arXiv research

Uniqueness proof for Calderón's problem on real-analytic vector bundles.

We solve the partial data Calderón problem for the connection Laplacian on Riemann surfaces.

The Stone-Weierstrass theorem aids in solving inverse problems on specific manifolds.

We outline an approach to the inverse problem of Calderón that highlights the role of microlocal normal forms and propagation of singularities and extends a number of earlier results also in the anisotropic case. The main result states that from the boundary measurements it is possible to recover integrals of the unkno…

Researchers solve a nonlocal parabolic equation on manifolds using source-to-solution maps.

Study improves Calderón problem for systems, uniquely determining connections and potentials.

Study magnetic perturbations in Riemannian and Lorentzian Calderón problems.

Researchers solve the Calderón problem for fractional Dirac operators.

Method solves Calderón problem for surfaces near disks.

Study adiabatic limits of Calderon projector on manifolds with cylindrical ends.

Two different spacetimes can mimic each other's boundary measurements.

Researchers solve a formally determined inverse problem in Lorentzian geometry.

Establishes Calderón-Zygmund inequalities on evolving Riemannian manifolds.

Sharp threshold found for metric uniqueness in Riemannian Calderón-type problems.

Being motivated by the problem of deducing $L^p$-bounds on the second fundamental form of an isometric immersion from $L^p$-bounds on its mean curvature vector field, we prove a (nonlinear) Calderón-Zygmund inequality for maps between complete (possibly noncompact) Riemannian manifolds.

We consider a conformally invariant version of the Calderón problem, where the objective is to determine the conformal class of a Riemannian manifold with boundary from the Dirichlet-to-Neumann map for the conformal Laplacian. The main result states that a locally conformally real-analytic manifold in dimensions $\geq …

In this note we show that on any compact subdomain of a Kähler manifold that admits sufficiently many global holomorphic functions, the products of harmonic functions form a complete set. This gives a positive answer to the linearized anisotropic Calderón problem on a class of complex manifolds that includes compact su…

Anisotropic metric on manifolds uniquely determined by boundary data.

We consider Calderon's inverse problem with partial data in dimensions $n \geq 3$. If the inaccessible part of the boundary satisfies a (conformal) flatness condition in one direction, we show that this problem reduces to the invertibility of a broken geodesic ray transform. In Euclidean space, sets satisfying the flat…

Comment on a theorem about surface determination from map data.

Two-dimensional Riemannian manifolds uniquely determined by boundary data.

Study on determining metrics via Dirichlet-to-Neumann map for harmonic maps.

In this article we study the linearized anisotropic Calderon problem. In a compact manifold with boundary, this problem amounts to showing that products of harmonic functions form a complete set. Assuming that the manifold is transversally anisotropic, we show that the boundary measurements determine an FBI type transf…

The paper addresses geometric analysis on non-compact Riemannian manifolds, proving Calderón-Zygmund inequalities.

Derives properties of heat kernel for Rumin complex on Heisenberg groups.

We consider the anisotropic Calderon problem of recovering a conductivity matrix or a Riemannian metric from electrical boundary measurements in three and higher dimensions. In the earlier work \cite{DKSaU}, it was shown that a metric in a fixed conformal class is uniquely determined by boundary measurements under two …

We prove uniqueness results for a Calderon type inverse problem for the Hodge Laplacian acting on graded forms on certain manifolds in three dimensions. In particular, we show that partial measurements of the relative-to-absolute or absolute-to-relative boundary value maps uniquely determine a zeroth order potential. T…

We introduce a method for solving Calderón type inverse problems for semilinear equations with power type nonlinearities. The method is based on higher order linearizations, and it allows one to solve inverse problems for certain nonlinear equations in cases where the solution for a corresponding linear equation is not…

New example shows non-compact manifolds can lack $L^p$-Calderón-Zygmund inequalities.

Calderón projector extended to fibred cusp operators.

Inverse scattering result on AH manifolds determines metric up to diffeo and conformal factor.

The study shows that close hypersurfaces have uniformly bounded inequalities.

This paper solves a Calderón problem for Beltrami fields on manifolds.

We construct the Calderon projection on the space of Cauchy datas for a twisted Dirac operator in the Mischenko--Fomenko pseudodifferential calculus for operators acting on bundles of finitely generated $C^*$--Hilbert modules on a compact manifold with boundary. In particular an invertible double is constructed general…

Based on a construction due to B. Güneysu and S. Pigola (\textit{Adv. Math.} \textbf{281} (2015), pp.353--393), for each $p \in [1,\infty]$ and $m \in \mathbb{Z}_{\geq 2}$, we exhibit an $m$-dimensional Riemannian open manifold $\mathcal{M}$ on which the $L^p$-Calderón--Zygmund estimate \begin{equation*} \|\nabla \nabl…

The paper shows connections can be uniquely determined by their boundary data.

We consider an arbitrary linear elliptic first--order differential operator A with smooth coefficients acting between sections of complex vector bundles E,F over a compact smooth manifold M with smooth boundary N. We describe the analytic and topological properties of A in a collar neighborhood U of N and analyze vario…

The existence and continuity for the Calderon projector of the perturbed odd signature operator on a 3-manifold is established. As an application we give a new proof of a result of Taubes relating the mod 2 spectral flow of a family of operators on a homology 3-sphere with the difference in local intersection numbers o…

We introduce the concept of Calderón-Zygmund inequalities on Riemannian manifolds. For $1<p<\infty$, these are inequalities of the form $\left\Vert \mathrm{Hess}\left( u\right) \right\Vert _{L^p}\leq C_{1}\left\Vert u\right\Vert _{L^p}+C_{2}\left\Vert Δu\right\Vert _{L^p},$ valid a priori for all smooth functions $…

First, we review the Dirac operator folklore about basic analytic and geometrical properties of operators of Dirac type on compact manifolds with smooth boundary and on closed partitioned manifolds and show how these properties depend on the construction of a canonical invertible double and are related to the concept o…

The paper explains how microlocal analysis solves geometric inverse problems.

For a Dirac operator $D_{\bar{g}}$ over a spin compact Riemannian manifold with boundary $(\bar{X},\bar{g})$, we give a natural construction of the Calderón projector and of the associated Bergman projector on the space of harmonic spinors on $\bar{X}$, and we analyze their Schwartz kernels. Our approach is based on th…

The paper studies elliptic operators on manifolds with boundary.

We reconstruct a Riemannian manifold and a Hermitian vector bundle with compatible connection from the hyperbolic Dirichlet-to-Neumann operator associated with the wave equation of the connection Laplacian. The boundary data is local and the reconstruction is up to the natural gauge transformations of the problem. As a…

The study bounds Riesz transforms on manifolds with controlled curvature.

Continuous family of elliptic operators' projections maintain Cauchy data spaces.

Survey of global Calderón-Zygmund inequalities on Riemannian manifolds.

We reduce boundary determination of an unknown function and its normal derivatives from the (possibly weighted and attenuated) broken ray data to the injectivity of certain geodesic ray transforms on the boundary. For determination of the values of the function itself we obtain the usual geodesic ray transform, but for…