic manifold. Let us recall the definition of a transversally anisotropic manifold.
Definition 1.1. Let (M, g) be a compact oriented manifold with a C∞ boundary and with dim M ≥ 3. (M, g) is called transversally anisotropic if (M, g) ⊂⊂ (T, g), where T = R × M0 and g(x) = g(x1, x0) = e(x1) ⊕ g0(x0) for x1∈ R and x0∈ M0. Here (R, e) is the Euclidean line and (M0, g0) is an (n − 1)-dimensional compact manifold with a smooth boundary.
Figure 1. An example of a transversally anisotropic manifold (M, g).
An example of a transversally anisotropic manifold is visualized in Figure1. Espe- cially, if Ω is a domain in Rn and g0 is any Riemannian metric on Ω, then R × Ω equipped with the metric
g(x1, x0) =
1 0
0 g0(x0)
is transversally anisotropic. For more details of inverse problems in transversally anisotropic geometries for linear equations, we refer readers to [FKLS16,FKL+17].
We prove the following.
Theorem 1.3. Let (M, g) be a transversally anisotropic manifold, let qj ∈ C∞(M ), and let Λqj be the DN maps for the equations
∆gu + qjum= 0 in M for j = 1, 2, where we assume that
m ∈ N, m ≥ 3.
If the DN maps satisfy
Λq1(f ) = Λq2(f )
for all sufficiently small f , then q1= q2 in M .
The higher order linearization method in this case reduces the proof of Theo- rem1.3to showing for any m ≥ 3 that the identity
(1.7)
Z
M
f v1· · · vm+1dV = 0
holding for any vj ∈ C∞(M ) with ∆gvj = 0 in M , implies f ≡ 0. Thus we prove that the products of at least four harmonic functions on a transversally anisotropic manifold form a complete set. The main point is that the argument works for arbi- trary transversally anisotropic manifolds without any restriction on the transversal geometry.
The solution to the analogous inverse problem for a linear equation ∆gu+qu = 0 on transversally anisotropic manifolds is only known under the additional assump- tion that the transversal manifold (M0, g0) has injective geodesic X-ray trans- form [FKLS16]. In the linearized version of that problem, the identity (1.7) only holds for m = 1 and one needs to prove that products of pairs of harmonic func- tions form a complete set. In [FKLS16] this is done by using complex geometrical optics solutions that concentrate near two-dimensional surfaces that are translates of geodesics on M0. Using products of such solutions and their complex conjugates recovers certain integrals over geodesics in M0, but does not yield pointwise in- formation. In [FKL+17] products of solutions concentrating near two intersecting geodesics were used instead to recover microlocal information in the linearized in- verse problem. The products are supported near finitely many points in M0, but there is oscillation that prevents recovering more information. We also mention [GST19] that deals with the linearized problem on certain complex manifolds.
The idea behind the proof of Theorem1.3is that since one can use products of at least four harmonic functions, we can use solutions related to two intersecting geodesics on M0 as well as their complex conjugates. The product of these four solutions is supported near finitely many points in M0 and the product does not have high oscillations. This allows one to recover the potential completely.
We mention that the aim of this paper is not to work in the highest possible generality or to provide an extensive list of all possible applications of the higher order linearization method. For example, it is clear that the method applies to certain more general nonlinearities and less regular coefficients. These are left to forthcoming works. Here we have included applications that illustrate the power of the higher order linearization method.
Finally, we mention that before submitting this paper we became aware of an up- coming preprint of Ali Feizmohammadi and Lauri Oksanen, which simultaneously and independently proves a result similar to Theorem1.3, and we agreed with them to publish the preprints of the results at the same time on the same preprint server.
See [FO19].
The paper is organized as follows. In Section 2, we lay out the basic properties for semilinear elliptic equations that we use. This includes the well-posedness of the Dirichlet problem and higher order linearizations of the DN map. We use the higher order linearization approach to prove Theorem 1.1 in Section 3, Theorem 1.2 in Section4, and Theorem1.3in Section5, respectively.
