arXiv subjects

Longyue Tao

Publications and source records attributed to Longyue Tao.

9 recordsLinked to original sources

arXiv2026

Quasianalyticity and geometric rigidity in anisotropic Calderón's problem

The anisotropic Calderón problem of determining a smooth Riemannian metric from boundary measurements, up to a boundary-fixing diffeomorphism, remains open in dimensions $n\ge3$~\cite{uhlmann2009electrical}. We establish unique identifiability results in two complementary regimes. In the first, the identity principle for quasianalytic functions propagates boundary information and yields unique identifiability on compact manifolds without a prescribed product structure, including a partial-data consequence; under a prescribed normal geometry, quasianalyticity is needed only in the distinguished direction. In the second, suitable symmetry or one-sided ordering assumptions lead to unique identifiability at $C^\infty$ regularity with full or restricted boundary access. Taken together, the results exhibit a tradeoff among regularity, geometric structure, and boundary access: quasianalyticity supplies continuation when no global product structure is prescribed, while symmetry or one-sided order replaces that continuation at $C^\infty$ regularity.

math.AP↗

arXiv2026

On an inverse boundary problem in elastodynamics

We study the simultaneous identification of the Lamé parameters $λ$ and $μ$ in three-dimensional static isotropic elasticity. For smooth coefficients, we establish uniqueness from the full Dirichlet-to-Neumann map when $μ$ is axisymmetric, admits a multiplicative or additive separation on a product domain, or is quasianalytic in one fixed direction. These results require neither smallness of $\nablaμ$ nor real analyticity of the coefficients, and impose no corresponding structural condition on $λ$. In the axisymmetric and separated cases, partial boundary data suffice to determine $μ$.

math.AP↗

arXiv2026

Determining Schrodinger Potentials on Riemannian Manifolds

We consider the inverse problem of determining a potential for the stationary Schrödinger equation on certain Riemannian manifolds with boundary from the Dirichlet-to-Neumann map. We derive two unique identifiability results. The first assumes that the Riemannian metric is close to the Euclidean metric, with the closeness quantified by a relative Fourier condition imposed on the potential function. The second assumes that the manifold is of product type and that the potential function is additively separable. Both the results and the technical arguments are new, contributing to the advancement of this challenging open inverse problem.

math.AP↗

arXiv2026

Exact high-order GPT cancellation in noncircular multilayers

This paper is concerned with exact high-order generalized polarization tensor (GPT) cancellation in two-dimensional noncircular multilayers with finite positive isotropic coating conductivities. We first prove that, for every admissible cyclically symmetric core--shell geometry, there exists a unique positive coating conductivity for which the structure is weakly neutral to every uniform incident field. For higher orders, we establish locally unique coating conductivities that cancel the complete $K$-contracted GPTs (K-CGPTs) block on a broad class of fixed noncircular multilayers with cyclic symmetry. This includes homothetic multilayers generated by strictly star-shaped symmetric domains, without assuming that the geometry is close to concentric disks. We also replace the near-background condition by a small-core condition covering finite, insulating, and perfectly conducting cores. By varying selected interface modes together with the coating conductivities near a radial GPT-vanishing structure, we further construct exact families with neither rotational nor reflection symmetry. For three coatings, the relevant nondegeneracy conditions and an explicit chiral example are rigorously certified. Finally, we show that the resulting cancellation yields enhanced near-cloaking for every entire harmonic incident field. A set of certified numerical examples fully corroborates the theoretical predictions established in this work.

math.AP↗

arXiv2026

Global recovery of Lorentzian principal geometry from the hyperbolic Dirichlet-to-Neumann map

We prove that the full Dirichlet-to-Neumann map on a finite time interval globally determines the Lorentzian metric $\mathbf G_{g,c}=-c(x,t)^2\mathrm{d}t^2+g(x)$ encoded by the principal symbol of the wave operator. The result holds in spatial dimensions $n\geq3$ for multiplicatively separable wave speeds $c(x,t)=a(x)b(t)$, provided the accumulated effective time during the experiment exceeds the maximal travel time from the boundary to the interior and back. The data determine both the unknown Riemannian metric $g(x)$ and the wave speed $c(x,t)$ throughout the observation cylinder, up to a spatial change of coordinates that fixes the boundary and leaves the measured time unchanged. In particular, the temporal factor $b$ is not prescribed but is recovered from the same measurements. We also construct several examples demonstrating that the dimensional and visibility conditions for uniqueness are sharp.

math.AP↗

arXiv2026

Inverse Scattering from Conformal Infinity for Totally Geodesic Defects in Hyperbolic Space

