An inverse problem for a semi-linear elliptic equation in Riemannian geometries
Introduction
Let be a smooth compact Riemannian manifold with a smooth boundary and . Let and consider an a priori unknown function . We make the following standing assumptions.
- (i)
,
- (ii)
,
- (iii)
V is analytic with respect to z in the topology,
- (iv)
0 is not a Dirichlet eigenvalue for the operator on .
In this paper, we consider the semi-linear elliptic equation where . In Section 2.1, we show that, given fixed sufficiently small, equation (2) admits a unique solution . Moreover, there exists a constant depending only on such that We subsequently define the Dirichlet-to-Neumann (DN) map, , for equation (2) through the expression where ν denotes the unit outward normal vector field on . This paper is concerned with the following question. Question 1 Given the map , can one uniquely determine the function V? Definition 1 Let be a compact oriented smooth Riemannian manifold with smooth boundary and dimension n. We say that is conformally transversally anisotropic, if and the following embedding holds: where I is a finite interval, is a smooth function and is a smooth compact orientable manifold of dimension with a smooth boundary ∂M.
In [7] it was proved that in the linear case , the Dirichlet-to-Neumann map uniquely determines a bounded function , under the strong assumption that the transversal manifold is simple, that is to say 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 uniquely determines , 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 is a CTA manifold. Before stating our results let us briefly review some notations for geodesic dynamics on . Let denote the unit sphere bundle on and be the unit speed geodesic with initial data . For all , we define the exit times and subsequently call a geodesic γ to be maximal, if and only if . Next, we define an admissibility condition on the transversal manifold as follows. Definition 2 Let be a smooth compact Riemannian manifold with boundary. We say that is admissible if there exists a dense set of points such that given any point there exists a non-self-intersecting maximal geodesic γ through p that contains no conjugate points to p. Theorem 1 Let be a CTA manifold such that the transversal manifold M is admissible. Suppose that satisfies conditions (i)–(iv), that is smooth and that are a priori known. Then, the Dirchlet-to-Neumann map 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 is assumed to be known, the complex geometric optic solutions will be known as well. The smoothness assumption on 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 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 is known, in the cases where the manifold is three or four dimensional. Theorem 2 Let 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 satisfies conditions (i)-(iv) and that is a priori known and smooth. Then the Dirichlet-to-Neumann map uniquely determines the function V.
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 that satisfy the homogeneity property (ii), and showed that under a monotonicity condition for V and suitable bounds on V, and , the non-linearity can be uniquely recovered on a specific subset of . 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 -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 on arbitrary open subsets of the boundary in two dimensions. There it was shown that if , 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 is assumed to be conformally transversally anisotropic. Indeed, uniqueness results for the linearized problem rely on injectivity of the geodesic ray transform on that is known to be true under strong geometric assumptions such as simplicity of the transversal manifold 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 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 (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 depending on , such that equation (2) admits a unique solution . Moreover, there holds for some constant C that depends on , and .
Let us define the Schrödinger operator , and consider the linear equation where . We introduce the solution operators so that the function is the unique
The Jacobi weighted ray transform
This section is concerned with the introduction of a geometrical data related to the transversal manifold 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 with , we define the orthogonal complement, , at the point as the set We also define the -tensor to be the projection from
Complex geometric optics
The main aim of this section is to construct a pair of so called complex geometric optics solutions , with for the equation Here, we have smoothly extended the known function from to the larger set such that . Recall from Section 2 that we have assumed without loss of generality that . We construct solutions that take the form Here, the functions 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 uniquely determines the function .
Proof We start by choosing a point 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 can be uniquely reconstructed from the knowledge of . Indeed, the recovery of the higher order derivatives with , follows by using induction as in the proof of Theorem 1. We start by choosing an arbitrary point and assume that γ is a maximal non-self intersecting geodesic passing through p. Let with σ fixed and construct a family of complex geometric optic solutions and as in Section 4 with a decay rate given by an integer 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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