[Paper Review] A survey of non-uniqueness results for the anisotropic Calder{\'o}n problem with disjoint data

Thierry Daudé, Niky Kamran+1 more|arXiv (Cornell University)|Mar 1, 2018

Nonlinear Partial Differential Equations8 citations

TL;DR

This paper establishes new non-uniqueness results for the anisotropic Calderón problem with disjoint Dirichlet and Neumann data on Riemannian manifolds of dimension $n \geq 3$. It introduces a novel gauge invariance for the Dirichlet-to-Neumann map when data are measured on disjoint boundary subsets, and constructs explicit counterexamples via warped product metrics on cylindrical manifolds, showing that infinitely many distinct conformal metrics can yield identical partial DN maps at fixed frequency. The results hold modulo this new gauge symmetry, demonstrating fundamental limitations in uniqueness for inverse problems with partial boundary data.

ABSTRACT

After giving a general introduction to the main known results on the anisotropic Calder{\'o}n problem on n-dimensional compact Riemannian manifolds with boundary, we give a motivated review of some recent non-uniqueness results obtained in [5, 6] for the anisotropic Calder{\'o}n problem at fixed frequency, in dimension n $\ge$ 3, when the Dirichlet and Neumann data are measured on disjoint subsets of the boundary. These non-uniqueness results are of the following nature: given a smooth compact connected Riemannian manifold with boundary (M, g) of dimension n $\ge$ 3, we first show that there exist in the conformal class of g an infinite number of Riemannian metrics gmetrics metrics g such that their corresponding Dirichlet-to-Neumann maps at a fixed frequency coincide when the Dirichlet data $\Gamma$D and Neumann data $\Gamma$N are measured on disjoint sets and satisfy $\Gamma$D $\cup$ $\Gamma$N = $\partial$M. The corresponding conformal factors satisfy a nonlinear elliptic PDE of Yamabe type on (M, g) and arise from a natural but subtle gauge invariance of the Calder{\'o}n when the data are given on disjoint sets. We then present counterexamples to uniqueness in dimension n $\ge$ 3 to the anisotropic Calder{\'o}n problem at fixed frequency with data on disjoint sets, which do not arise from this gauge invariance. They are given by cylindrical Riemannian manifolds with boundary having two ends, equipped with a suitably chosen warped product metric. This survey concludes with some remarks on the case of manifolds with corners.

Motivation & Objective

• Address the fundamental challenge of non-uniqueness in the anisotropic Calderón problem when Dirichlet and Neumann data are measured on disjoint subsets of the boundary.
• Identify and formalize a new gauge invariance for the Dirichlet-to-Neumann map under disjoint data conditions, arising from a nonlinear elliptic PDE of Yamabe type.
• Construct explicit counterexamples to uniqueness in the anisotropic Calderón problem at fixed frequency, independent of the newly identified gauge invariance.
• Extend the non-uniqueness results to manifolds with corners by relaxing the smoothness assumption on the boundary, showing counterexamples persist even with connected boundaries.
• Provide a comprehensive survey of recent non-uniqueness results and their geometric underpinnings in the context of inverse problems on Riemannian manifolds.

Proposed method

• Introduce the partial Dirichlet-to-Neumann (DN) map $\Lambda_{g,\Gamma_D,\Gamma_N}(\lambda)$ for the Laplace-Beltrami operator on a compact Riemannian manifold with boundary, where $\Gamma_D$ and $\Gamma_N$ are disjoint open subsets of $\partial M$.
• Establish a new gauge invariance for the partial DN map under pullback by diffeomorphisms that fix $\Gamma_D \cup \Gamma_N$, and identify a conformal gauge invariance for $n \geq 3$ arising from a nonlinear Yamabe-type PDE for the conformal factor.
• Construct counterexamples using cylindrical manifolds $M = [0,1] \times K$ equipped with a warped product metric $g = dt^2 + a(t)^2 g_K$, where $a(t)$ is a smooth positive function.
• Use separation of variables in the spectral decomposition of the Laplacian on $K$ to reduce the problem to a family of one-dimensional boundary value problems parametrized by the Dirichlet spectrum of $-\Delta_K$.
• Apply the method of sub- and super-solutions to prove existence of smooth positive solutions to the nonlinear Dirichlet problem $\Delta_g w + (\lambda - V)w - \lambda w^{\frac{n+2}{n-2}} = 0$ on $M$, with $w = \eta$ on $\partial M$, where $\eta = 1$ on $\Gamma_D \cup \Gamma_N$.
• Show that distinct conformal factors $c$ and $\tilde{c}$, corresponding to different potentials $V$ and $\tilde{V}$, yield identical partial DN maps $\Lambda_{c^4g, \Gamma_D, \Gamma_N}(\lambda) = \Lambda_{\tilde{c}^4g, \Gamma_D, \Gamma_N}(\lambda)$, proving non-uniqueness modulo the new gauge invariance.

Experimental results

Research questions

• RQ1 Does the partial Dirichlet-to-Neumann map $\Lambda_{g,\Gamma_D,\Gamma_N}(\lambda)$ on a compact Riemannian manifold with $\Gamma_D \cap \Gamma_N = \emptyset$ determine the metric $g$ uniquely at a fixed frequency $\lambda$?
• RQ2 What geometric or analytic structure underlies the non-uniqueness of the anisotropic Calderón problem when data are measured on disjoint boundary subsets?
• RQ3 Can non-uniqueness arise independently of the standard diffeomorphism and conformal gauge invariances in the case of disjoint data?
• RQ4 How do the non-uniqueness results extend to manifolds with corners, where the boundary is connected but the manifold is not smooth?
• RQ5 Can explicit counterexamples be constructed using warped product metrics on cylindrical manifolds that yield identical partial DN maps for distinct conformal factors?

Key findings

• There exists an infinite family of conformal metrics $\tilde{g} = c^4 g$ on a cylindrical manifold $M = [0,1] \times K$ with $n \geq 3$, such that $\Lambda_{\tilde{g}, \Gamma_D, \Gamma_N}(\lambda) = \Lambda_{g, \Gamma_D, \Gamma_N}(\lambda)$ for disjoint $\Gamma_D, \Gamma_N \subset \partial M$, even though $\tilde{g} \neq g$.
• The non-uniqueness arises from a new gauge invariance tied to a nonlinear Yamabe-type PDE for the conformal factor, which is not captured by standard diffeomorphism or conformal invariance alone.
• Counterexamples to uniqueness exist that are not related by the new gauge invariance: distinct conformal factors $c$ and $\tilde{c}$ yield the same partial DN map $\Lambda_{c^4g, \Gamma_D, \Gamma_N}(\lambda)$, but correspond to different potentials $V = V_{g,c,\lambda}$ and $\tilde{V} = V_{g,\tilde{c},\lambda}$.
• Non-uniqueness persists in the case of manifolds with corners: even when the boundary $\partial M$ is connected (e.g., $M = [0,1] \times K$ with $K$ having boundary), counterexamples to uniqueness exist for the partial DN map with disjoint data.
• Existence of smooth positive solutions to the nonlinear Dirichlet problem $\Delta_g w + (\lambda - V)w - \lambda w^{\frac{n+2}{n-2}} = 0$ is proven via sub- and super-solution methods for $\lambda = 0$, $\lambda > 0$, and $\lambda < 0$, under appropriate conditions on $V$ and the boundary data $\eta$.