arXiv subjects

Tony Liimatainen

Publications and source records attributed to Tony Liimatainen.

At least 37 records · Page 2Linked to original sources

arXiv2022

An Inverse Problem for the Relativistic Boltzmann Equation

We consider an inverse problem for the Boltzmann equation on a globally hyperbolic Lorentzian spacetime $(M,g)$ with an unknown metric $g$. We consider measurements done in a neighbourhood $V\subset M$ of a timelike path $μ$ that connects a point $x^-$ to a point $x^+$. The measurements are modelled by a source-to-solution map, which maps a source supported in $V$ to the restriction of the solution to the Boltzmann equation to the set $V$. We show that the source-to-solution map uniquely determines the Lorentzian spacetime, up to an isometry, in the set $I^+(x^-)\cap I^-(x^+)\subset M$. The set $I^+(x^-)\cap I^-(x^+)$ is the intersection of the future of the point $x^-$ and the past of the point $x^+$, and hence is the maximal set to where causal signals sent from $x^-$ can propagate and return to the point $x^+$. The proof of the result is based on using the nonlinearity of the Boltzmann equation as a beneficial feature for solving the inverse problem.

math.AP↗

arXiv2022

An inverse problem for the Riemannian minimal surface equation

In this paper we consider determining a minimal surface embedded in a Riemannian manifold $Σ\times \mathbb{R}$. We show that if $Σ$ is a two dimensional Riemannian manifold with boundary, then the knowledge of the associated Dirichlet-to-Neumann map for the minimal surface equation determine $Σ$ up to an isometry.

math.AP↗

arXiv2022

An inverse problem for a semi-linear wave equation: a numerical study

We consider an inverse problem of recovering a potential associated to a semi-linear wave equation with a quadratic nonlinearity in $1 + 1$ dimensions. We develop a numerical scheme to determine the potential from a noisy Dirichlet-to-Neumann map on the lateral boundary. The scheme is based on the recent higher order linearization method [20]. We also present an approach to numerically estimating two-dimensional derivatives of noisy data via Tikhonov regularization. The methods are tested using synthetic noisy measurements of the Dirichlet-to-Neumann map. Various examples of reconstructions of the potential functions are given.

math.AP↗

arXiv2021

An inverse problem for a semilinear elliptic equation on conformally transversally anisotropic manifolds

Given a conformally transversally anisotropic manifold $(M,g)$, we consider the semilinear elliptic equation $$(-Δ_{g}+V)u+qu^2=0\quad \text{on $M$}.$$ We show that an a priori unknown smooth function $q$ can be uniquely determined from the knowledge of the Dirichlet-to-Neumann map associated to the semilinear elliptic equation. This extends the previously known results of the works [FO20, LLLS21a]. Our proof is based on analyzing higher order linearizations of the semilinear equation with non-vanishing boundary traces and also the study of interactions of two or more products of the so-called Gaussian quasimode solutions to the linearized equation.

math.AP↗

arXiv2021

Stability estimates for inverse problems for semi-linear wave equations on Lorentzian manifolds

This paper concerns an inverse boundary value problem of recovering a zeroth order time-dependent term of a semi-linear wave equation on a globally hyperbolic Lorentzian manifold. We show that an unknown potential $q$ in the non-linear wave equation $\square_g u +q u^m=0$, $m\geq 4$, can be recovered in a Hölder stable way from the Dirichlet-to-Neumann map. Our proof is based on the higher order linearization method and the use of Gaussian beams. Unlike some related works, we do not assume that the boundary is convex or that pairs of lightlike geodesics can intersect only once. For this, we introduce some general constructions in Lorentzian geometry. We expect these constructions to be applicable to studies of related problems as well.

