Elsevier

Journal of Differential Equations

Volume 269, Issue 6, 5 September 2020, Pages 4683-4719
Journal of Differential Equations

An inverse problem for a semi-linear elliptic equation in Riemannian geometries

https://doi.org/10.1016/j.jde.2020.03.037Get rights and content

Abstract

We study the inverse problem of unique recovery of a complex-valued scalar function V:M×CC, defined over a smooth compact Riemannian manifold (M,g) with smooth boundary, given the Dirichlet-to-Neumann map, in a suitable sense, for the elliptic semi-linear equation Δgu+V(x,u)=0. We show that uniqueness holds for a large class of non-linearities when the manifold is conformally transversally anisotropic. The proof is constructive and is based on a multiple-fold linearization of the semi-linear equation near complex geometric optic solutions for the linearized operator and the resulting non-linear interactions. These interactions result in the study of a weighted integral transform along geodesics, that we call the Jacobi weighted ray transform.

Introduction

Let (M,g) be a smooth compact Riemannian manifold with a smooth boundary M and dimM:=n3. Let α(0,1) and consider an a priori unknown function V:M×CC. We make the following standing assumptions.

  • (i)

    V(,z)Cα(M),zC,

  • (ii)

    V(x,0)=0,xM,

  • (iii)

    V is analytic with respect to z in the Cα(M) topology,

where Cα(M) is the space of Hölder continuous complex-valued functions with exponent α. By analyticity with respect to zC we mean that the following limit exists in the Cα(M) topology,zV(x,z):=limh0V(x,z+h)V(x,z)h. As a result of analyticity, the function V admits a power series representation in the Cα(M) topology given by the expressionV(x,z)=k=1Vk(x)zkk!, where Vk(x):=zkV(x,0)Cα(M). We additionally impose the following conditions on the set of admissible functions V(x,z):
  • (iv)

    0 is not a Dirichlet eigenvalue for the operator Δg+V1(x) on (M,g).

Here, Δg denotes the Laplace-Beltrami operator on (M,g) given in local coordinates by the expression Δg=j,k=1n1gxj(ggjkxk).

In this paper, we consider the semi-linear elliptic equation{Δgu+V(x,u)=0,xMu=fBr0α(M),xM where Br0α(M):={hC2,α(M)|hC2,α(M)r0}. In Section 2.1, we show that, given fixed r0,r1>0 sufficiently small, equation (2) admits a unique solution uBr1α(M). Moreover, there exists a constant C>0 depending only on r0,r1 such thatuC2,α(M)CfC2,α(M)fBr0α(M). We subsequently define the Dirichlet-to-Neumann (DN) map, ΛV, for equation (2) through the expressionC1,α(M)ΛVf:=νu|M,fBr0α(M), where ν denotes the unit outward normal vector field on M. This paper is concerned with the following question.

Question 1

Given the map ΛV, can one uniquely determine the function V?

We will briefly review the history related to inverse problems for non-linear elliptic equations in Section 1.2. For now, let us recall some facts about the case where V(x,z)V1(x)z. In this case, the problem reduces to a version of the Calderón conjecture [2]. This formulation of the conjecture has been extensively studied but remains open in general geometries (M,g) with dimension n3. Uniqueness of the coefficient V1 has been proved for analytic metrics with an analytic function V1 [29], the Euclidean metric [33], [42] and the hyperbolic metric [19]. Beyond these cases, the most general uniqueness result is obtained in the so-called conformally transversally anisotropic (CTA) geometries defined as follows.

Definition 1

Let (M,g) be a compact oriented smooth Riemannian manifold with smooth boundary and dimension n. We say that (M,g) is conformally transversally anisotropic, if n3 and the following embedding holds:MIint×Mintandg(x0,x)=c(x0,x)((dx0)2g(x)), where I is a finite interval, c(x0,x)>0 is a smooth function and (M,g) is a smooth compact orientable manifold of dimension n1 with a smooth boundary ∂M.