Acknowledgements. All authors were supported by the Finnish Centre of Ex- cellence in Inverse Modelling and Imaging (Academy of Finland grant 284715).
M.S. was also supported by the Academy of Finland (grant 309963) and by the European Research Council under Horizon 2020 (ERC CoG 770924). Y.-H. L. is partially supported by the Ministry of Science and Technology Taiwan, under the Columbus Program: MOST-109-2636-M-009-006, 2020-2025. The authors would
like to thank the anonymous referees for some useful comments to improve this paper.
2. Preliminaries
In this section, we prove well-posedness of the Dirichlet problem for semilinear elliptic equations on Riemannian manifolds with small boundary data, and study higher order linearizations of the DN map. We assume that the Riemannian man- ifolds we consider are compact, C∞smooth and have C∞boundary.
We state the first result of this section for a general nonlinearity satisfying two conditions: Let Q be the semilinear elliptic operator
Q(u) := ∆gu + a(x, u), (2.1)
where a ∈ C∞(M × R) satisfies the following two conditions:
a(x, 0) = 0, (2.2)
The map v 7→ ∆gv + ∂ua( · , 0)v is injective on H01(M ).
(2.3)
The first condition ensures that u ≡ 0 is a solution, and the second states that the equation linearized at u ≡ 0 is well-posed.
The next result considers mappings between Banach spaces which are Fr´echet differentiable. We refer the reader to [RR06, Section 10] and [Hor85, Section 1.1]
for basics about Fr´echet differentiability.
Proposition 2.1 (Well-posedness). Let (M, g) be a compact Riemannian manifold with C∞ boundary ∂M and let Q be the semilinear elliptic operator given by (2.1) satisfying (2.2) and (2.3). Let s > 2 with s /∈ Z. There exist δ, C > 0 such that for any f in the set
Uδ := {h ∈ Cs(∂M ) ; khkCs(∂M )< δ}, there is a solution u = uf of
(∆gu + a(x, u) = 0 in M,
u = f on ∂M,
(2.4)
which satisfies
kukCs(M )≤ Ckf kCs(∂M ).
The solution uf is unique within the class {w ∈ Cs(M ) ; kwkCs(M ) ≤ Cδ}, and if f ∈ C∞(∂M ), then uf ∈ C∞(M ). Moreover, there are C∞ Fr´echet differentiable maps
S : Uδ → Cs(M ), f 7→ uf, Λ : Uδ → Cs−1(∂M ), f 7→ ∂νuf|∂M.
Proof. We prove the existence of solutions by using the implicit function theorem in Banach spaces [RR06, Theorem 10.6]. Let
X = Cs(∂M ), Y = Cs(M ), Z = Cs−2(M ) × Cs(∂M ).
Consider the map
F : X × Y → Z, F (f, u) = (Q(u), u|∂M − f ).
We wish to show that F indeed maps to Z and is a C∞map. Note that since a is smooth, the map
u 7→ a(x, u)
takes Cs(M ) to Cs(M ), and if kukCs(M ) ≤ K then ka(x, u)kCs(M ) ≤ C(a, s, K) (these facts follow from a local coordinate computation). Thus F is well defined.
If u, v ∈ Cs(M ) we use the Taylor formula a(x, u + v) =
m
X
j=0
∂uja(x, u) j! vj+
Z 1 0
∂m+1u a(x, u + tv)
m! vm+1(1 − t)mdt.
Since Cs(M ) is an algebra, we have that when kvkCs(M )≤ 1 one has
a(x, u + v) −
m
X
j=0
∂uja(x, u) j! vj
Cs(M )
≤ Cm,a,ukvkm+1Cs(M ).
This shows that u 7→ a(x, u) is a C∞map Cs(M ) → Cs(M ). Since the other parts of F are linear, F is a C∞map in the standard sense of [RR06, Definition 10.2].
Note that F (0, 0) = 0 by (2.2). The linearization of F at (0, 0) in the u-variable is
DuF |(0,0)(v) = (∆gv + ∂ua(x, 0)v, v|∂M).
