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Gunther Uhlmann

Publications and source records attributed to Gunther Uhlmann.

At least 19 recordsLinked to original sources
arXiv2026

Determination of Wave Speed from interior sources

We consider the wave equation with variable wave speed in Euclidean space, with point sources modeled by Dirac delta initial displacement data. There are three special subsets of the whole space: (1) the source set where the delta initial conditions are supported, (2) the unknown set where the wave speed is not known a priori, and (3) the receiver set where the waves are measured. The inverse problem is to reconstruct the wave speed uniquely in the unknown set from an unlabeled collection of waves generated by point sources and measured in the receiver set. We use propagation of singularities and sharp finite speed of propagation to reduce this data to geometric travel-time data, whose form depends on how the three sets lie in relation to each other. We give three scenarios where this procedure leads to unique determination of the wave speed.

math.AP↗
arXiv2026

The Calder\'on problem for near-Euclidean metrics

In this article, we consider the anisotropic Calder\'on problem of determining the potential from boundary measurements of the time-independent Schr\"odinger equation on compact Riemannian manifolds with boundary. We prove a global uniqueness theorem for small perturbations of the Euclidean metric.

math.AP↗
arXiv2026

The linearized minimal surfaces problem

We characterize the kernel of the linearization $R$ of the minimal surface problem about the Euclidean metric in a bounded smooth domain $\Omega\subset\mathbb{R}^n$, $n\ge2$, with the background minimal surfaces being the Euclidean planes. We show that, in the whole-space Euclidean decomposition, the kernel consists of potential fields and TT fields. For bounded domains, a similar phenomenon appears with additional boundary coupling conditions; in particular, the TT part may be coupled to a harmonic conformal component.

math.DG↗
arXiv2026

Inverse Problems for the Monge--Amp\`ere Equation: Linearization and Nonlinear Recovery

We study inverse boundary value problems for the nonlinear Monge--Amp\`ere equation \[ \det D^2u=a(x,u,\nabla u) \] in a bounded domain. We introduce a nonlinear Cauchy data set and investigate the recovery of the nonlinearity from boundary measurements. Linearizing around a strictly convex background solution, we establish local well-posedness, smooth dependence on boundary data, and a higher-order linearization framework. We show that the first variation of the nonlinear Cauchy data is governed by a linear elliptic operator whose principal coefficient is the cofactor matrix of the background Hessian. An Alessandrini-type identity then yields a reduction principle from the nonlinear inverse problem to an anisotropic Calder\'on-type inverse problem. As a consequence, under suitable uniqueness assumptions for the associated linear problem, the nonlinear Cauchy data determine the first-order derivatives of the nonlinearity along the background jet. Higher-order linearization identities provide recovery of higher derivatives and, under suitable density assumptions, determine the full Taylor expansion of the nonlinearity along the background solution. We also discuss applications to semilinear equations and nonlinearities arising in optimal transport.

math.AP↗
arXiv2026

The partial data Calder\'on problem in dimension three

We consider an inverse boundary value problem for the time-independent Schr\"odinger equation in dimension three. We prove that the local Dirichlet-to-Neumann map defined near a boundary point uniquely determines the potential in a neighborhood of the boundary point in the interior. In particular, we show that the uniqueness question can be reduced to the injectivity of a weighted X-ray transform, which links inverse boundary value problems to integral geometry.

math.AP↗
arXiv2026

An inverse problem for compressible Euler's equations

We consider an inverse problem for the compressible Euler's equations in polytropic fluid. We show that by taking active measurements near a particle trajectory one can determine the background flow in a set where pressure waves can propagate from and return to the particle trajectory, under the additional assumption that the flow has nonzero vorticity.

math.AP↗
arXiv2025

The Dirichlet-to-Neumann map on asymptotically anti-de Sitter spaces and holography

We consider the Klein-Gordon equation on asymptotically anti-de Sitter spacetimes, and show that the forward Dirichlet-to-Neumann map (or scattering matrix) is a fractional power of the boundary wave operator modulo lower order terms in the sense of paired Lagrangian distributions. We use it to show that, outside of a countable set of mass parameters, the Dirichlet-to-Neumann map determines the Taylor series of the bulk metric at the boundary, and hence allows the recovery of a real analytic metric or Einstein metric modulo isometries. Furthermore, we prove a Lorentzian version of the Graham-Zworski theorem relating poles of the Dirichlet-to-Neumann map to conformally invariant powers of the boundary wave operator.

math.AP↗
arXiv2025

Fractional anisotropic Calderón problem with external data

In this paper, we solve the fractional anisotropic Calderón problem with external data in the Euclidean space, in dimensions two and higher, for smooth Riemannian metrics that agree with the Euclidean metric outside a compact set. Specifically, we prove that the knowledge of the partial exterior Dirichlet--to--Neumann map for the fractional Laplace-Beltrami operator, given on arbitrary open nonempty sets in the exterior of the domain in the Euclidean space, determines the Riemannian metric up to diffeomorphism, fixing the exterior. We provide two proofs of this result: one relies on the heat semigroup representation of the fractional Laplacian and a pseudodifferential approach, while the other is based on a variable-coefficient elliptic extension interpretation of the fractional Laplacian.