In [7] it was proved that in the linear case V(x,z)=V1(x)z, the Dirichlet-to-Neumann map ΛV uniquely determines a bounded function V1, under the strong assumption that the transversal manifold is simple, that is to say (M,g) has a strictly convex boundary and given any two points in M there exists a unique geodesic connecting them. This result was subsequently strengthened in [8] where the authors showed that ΛV1 uniquely determines V1, if the geodesic ray transform is injective on the transversal manifold. The inversion of the geodesic ray transform is open in general, and has only been proved under certain geometrical assumptions, see for example the discussion in [8, Section 1]. For a broad review of the Calderón conjecture, and alternative formulations with the presence of non-linear coefficients, we refer the reader to survey articles [43], [44].

Let us return to Question 1. We will consider only the case where (M,g) is a CTA manifold. Before stating our results let us briefly review some notations for geodesic dynamics on (M,g). Let SMTM denote the unit sphere bundle on (M,g) and γ(,x,θ) be the unit speed geodesic with initial data (x,θ). For all (x,θ)SMint, we define the exit timesτ±=sup{r>0|γ(±r;x,θ)M,γ˙(±r;x,θ)TM}, and subsequently call a geodesic γ to be maximal, if and only if τ±<. Next, we define an admissibility condition on the transversal manifold (M,g) as follows.

Definition 2

Let (M,g) be a smooth compact Riemannian manifold with boundary. We say that (M,g) is admissible if there exists a dense set of points TM such that given any point pT there exists a non-self-intersecting maximal geodesic γ through p that contains no conjugate points to p.

The first result in this paper can now be stated as follows.

Theorem 1

Let (M,g) be a CTA manifold such that the transversal manifold M is admissible. Suppose that V(x,z) satisfies conditions (i)–(iv), that V1 is smooth and that V1,V2 are a priori known. Then, the Dirchlet-to-Neumann map ΛV uniquely determines the function V.

The proof of this theorem relies on a multiple-fold linearization of (2) that results in the interaction of the so called complex geometric optic solutions for the corresponding linearized equation. Since V1 is assumed to be known, the complex geometric optic solutions will be known as well. The smoothness assumption on V1 is imposed in order to make these solutions smooth and also to simplify the task of proving suitable decay rates (see Proposition 5). Under the assumption that V2 is assumed to be known, the non-linear interaction of the complex geometric optic solutions will result in a weighted ray transform along geodesics on the transversal manifold M. This weighted transform will be shown to be invertible along a single geodesic (see Proposition 4).

Our second main result is concerned with the recovery of the function V without imposing the assumption that the coefficient V2 is known, in the cases where the manifold is three or four dimensional.

Theorem 2

Let (M,g) be a three or four dimensional CTA manifold such that given any point on the transversal manifold M there exists a maximal non-self-intersecting geodesic without conjugate points through that point. Suppose that V(x,z) satisfies conditions (i)-(iv) and that V1 is a priori known and smooth. Then the Dirichlet-to-Neumann map ΛV uniquely determines the function V.

The proof of this theorem mostly follows the same technique as the previous theorem. However, due to the weaker assumption on the coefficient V, namely that V2 is unknown, the non-linear interaction of the complex geometric optic solutions results in a different ray transform along geodesics on the transversal manifold M. The inversion of this transform along a single geodesic is proved when the transversal manifold is two or three dimensional and left open in higher dimensions (see Proposition 3). We also refer the reader to Remark 1 in Section 3 where the restriction to three and four dimensions is discussed further.

The study of non-linear partial differential equations is an interesting topic in its own right, due to the complexity of the subject matter and as such, the corresponding inverse problems also carry significant mathematical interest. However, let us point out that there are applications for these inverse problems outside the realm of mathematics as well. Indeed, a large class of inverse problems for elliptic nonlinear equations can be seen as the study of stationary solutions to nonlinear equations describing physical phenomena. For example, we mention the nonlinear Schrödinger equation that arises as nonlinear variations of the classical field equations and has applications in the study of nonlinear optical fibers, planar wave guides and Bose Einstein condensates [30]. Other examples include nonlinear Klein-Gordon or Sine-Gordon equations with applications to the study of general relativity [34] and relativistic super-fluidity [46] respectively.