This is a homeomorphism Y → Z by (2.3). To see this, let (w, φ) ∈ Z = Cs−2(M )×
Cs(∂M ), and consider the Dirichlet problem
((∆g+ ∂ua(x, 0))v = w in M,
v = φ on ∂M.
(2.5)
If a solution to (2.5) exists, it is unique by (2.3). Consequently, by using the Fredholm alternative (see e.g. [Tay11a, Proposition 1.9]), we may solve (2.5) in H01(M ) for any source in H−1(M ) and zero boundary value. Thus, we have solutions v1 and v2 in H01(M ) to (2.5) with sources w and −(∆g+ ∂ua(x, 0))Φ respectively, where Φ ∈ Cs(M ) is a function with Φ|∂M = φ ∈ Cs(∂M ). Then v = v1+ v2+ Φ is the unique solution in H1(M ) to (2.5). We have the well-known Schauder estimate
kvkCs(M )≤ C
kwkCs−2(M )+ kΦkCs(M )
,
for some constant C > 0 independent of w ∈ Cs−2(M ) and Φ ∈ Cs(M ), which shows that solutions to (2.5) depend continuously on w and φ. (We have included a proof of the Schauder estimate in the manifold setting in AppendixB.)
The implicit function theorem in Banach spaces [RR06, Theorem 10.6 and Re- mark 10.5] now yields that there is δ > 0 and an open ball Uδ = BX(0, δ) ⊂ X and a C∞map S : Uδ → Y such that whenever kf kCs(∂M )≤ δ we have
F (f, S(f )) = (0, 0).
Since S is Lipschitz continuous and S(0) = 0, u = S(f ) satisfies kukCs(M )≤ Ckf kCs(∂M ).
Moreover, by redefining δ if necessary u = S(f ) is the only solution to F (f, u) = (0, 0) whenever kf kCs(∂M )≤ δ and kukCs(M )≤ Cδ. We have proven the existence of unique small solutions of the Dirichlet problem (2.4) and the fact that the solution operator S : Uδ → Cs(M ) is a C∞ map. Since the normal derivative is a linear map Cs(M ) → Cs−1(∂M ), it follows that also Λ is a well defined C∞ map Uδ →
Cs−1(∂M ).
In the rest of the paper, we consider power type nonlinearities of the form a(x, u) = q(x)um, where m ∈ N and m ≥ 2. For such nonlinearities, the higher order linearizations of the DN map will be particularly simple. We will consider complex solutions u to the boundary value problem (2.4). We remark that even
though the Proposition 2.1was proven for real valued solution the proposition re- mains valid for the nonlinearity a(x, u) = q(x)umby analyticity in u. In the rest of the work we will consider complex valued solutions without separate notice.
The next proposition justifies the formal calculation that we may differentiate the equation
(2.6) ∆guf+ q(x)umf = 0 in M , uf|∂M = 1f1+ · · · + mfm
in the j variables to have equations corresponding to first and mth linearizations,
∆gvfk = 0 and ∆gw = −(m!)qvf1· · · vfm.
The normal derivative of w is the mth linearization of the DN map of (2.6). Below, we write
(Dkf )x(y1, . . . , yk)
to denote the kth derivative at x of a mapping f between Banach spaces, considered as a symmetric k-linear form acting on (y1, . . . , yk). We refer to [Hor85, Section 1.1], where the notation f(k)(x; y1, . . . , yk) is used instead of (Dkf )x(y1, . . . , yk).
For f ∈ Cs(∂M ) with s > 2, s /∈ N, let us denote by vf the unique solution of the Laplace equation
(2.7) ∆gvf = 0 in M , vf|∂M = f.
By using this notation, we have the following result.
Proposition 2.2. Let q ∈ C∞(M ), and let Λq be the DN map for the semilinear elliptic equation
(2.8) ∆gu + q(x)um= 0 in M ,
where
m ∈ N and m ≥ 2.
