Inverse problems for the Bakry-Émery Laplacian on manifolds with boundary -- uniqueness and non-uniqueness

TL;DR

This work studies inverse boundary problems for the Bakry-Émery Laplacian $- abla_{ ext{E}} = -\Delta_g + g(dV,\cdot)$ on manifolds with boundary, focusing on how much boundary Taylor data of the metric $g$ and the weight $V$ can be recovered from Dirichlet-to-Neumann maps. It rewrites $- abla_{ ext{E}}$ as a gauge-invariant Schrödinger operator and performs a Lee–Uhlmann-type symbol analysis of two DN maps, $\Lambda^0$ (scalar) and $\Lambda^1$ (gauge-natural), via a boundary-normal-coordinate factorisation to relate boundary data to the DN maps. The main results show that $\Lambda^1$ determines boundary $g_{\alpha\beta}$ and $\partial_r g_{\alpha\beta}$ and $V$ up to an additive constant, while $\Lambda^0$ determines the boundary data of $V$ (and, with extra information such as known $\omega_g$ near the boundary, all radial derivatives of $V$); in the non-gauge setting, $V$ is uniquely determined in the real-analytic category. Collectively, the paper clarifies the extent to which geometry and measure decouple in these inverse problems and delineates when unique recovery is possible under different DN-map formulations and gauge choices.

Abstract

We study the questions of uniqueness and non-uniqueness for a pair of closely related inverse problems for the Bakry-Émery Laplacian $-Δ_{\mathcal E}$ on a smooth compact and oriented Riemannian manifold with boundary $(\overline{M},g)$, endowed with a volume form $\mathfrak{m}=e^{-V}ω_g$. These consist in recovering the Taylor coefficients of metric $g$ and weight $V$ along the boundary of $\overline{M}$ from the knowledge of a pair of operators that can be viewed as geometrically natural Dirichlet-to-Neumann maps associated to $-Δ_{\mathcal E}$.

Theorems & Definitions (18)

• Proposition 1
• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 1
• proof
• Remark
• Lemma 2
• Lemma 3
• Lemma 4