We study inverse scattering from conformal infinity for impenetrable topological defects whose interaction surfaces are totally geodesic in hyperbolic space $\mathbb H^n$, $n \geq 2$. For a fixed spectral parameter $λ_0 > 0$, the prescribed inputs are boundary labels $ξ\in \partial_\infty \mathbb H^n$. Each label selects an incoming Helgason mode in the hyperbolic interior. Given a defect $\mathcal P \Subset \mathbb H^n$, this mode generally fails to satisfy the homogeneous trace condition on the interaction surface and hence generates an outgoing correction. The leading coefficient of this correction at conformal infinity defines the measured far-field pattern. The central question is whether $\mathcal P$ can be recovered from the far-field patterns corresponding to one or finitely many prescribed boundary labels. This gives a formally determined inverse problem at one fixed spectral parameter. Our first main result establishes the unique determination of totally geodesic defects, an admissible class that includes both bulk components and hypersurface-supported ones. For Dirichlet-type defects, a single boundary label suffices. For Neumann-type defects, $n+1$ boundary labels are sufficient and in general necessary. These labels are required to satisfy the natural affine-independence condition at conformal infinity. Our second main result provides quantitative stability estimates within the same framework. The hyperbolic Hausdorff distance between two defects is controlled by the discrepancy of their far-field patterns at conformal infinity. The proof combines continuation from conformal infinity with quantitative geodesic reflection across totally geodesic hypersurfaces. Taken together, these results yield a qualitative and quantitative far-field inverse scattering theory at conformal infinity, based on formally determined data.

math.AP↗

arXiv2026

Determining evolutionary equations from a single passive boundary observation

We study inverse boundary problems for evolutionary PDEs using only a single passive boundary observation, where data from an unknown internal source propagate through an unknown medium without active inputs. The goal is the simultaneous recovery of coupled unknowns (sources and coefficients) from severely limited data. Unlike active methods with rich, structured inputs, passive observation poses two core challenges: minimal information and intrinsic coupling of multiple unknowns. Consequently, such problems remain largely open and unsystematically studied. We develop a unified framework based on integral identities, harmonic and microlocal analysis, and low-/high-frequency asymptotics. This approach yields the first systematic resolution for second-order hyperbolic, parabolic, and Schrödinger equations under a single coherent method. The key condition requires the measurement dataset's cardinality to exceed the unknowns' by at least one dimension, providing room to decouple unknowns and linearize the nonlinear inverse problem. Our unique identifiability results subsume all existing literature and cover more general configurations of practical interest. This framework complements classical theories and opens a promising new direction for future development.

math.AP↗

arXiv2026

Stably Determining a generalised Impedance Obstacle from a Single Far-Field Pattern

Inverse scattering focuses on recovering unknown scatterers from wave measurements. A fundamental challenge is determining whether an inverse obstacle problem can be resolved from a single far-field measurement, a task particularly demanding for non-convex polytope obstacles under generalized impedance boundary conditions and closely linked to the long-standing Schiffer problem. In this paper, we develop a novel \emph{Artificial Test Domain} (ATD) framework for single-measurement inverse scattering of impenetrable polytope obstacles. Based on microlocal analysis near exterior-visible flat boundary patches, this approach transcends traditional methods reliant on observable corners. The ATD framework establishes two primary conceptual advancements: a unified \emph{generalized impedance hyperplane (GIH) exclusion mechanism}, which clarifies the structural role of uniqueness mechanisms, and a unified \emph{qualitative--quantitative principle} for the generalized impedance setting. Quantitatively, the method yields a \emph{far-field--geometry relation} where geometric discrepancy is controlled by far-field error, scaled by a leading ATD coefficient. Qualitatively, the non-vanishing of this coefficient reduces to the exclusion of exterior generalized impedance hyperplanes. Once uniqueness is established, this relation produces sharp stability estimates. Within this framework, the classical stability estimates for the sound-soft and sound-hard cases are recovered as special instances of a much more general stability theory. At the same time, we obtain several new sharp stability results that are of significant importance. These results unify currently available single-measurement uniqueness regimes for polytope geometry and provide new insights into the Schiffer problem across multiple generalized impedance settings.

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arXiv2023

Stable determination of an impedance obstacle by a single far-field measurement

We establish sharp stability estimates of logarithmic type in determining an impedance obstacle in $\mathbb{R}^2$. The obstacle is of general polygonal shape and the impedance parameter can be variable. We establish the stability results by using a single far-field pattern, which constitutes a longstanding problem in the inverse scattering theory. This is the first stability result in the literature in determining an impedance obstacle by a single far-field measurement. If the obstacle is of a generally polygonal shape, the stability in determining the obstacle is established in terms of a modified Hausdorff distance and is independent of the boundary impedance parameter. If the obstacle is further known to be convex, the stability in simultaneously determining the obstacle and the boundary impedance is established in terms of the classical Hausdorff distance. There are several technical novelties and development in the mathematical strategy developed for establishing the aforementioned stability results. First, the stability analysis is conducted around a corner point in a micro-local manner. Second, our stability estimates establish explicit relationships among the geometric configurations of the obstacle and the vanishing order of the wave field at the corner point. Third, we develop novel error propagation techniques to tackle singularities of the wave field at a corner as well as to tackle the impedance boundary condition.

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