The first linearization (DΛq)0 of Λq at f = 0 is the DN map of the Laplace equation (2.7) such that
(DΛq)0: Cs(∂M ) → Cs−1(∂M ), f 7→ ∂νvf|∂M.
The higher order linearizations (DjΛq)0 are identically zero for 2 ≤ j ≤ m − 1.
The m-th linearization (DmΛq)0of Λq at f = 0 is characterized by the following identity: for any f1, . . . , fm+1∈ Cs(∂M ) one has
(2.9)
Z
∂M
(DmΛq)0(f1, . . . , fm)fm+1dS = −(m!) Z
M
qvf1· · · vfm+1dV here each vfk, k = 1, . . . , m + 1, is the solution to (2.7) with boundary value f = fk. Proof. The nonlinearity a(x, u) = q(x)umsatisfies the conditions in Proposition2.1, and thus the DN map Λq = ∂νS|∂M is well defined for small data. Here S : f 7→ uf
is the solution operator for the Dirichlet problem of the equation (2.8). To compute the derivatives of Λq at 0, it is enough to consider the derivatives of S. Let us write f = f (1, . . . , k) := 1f1+ · · · + kfk. The function uf = S(1f1+ · · · + kfk) ∈ Cs(M ) depends smoothly on 1, . . . , k since S : Uδ → Cs(M ) is C∞ Fr´echet differentiable by Proposition 2.1. Applying ∂1· · · ∂k|1=···=k=0 to the Taylor’s formula for C∞ Fr´echet differentiable mappings (see e.g. [Hor85, Equation 1.1.7])
S(f ) =
k
X
j=0
(DjS)0(f, . . . , f )
j! +
Z 1 0
(Dk+1S)tf(f, . . . , f )
k! (1 − t)kdt implies that (DkS)0 may be computed using the formula
(DkS)0(f1, . . . , fk) = ∂1· · · ∂kuf|1=···=k=0.
Moreover, since uf is smooth in the j variables and ∆gis linear, we may differen- tiate the equation
(2.10) ∆guf+ q(x)umf = 0, uf|∂M = f freely in the j variables.
Let first k = 1, so that u = u1f1. Since u0 = 0 and m ≥ 2, the derivative of (2.10) in 1 evaluated at 1= 0 satisfies
∆g(∂1uf|1=0) = 0, ∂1uf|∂M = f1. Thus the first linearization of the map S at f = 0 is
(DS)0(f1) = ∂1u1f1|1=0= vf1, where vf1 satisfies (2.7) with f = f1.
For 2 ≤ k ≤ m − 1, applying ∂1· · · ∂k|1=···=k=0 to (2.10) gives that
∆g(∂1· · · ∂kuf|1=···=k=0) = 0, ∂1· · · ∂kuf|∂M = 0,
since ∂1· · · ∂k(q(x)umf) is a sum of terms containing positive powers of uf, which are equal to zero when f = 0. Uniqueness of solutions for the Laplace equation implies that
(DkS)0(f1, . . . , fk) = 0, 2 ≤ k ≤ m − 1.
When k = m, the only term in ∂1· · · ∂m(q(x)umf) which does not contain second or higher order power of uf is q(x)(m!)(∂1uf) · · · (∂muf). This is the only nonzero term after setting 1= . . . = m= 0, and thus the function
w := (DmS)0(f1, . . . , fm) = ∂1· · · ∂muf|1=...=m=0
solves
(2.11) ∆gw = −q(x)(m!)vf1· · · vfm in M with zero Dirichlet boundary values.
By linearity one has
(DkΛq)0= ∂ν(DkS)0|∂M.
The claims for (DkΛq)0 when 1 ≤ k ≤ m − 1 follow immediately. For k = m we observe that (DmΛq)0(f1, . . . , fm) = ∂νw|∂M satisfies
Z
∂M
(∂νw)fm+1dS = Z
M
(hdw, dvfm+1ig+ (∆gw)vfm+1) dV,
where d denotes the exterior derivative on M . The integral of hdw, dvfm+1igvanishes since w|∂M = 0 and vfm+1is harmonic. The proposition follows by using (2.11).