math.AP↗
arXiv2024

Inverse Nonlinear Scattering by a Metric

We study the inverse problem of determining a time-dependent globally hyperbolic Lorentzian metric from the scattering operator for semilinear wave equations.

math.AP↗
arXiv2024

Partial data inverse problems for the nonlinear magnetic Schrödinger equation

In this paper, we study the partial data inverse problem for nonlinear magnetic Schrödinger equations. We show that the knowledge of the Dirichlet-to-Neumann map, measured on an arbitrary part of the boundary, determines the time-dependent linear coefficients, electric and magnetic potentials, and nonlinear coefficients, provided that the divergence of the magnetic potential is given. Additionally, we also investigate both the forward and inverse problems for the linear magnetic Schrödinger equation with a time-dependent leading term. In particular, all coefficients are uniquely recovered from boundary data.

math.AP↗
arXiv2024

On the anisotropic Calderón's problem

We prove that the Riemannian metric on a compact manifold of dimension $n\geq 3$ with smooth boundary can be uniquely determined, up to an isometry fixing the boundary, by the Dirichlet-to-Neumann map associated to the Laplace-Beltrami operator.

math.AP↗
arXiv2024

An Inverse Hyperbolic Problem with Application to Joint Photoacoustic Parameter Determination

We consider an inverse problem of recovering a parameter appearing in all levels in a second-order hyperbolic equation from a single boundary measurement. The model is motivated from applications in photoacoustic tomography when one seeks to recover both the wave speed and the initial ultrasound pressure from a single ultrasound signal. In particular, our result shows that the ratio of the initial ultrasound pressure and the wave speed squared uniquely determines both of them respectively.

math.AP↗
arXiv2024

Calder\'{o}n problem for fractional Schr\"{o}dinger operators on closed Riemannian manifolds

We study an analog of the anisotropic Calder\'on problem for fractional Schr\"odinger operators $(-\Delta_g)^\alpha + V$ with $\alpha \in (0,1)$ on closed Riemannian manifolds of dimensions two and higher. We prove that the knowledge of a Cauchy data set of solutions of the fractional Schr\"odinger equation, given on an open nonempty a priori known subset of the manifold determines both the Riemannian manifold up to an isometry and the potential up to the corresponding gauge transformation, under certain geometric assumptions on the manifold as well as the observation set. Our method of proof is based on: (i) studying a new variant of the Gel'fand inverse spectral problem without the normalization assumption on the energy of eigenfunctions, and (ii) the discovery of an entanglement principle for nonlocal equations involving two or more compactly supported functions. Our solution to (i) makes connections to antipodal sets as well as local control for eigenfunctions and quantum chaos, while (ii) requires sharp interpolation results for holomorphic functions. We believe that both of these results can find applications in other areas of inverse problems.

math.AP↗
arXiv2024

Partial data inverse problems for reaction-diffusion and heat equations

We study partial data inverse problems for linear and nonlinear parabolic equations with unknown time-dependent coefficients. In particular, we prove uniqueness results for partial data inverse problems for semilinear reaction-diffusion equations where Dirichlet boundary data and Neumann measurements of solutions are restricted to any open subset of the boundary. We also prove injectivity of the Fréchet derivative of the partial Dirichlet-to-Neumann map associated to heat equations. Our proof consists of two crucial ingredients; (i) we introduce an asymptotic family of spherical quasimodes that approximately solve heat equations modulo an exponentially decaying remainder term and (ii) the asymptotic study of a weighted Laplace transform of the unknown coefficient along a straight line segment in the domain where the weight may be viewed as a semiclassical symbol that itself depends on the complex-valued frequency. The latter analysis will rely on Phragmén-Lindelöf principle and Grönwall inequality.

math.AP↗
arXiv2024

Recovery of coefficients in semilinear transport equations

We consider the inverse problem for time-dependent semilinear transport equations. We show that time-independent coefficients of both the linear (absorption or scattering coefficients) and nonlinear terms can be uniquely determined, in a stable way, from the boundary measurements by applying a linearization scheme and Carleman estimates for the linear transport equations. We establish results in both Euclidean and general geometry settings.

math.AP↗
arXiv2024

Invertibility of local geodesic transverse and mixed ray transforms II: higher order tensors

Consider a compact Riemannian manifold in dimension $n$ with strictly convex boundary. We show the local invertibility near a boundary point of the transverse ray transform of $2$ tensors for $n\geq 3$ and the mixed ray transform of $2+2$ tensors for $n=3$. When the manifold admits a strictly convex function, this local invertibility result leads to global invertibility.

math.DG↗
arXiv2024

Determination of the density in a nonlinear elastic wave equation

This is a continuation of our study [Uhlmann-Zhai, JMPA, 2021] on an inverse boundary value problem for a nonlinear elastic wave equation. We prove that all the linear and nonlinear coefficients can be recovered from the displacement-to-traction map, including the density, under some natural geometric conditions on the wavespeeds.

math.AP↗