The majority of the literature dealing with inverse problems for non-linear elliptic equations is in the Euclidean geometry. The first uniqueness result was obtained by Isakov and Sylvester in [18] where the authors considered a Euclidean domain of dimension greater than or equal to three with non linear functions V(x,u) that satisfy the homogeneity property (ii), and showed that under a monotonicity condition for V and suitable bounds on V, uV and u2V, the non-linearity can be uniquely recovered on a specific subset of M×R. There, it was also proved that under a stronger bound on V, it could be recovered everywhere. Removing the homogeneity property (ii) introduces a natural gauge for the uniqueness of the non-linearity. This was studied by Sun in [41] under similar smoothness and monotonicity assumptions. There, a similar uniqueness result as in [18] was proved (up to the natural gauge), under the additional assumption that a common solution exists.

In dimension two, the problem was first solved by Sylvester and Nachman in [17], where the authors considered a domain in two-dimensional Euclidean space with a Carathéodory type non-linearity that has a continuous bounded Lp-valued derivative in the u variable and proved unique recovery of the non-linearity. In [32] uniqueness is proved for yet another family of admissible non-linearities in two dimensional Euclidean domains. There, a connection is also made between the theoretical study of these types of semi-linear inverse problems and the physical study of semi-conductor devices and ion channels. We also mention the work of Imanuvilov and Yamamoto in [13] where the authors considered the partial data problem for the operator Δu+q(x)u+V(x,u) on arbitrary open subsets of the boundary in two dimensions. There it was shown that if V(x,0)=uV(x,0)=0, it is possible to uniquely recover q everywhere and also that it is possible to recover V in certain subsets of the domain, under suitable bounds on the non-linear function V.

Aside from the study of inverse problems for semi-linear equations in Euclidean geometries, let us also mention that there are several works related to inverse problems for quasi-linear elliptic equations (see for example [3], [9], [16], [31], [38], [39], [40]). It should be emphasized that the key idea in all of these results has been a linearization technique introduced by Isakov in [14] in the context of semi-linear parabolic equations and developed further in [15], [17], [18], [38], [39]. This linearization technique together with the uniqueness results for the Calderón conjecture in Euclidean domains leads to the unique recovery of the non-linear terms.

The main novelty of this paper is to extend uniqueness results for non-linear elliptic equations to a wider class of Riemannian manifolds, known as conformally transversally anisotropic manifolds (see Definition 1). We consider local solutions about the trivial solution, but our proof is based on a multiple-fold linearization technique that differs from most of the previously mentioned works. As already discussed, the results in the Euclidean setting rely on the fact that uniqueness holds for the linearized inverse problem. This is no longer the case when M is assumed to be conformally transversally anisotropic. Indeed, uniqueness results for the linearized problem rely on injectivity of the geodesic ray transform on (M,g) that is known to be true under strong geometric assumptions such as simplicity of the transversal manifold (M,g) or existence of a strictly convex foliation [45]. The strength of our results lies in removing such strong geometric assumptions. On the other hand, contrary to the Euclidean cases, the results here assume analyticity of V(x,u) with respect to u.

The multiple-fold linearization technique in this paper is inspired by the study of similar types of non-linear problems for hyperbolic equations that was developed by Kurylev, Lassas and Uhlmann in [24], [25] in the context of Einstein scalar field equations and used in subsequent works in the context of semi-linear wave equations (see for example [4], [11], [27], [28], [47]). However, these works are based on the study of propagation of singularities for linear wave equations and the non-linear interactions of these singularities, making it difficult to apply them to an elliptic problem. Another key difference with all previous works in the hyperbolic setting is that we study non-linear interaction of localized solutions that correspond to a single geodesic. This will lead us to the study of a weighted transform along geodesics that we call the Jacobi ray transforms of the first and second kind. We show that it is possible to invert these transforms along a single geodesic (see Proposition 3, Proposition 4).