3. Proof of Theorem 1.1
In this section, we use the higher order linearization approach (in fact, the second order linearization of the DN map) to prove Theorem 1.1. We could use Propo- sition 2.2 to have the integral equation (3.6) below directly, even for the product of three harmonic functions instead of two (this is a stronger statement since one can always take the third harmonic function to be constant). The theorem would follow from this by using harmonic exponentials. However, we choose to give a direct hands-on approach that explains how to use the method.
Proof of Theorem 1.1. Let 1, 2 be sufficiently small numbers and let f1, f2 ∈ C∞(∂M ). Let the function uj:= uj(x; 1, 2) ∈ Cs(M ) be the unique small solution of
(∆uj+ qju2j = 0 in Ω, uj= 1f1+ 2f2 on ∂Ω, (3.1)
for j = 1, 2 provided by Proposition2.2. Let us differentiate (3.1) with respect to
`so that
(∆
∂
∂`uj
+ 2qjuj
∂
∂`uj
= 0 in Ω,
∂
∂`uj= f` on ∂Ω.
(3.2)
Inserting 1= 2= 0 into (3.2), shows that
∆vj(`)= 0 in Ω with v(`)j = f` on ∂Ω, where
vj(`)(x) = ∂
∂`
1=2=0
uj(x; 1, 2) .
Here we used uj(x; 0, 0) ≡ 0. The functions vj`are just harmonic functions defined in Ω with boundary data f`|∂Ω. By uniqueness of the Dirichlet problem for the Laplacian we have that
v(`) := v(`)1 = v(`)2 in Ω for ` = 1, 2.
(3.3)
Next, let us differentiate (3.2) with respect to k for k 6= `. Then we have that (∆
∂2
∂1∂2uj
+ 2qjuj
∂2
∂1∂2uj
+ 2qj
∂u
j
∂1
∂u
j
∂2
= 0 in Ω,
∂2
∂1∂2uj = 0 on ∂Ω.
(3.4)
Again, evaluating at 1= 2= 0, the equation (3.4) becomes (∆wj+ 2qjv(1)v(2)= 0 in Ω,
wj = 0 on ∂Ω,
(3.5)
where wj(x) =
∂2
∂1∂2uj
(x; 0, 0) and we used uj(x; 0, 0) ≡ 0 for j = 1, 2 again.
By using the fact that Λq1(1f1+ 2f2) = Λq2(1f1+ 2f2) for small 1, 2, we have
∂νu1|∂Ω= ∂νu2|∂Ω, and applying ∂1∂2|1=2=0 to this identity gives that
∂νw1|∂Ω= ∂νw2|∂Ω.
Thus, by integrating the equation (3.5) over Ω (i.e. integrating against the harmonic function v(3)= 1) and by using integration by parts we have
0 = Z
∂Ω
(∂νw1− ∂νw2) dS = Z
Ω
∆(w1− w2) dx = 2 Z
Ω
(q2− q1)v(1)v(2)dx (3.6)
where v(1) and v(2) are defined in (3.3). Therefore, by choosing f1 and f2 as the boundary values of the Calder´on’s exponential solutions [Cal80],
v(1)(x) := exp((k + iξ) · x), v(2)(x) := exp((−k + iξ) · x), (3.7)
where k, ξ ∈ Rn, k ⊥ ξ and |k| = |ξ|, we obtain that the Fourier transformation of the difference q2−q1at −2ξ vanishes. As ξ ∈ Rnis arbitrary, we obtain q1= q2. In the proof above we did not need to construct special solutions for an elliptic equation with unknown coefficients, such as complex geometrical optics solutions.
The linearization technique allowed us to simply use known harmonic functions.
This fact gives an extremely simple reconstruction in the setting of Theorem1.1.
