Stability in the Inverse Steklov Problem on Warped Product Riemannian Manifolds

Abstract

In this paper, we study the amount of information contained in the Steklov spectrum of some compact manifolds with connected boundary equipped with a warped product metric. Examples of such manifolds can be thought of as deformed balls in R d . We first prove that the Steklov spectrum determines uniquely the warping function of the metric. We show in fact that the approximate knowledge (in a given precise sense) of the Steklov spectrum is enough to determine uniquely the warping function in a neighbourhood of the boundary. Second, we provide stability estimates of logtype on the warping function from the Steklov spectrum. The key element of these stability results relies on a formula that, roughly speaking, connects the inverse data (the Steklov spectrum) to the Laplace transform of the difference of the two warping factors.

Stability estimates for an inverse Steklov problem in a class of hollow spheres

“…We would like to answer the following question : if the data of the Steklov spectrum is known up to some exponentially decreasing sequence, is it possible to recover the conformal factor f in the neigbourhood of the boundary (or one of its component) up to a natural gauge invariance ? The main difficulty that appears here is due to the presence of two sets of eigenvalues, in each spectrum σ Λ g (ω) and σ Λ g(ω) , instead of one as in [5]. With the previous definitions of closeness, it is not clear that we can get, for example, this kind of implication :…”

“…This work is based on ideas developped by Daudé, Kamran and Nicoleau in [5]. However, due to the specific structure of our model that possesses a disconnected boundary (contrary to the model studied in [5]), some new difficulties arise.…”

Uniqueness results in the inverse spectral Steklov problem

“…In this setting, we want to answer the following question : does the Steklov spectrum determine uniquely the warping function f (x)? This question has been answered positively in [5] when K = [0, +∞[×S n−1 equipped with the metric (4). The difference between this case and the one we are studying here is that the boundary of K is connected whereas that of M is made of two connected components.…”

“…has a simple interpretation involving the so-called Weyl-Titchmarsh theory associated to the Sturm-Liouville equation (5). More precisely, if we denote…”

“…where M (z), N (z) are the Weyl-Titchmarsh functions and ∆(z) is the characteristic function associated to (5). In particular, the trace (respectively the determinant) of these operators Λ m g (λ) are meromorphic functions evaluated in µ m .…”