We conclude this introductory section by remarking that while writing this paper we became aware of an upcoming preprint by Matti Lassas, Tony Liimatainen, Yi-Hsuan Lin and Mikko Salo, which simultaneously and independently proves a similar result. We agreed to post our respective preprints to arXiv at the same time. See [26] for their preprint.

This paper is organized as follows. Section 2 is concerned with some preliminary discussions. We show that the Dirichlet-to-Neumann map ΛV (see (4)) is well-defined. We also discuss the linearization method for solutions to equation (2) near the trivial solution, in particular showing the appearance of what we call a nonlinear interaction. The rest of Section 2 is concerned with some lemmas and notations that will be needed throughout the paper. In Section 3 we define the Jacobi weighted transform of the first and second kind along a fixed geodesic, and subsequently prove injectivity results for these two transforms, see Proposition 3, Proposition 4. Section 4 starts with a review of the well-known Gaussian quasi modes for the linearized operator following [8]. In the remainder of this section we use this construction, together with a Carleman estimate to produce a family of complex geometric optic solutions for the linearized operator. In Section 5, we use an induction argument, based on the application of our linearization technique near the complex geometric optic solutions, to complete the proof of Theorem 1. Section 6 is concerned with the proof of Theorem 2.

Section snippets

Direct problem

In this section we prove the following proposition for the direct problem (2).

Proposition 1

r0,r1>0 depending on (M,g), such that equation (2) admits a unique solution uBr1α(M). Moreover, there holdsuC2,α(M)CfC2,α(M),fBr0α(M), for some constant C that depends on (M,g), r0 and r1.

Let us define the Schrödinger operator PV1=Δg+V1(x), and consider the linear equation{PV1u=F,xMu=fxM where (f,F)C2,α(M)×Cα(M). We introduce the solution operators GV1D,GV1S so that the function GV1Df is the unique

The Jacobi weighted ray transform

This section is concerned with the introduction of a geometrical data related to the transversal manifold (M,g) that will appear later in the proof of Theorem 1, Theorem 2. Before proceeding, let us introduce some notation, following [5, Section 1.2]. Given a maximal unit speed geodesic γ(t)M with t[τ,τ+], we define the orthogonal complement, γ˙(t), at the point γ(t) as the setγ˙(t):={vTγ(t)M|g(γ˙(t),v)=0}. We also define the (1,1)-tensor Πγ(t)=Πij(t)xjdxi to be the projection from Tγ(t

Complex geometric optics

The main aim of this section is to construct a pair of so called complex geometric optics solutions Uρ±, withρ=λ+iσ,λ>λ0>0, for the equationPV1Uρ±=0on T=I×M. Here, we have smoothly extended the known function V1 from M to the larger set T=I×M such that V1Cc(T). Recall from Section 2 that we have assumed without loss of generality that c1. We construct solutions that take the formUρ±(x)=e±λx0(eiσx0Vρ±(x0,x)+Rρ±(x0,x)). Here, the functions Vρ± are directly related to Gaussian quasi modes for

Proof of Theorem 1

This section is concerned with the proof of Theorem 1. The proof will be built on an induction argument based on m, where m is the order of the linearization method discussed in Section 2.2. As the first step of induction and also to shed some light on the methodology, we start with a proposition.

Proposition 6

Let the assumptions of Theorem 1 hold. Then the DN map ΛV uniquely determines the function V3(x).

Proof

We start by choosing a point pT and choose γ to be an admissible geodesic passing through p in the

Proof of Theorem 2

Proof of Theorem 2

It suffices to prove that V2 can be uniquely reconstructed from the knowledge of ΛV. Indeed, the recovery of the higher order derivatives Vm with m=3,4,, follows by using induction as in the proof of Theorem 1.

We start by choosing an arbitrary point pM and assume that γ is a maximal non-self intersecting geodesic passing through p. Let ρ=λ+iσ with σ fixed and construct a family of complex geometric optic solutions Uρ± and U2ρ± as in Section 4 with a decay rate given by an integer sn6 where

Acknowledgements

A.F was supported by EPSRC grant EP/P01593X/1. L.O acknowledges support from EPSRC grants EP/R002207/1 and EP/P01593X/1.

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