Corollary 3.1. Let n ≥ 2, and let Ω ⊂ Rnbe a bounded domain with C∞boundary
∂Ω. Assume that q ∈ C∞(Ω), and let Λq be the DN map for the equation
(3.8) ∆u + qu2= 0 in Ω.
Then
bq(−2ξ) = −1 2
Z
∂Ω
∂2
∂1∂2
1=2=0
Λq(1f1+ 2f2) dS, (3.9)
where f1 and f2 are the boundary values of the exponential solutions (3.7) and qb stands for the Fourier transform of q.
Proof. The proof of the reconstruction can be directly read from (3.6) in the proof
of Theorem 1.1.
We end of this section with a remark about the stability of the reconstruction formula in Corollary 3.1.
Remark 3.2. Let us consider the stability of the solution of the inverse problem of Theorem 1.1, which regards determination of the potential q from the DN map of the equation ∆u+qu2= 0. Let us assume as in Corollary3.9and adopt its notation with the difference that we consider the equation (3.8) with two different potentials q1 and q2 and the corresponding DN maps Λq1 and Λq2. By the reconstruction formula (3.9) we have
qbj(−2ξ) = −1 2
Z
∂Ω
(D2Λqj)0(f1, f2) dS, for j = 1, 2.
By subtracting the above formula for j = 1 and j = 2 from each other, we obtain (qb1−bq2)(−2ξ) = −1
2 Z
∂Ω
((D2Λq1)0− (D2Λq2)0)(f1, f2) dS.
Now, we assume that
(1) kDk(Λq1− Λq2)0k∗ is sufficiently small for k = 0, 1, 2, and (2) kqjkH1(Ω)≤ R for j = 1, 2,
where
kT k∗= sup
kf1kCs (∂Ω)=···=kfkkCs (∂Ω)=1
kT (f1, . . . , fk)kCs−1(∂Ω)
for a bounded k-linear form T : Cs(∂Ω)×· · ·×Cs(∂Ω) → Cs−1(∂Ω). Next, by taking harmonic functions vf1 = v(1), vf2 = v(2)in Ω, where v(1), v(2) are the functions defined in (3.7), one can obtain that
kq1− q2kL2≤ ω kD2(Λq1− Λq2)0k∗ , (3.10)
where ω(t) is a modulus of continuity satisfying, for some C = C(R), ω(t) ≤ C| log t|−n+22 , 0 < t < 1
e.
One can directly prove the logarithmic stability (3.10) by using standard arguments in stability for the Calder´on problem, for example, see [Sal08, Section 4].
4. Simultaneous recovery on two-dimensional Riemannian surfaces We use the higher order linearization approach to simultaneously recover, from the DN map, the conformal class of a Riemannian surface and the potential of a semilinear Schr¨odinger operator up to the gauge symmetry. We use first order lin- earization to recover the conformal class of the manifold by using the result [LU01]
(see also [LLS19] for a recent alternative proof). Then by using the result [GT11]
we recover the potential on the known conformal manifold (up to gauge).
Proof of Theorem 1.2. The proof is divided into two steps. We first recover the manifold and the conformal class of the Riemannian metric. After that we recover the potential on a known manifold up to the gauge symmetry.
Step 1. Recovering the conformal manifold.
Notice first by Proposition2.2that the equality ΛM1,g1,q1(f ) = ΛM2,g2,q2(f ) for all f ∈ Cs(∂M ) with kf kCs(∂M )≤ δ, δ > 0, implies that
(DΛM1,g1,q1)0= (DΛM2,g2,q2)0.
By Proposition 2.2, the maps (DΛMj,gj,qj)0, for j = 1, 2, are the DN maps of the linearizations of the equations ∆gju + qjum = 0 in Mj at zero. The linearized equations are Laplace equations on (Mj, gj). Since D(ΛM1,g1,q1)0= D(ΛM2,g2,q2)0 we have that the DN maps of the Dirichlet problems
(∆gjvj= 0 in Mj, vj = f on ∂M
agree. We are in the setting of the standard anisotropic Calder´on problem on 2- dimensional Riemannian manifolds. We apply [LLS19, Theorem 5.1] (with Γ =
∂M ) to determine the manifold and the Riemannian metric up to a conformal transformation. That is, there exists a C∞ smooth diffeomorphism J : M1 → M2
such that
σJ∗g2= g1
with J |∂M = Id. Here σ ∈ C∞(M1) is a positive function with σ|∂M = 1. This completes the Step 1. of the proof.
Step 2. Recovering the potential.
We transform the equation (1.6) on the manifold (M2, g2) into an equation on the manifold (M1, g1) as follows. We denote
qe2= σ−1q2◦ J ≡ σ−1J∗q2.
Let f ∈ Cs(∂M ) with kf kCs(∂M )≤ δ and let u2 be the unique solution to
∆g2u2+ q2um2 = 0 in M2 with u2= f on ∂M given by Proposition 2.1. Let us denote
ue2:= J∗u2≡ u2◦ J.
Thenue2 solves
∆g1eu2+qe2(ue2)m= ∆σJ∗g2eu2+qe2(eu2)m
= σ−1∆J∗g2eu2+ σ−1(J∗q2)(eu2)m
= σ−1J∗(∆g2u2) + σ−1(J∗q2)(J∗u2)m
= σ−1J∗[∆g2u2+ q2um2 ] .
Here we used the conformal invariance of the Laplace-Beltrami operator in dimen- sion 2 in the second equality. In the third equality, the coordinate invariance of Laplace-Beltrami operator was used. Since u2solves ∆g2u2+ q2um2 = 0 in M2, we consequently have that
(∆g1ue2+qe2(ue2)m= 0 in M1,
ue2= f on ∂M.
(4.1)
Here we also used J |∂M = Id.
Next, let u1 be the unique solution to the nonlinear equation (1.6) on (M1, g1) with potential q1 and boundary value f . We show that the following equation (4.2) ∂ν1u1= ∂ν1eu2on ∂M,
holds by the assumption that ΛM1,g1,q1(f ) = ΛM2,g2,q2(f ). Since ΛM1,g1,q1 = ΛM2,g2,q2, it follows that if u1= u2= f on ∂M , then by definition
(4.3) ∂ν1u1= ∂ν2u2on ∂M.
We calculate
(4.4) ∂ν2u2= ν2· du2= ν2· d(u2◦ J ◦ J−1) = (J∗−1ν2) · d˜u2= ν1· d˜u2= ∂ν1ue2. Here · denotes the canonical pairing between vectors and covectors, and d is the exterior derivative of a function. For example ν2·du2= g(ν2, ∇u2) =P2
k=1ν2k∂ku2. In calculating (4.4), we used that J : M1 → M2 is conformal diffeomorphism, σJ∗g2 = g1, with J |∂M = Id and σ|∂M = 1. By combining (4.3) and (4.4) we have (4.2). Since the solutioneu2 is unique, we have that
(4.5) ΛM1,g1,q1(f ) = eΛM1,g1,˜q2(f ),
for all f ∈ Cs(∂M ) with kf kCs(∂M ) ≤ δ, where eΛM1,g1,˜q2 stands for the DN map of the Dirichlet problem (4.1).
We apply Proposition 2.2 on the single Riemannian manifold (M1, g1) for the DN maps ΛM1,g1,q1and eΛM1,g1,˜q2, which agree by (4.5). By Proposition2.2we have
(D2ΛM1,g1,q1)0= (D2ΛeM1,g1,˜q2)0
and
Z
M1
(q1−qe2)v1v2v3dV = 0,
where v1, v2, v3 ∈ Cs(M1) are harmonic functions in (M1, g1). Choosing v3 = 1, we get
Z
M1
(q1−qe2)v1v2dV = 0
for any harmonic functions v1 and v2 in (M1, g1). We choose v1 and v2 to be complex geometrical optics solutions constructed in [GT11]. (See the construction in the proof of Proposition 5.1 in [GT11]. We note that the construction can in fact be significantly simplified in our case where v1 and v2 are actually harmonic.
In this case, Carleman estimates are not needed and the construction in [GST19]
would suffice.) As in [GT11, Proposition 5.1], this yields that q1=eq2 in M1.
This concludes the proof.
5. Transversally anisotropic manifolds: simplified case
In this and the next section we prove Theorem1.3, which will be a consequence of the following proposition. The proof of the proposition is based on the existence of special harmonic functions on transversally anisotropic manifolds. These harmonic functions were constructed in [FKLS16]. They have the property that if (M, g) is a transversally anisotropic manifold, i.e. (M, g) ⊂⊂ R × M0, g = e ⊕ g0, then on the transversal manifold M0 these harmonic functions concentrate near the geodesics of (M0, g0).
Proposition 5.1. Let (M, g) be a transversally anisotropic manifold and assume that m ≥ 4. If f ∈ C1(M ) satisfies
Z
M
f u1· · · umdV = 0 (5.1)
for any uj∈ C∞(M ) with ∆guj= 0 in M , then f ≡ 0.
Theorem 1.3follows immediately from Proposition5.1:
Proof of Theorem 1.3. Let Λqj be the DN map for the equation ∆gu + qum = 0 in M . If Λq1(f ) = Λq2(f ) for small f , then (DmΛq1)0 = (DmΛq2)0. Thus by Proposition 2.2, one has
Z
M
(q1− q2)v1· · · vm+1dV = 0
where vj ∈ Cs(M ) are harmonic functions in M . Since m ≥ 3, it follows from
Proposition 5.1that q1= q2.
The harmonic functions u` for ` = 1, 2, . . . , m on M used in the proof of Propo- sition5.1are of the form
e−sx1(evs(x0) + rs(x))
where x1 is the coordinate along R and x = (x1, x0). They may be considered as an analogue of complex geometrical optics solutions for transversally anisotropic manifolds. Here evs, s = τ + iλ, is a so called Gaussian beam quasimode on M0, i.e. an approximate eigenfunction concentrating near a geodesic on (M0, g0) with (slightly complex) large frequency s. The function rs is a small correction term.
Such harmonic functions were introduced in [FKSU09] and [FKLS16]. Since we need some additional properties of these harmonic functions, we include a few statements regarding these functions. We have placed their proofs in AppendixA for readers’ convenience.
We say that a geodesic γ : [0, T ] → M is nontangential if γ(0) and γ(T ) are on
∂M , γ(t) ∈ Mint for 0 < t < T , and ˙γ(0) and ˙γ(T ) are not tangential to ∂M . We remark that we will apply the following proposition in the case where (M, g) is a transversal manifold (M0, g0).
Proposition 5.2 (Gaussian beams quasimodes). Let (M, g) be a compact Rie- mannian manifold with smooth boundary ∂M , dim(M ) = m. Let γ : [0, T ] → M be a nontangential geodesic, and let λ ∈ C. For any K ∈ N and k ∈ N, there is a family of functions (evs) ⊂ C∞(M ), where s = τ + iλ ∈ C and τ ≥ 1, such that
k(−∆g− s2)evskHk(M )= O(τ−K), kevskL4(M )= O(1), kevskL4(∂M )= O(1) (5.2)
as τ → ∞. The functions evs have the following properties: If p ∈ γ([0, T ]), then there is P ∈ N such that on a neighborhood U of p the function evs is a finite sum (5.3) evs=ev(1)+ · · · +ev(P )
where t1< . . . < tP are the times in [0, T ] such that γ(tl) = p. Eachev(l) has the form
(5.4) ev(l)= τ−m−18 eisΘ(l)a(l),
where each Θ = Θ(l) is a smooth complex function in U satisfying Θ(γ(t)) = t, ∇Θ(γ(t)) = ˙γ(t),
Im(∇2Θ(γ(t))) ≥ 0, Im(∇2Θ)(γ(t))|γ(t)˙ ⊥ > 0, (5.5)