math.AP↗

arXiv2021

Counterexamples to inverse problems for the wave equation

We construct counterexamples to inverse problems for the wave operator on domains in $\mathbb{R}^{n+1}$, $n \ge 2$, and on Lorentzian manifolds. We show that non-isometric Lorentzian metrics can lead to same partial data measurements, which are formulated in terms certain restrictions of the Dirichlet-to-Neumann map. The Lorentzian metrics giving counterexamples are time-dependent, but they are smooth and non-degenerate. On $\mathbb{R}^{n+1}$ the metrics are conformal to the Minkowski metric.

math.AP↗

arXiv2020

Inverse problems for elliptic equations with fractional power type nonlinearities

We study inverse problems for semilinear elliptic equations with fractional power type nonlinearities. Our arguments are based on the higher order linearization method, which helps us to solve inverse problems for certain nonlinear equations in cases where the solution for a corresponding linear equation is not known. By using a fractional order adaptation of this method, we show that the results of [LLLS20a, LLLS20b] remain valid for general power type nonlinearities.

math.AP↗

arXiv2020

Linearized Calderón problem and exponentially accurate quasimodes for analytic manifolds

In this article we study the linearized anisotropic Calderón problem on a compact Riemannian manifold with boundary. This problem amounts to showing that products of pairs of harmonic functions of the manifold form a complete set. We assume that the manifold is transversally anisotropic and that the transversal manifold is real analytic and satisfies a geometric condition related to the geometry of pairs of intersecting geodesics. In this case, we solve the linearized anisotropic Calderón problem. The geometric condition does not involve the injectivity of the geodesic X-ray transform. Crucial ingredients in the proof of our result are the construction of Gaussian beam quasimodes on the transversal manifold, with exponentially small errors, as well as the FBI transform characterization of the analytic wave front set.

math.AP↗

arXiv2020

Uniqueness and stability of an inverse problem for a semi-linear wave equation

We consider the recovery of a potential associated with a semi-linear wave equation on $\mathbb{R}^{n+1}$, $n\geq 1$. We show a Hölder stability estimate for the recovery of an unknown potential $a$ of the wave equation $\square u +a u^m=0$ from its Dirichlet-to-Neumann map. We show that an unknown potential $a(x,t)$, supported in $Ω\times[t_1,t_2]$, of the wave equation $\square u +a u^m=0$ can be recovered in a Hölder stable way from the map $u|_{\partial Ω\times [0,T]}\mapsto \langleψ,\partial_νu|_{\partial Ω\times [0,T]}\rangle_{L^2(\partial Ω\times [0,T])}$. This data is equivalent to the inner product of the Dirichlet-to-Neumann map with a measurement function $ψ$. We also prove similar stability result for the recovery of $a$ when there is noise added to the boundary data. The method we use is constructive and it is based on the higher order linearization. As a consequence, we also get a uniqueness result. We also give a detailed presentation of the forward problem for the equation $\square u +a u^m=0$.

math.AP↗

arXiv2019

Conformal harmonic coordinates

We study conformal harmonic coordinates on Riemannian manifolds. These are coordinates constructed as quotients of solutions to the conformal Laplace equation. We show their existence under general conditions. We find that conformal harmonic coordinates are a close conformal analogue of harmonic coordinates. We prove up to boundary regularity results for conformal mappings. We show that Weyl, Cotton, Bach, and Fefferman-Graham obstruction tensors become elliptic operators in conformal harmonic coordinates if one normalizes the determinant of the metric. We give a corresponding elliptic regularity result, which includes an analytic case. We prove a unique continuation result for local conformal flatness for Bach and obstruction flat manifolds. We discuss and prove existence of conformal harmonic coordinates on Lorentzian manifolds. We prove unique continuation results for conformal mappings both on Riemannian and Lorentzian manifolds.

math.DG↗

arXiv2019

Partial data inverse problems and simultaneous recovery of boundary and coefficients for semilinear elliptic equations

We study various partial data inverse boundary value problems for the semilinear elliptic equation $Δu+ a(x,u)=0$ in a domain in $\mathbb R^n$ by using the higher order linearization technique introduced in [LLS 19, FO19]. We show that the Dirichlet-to-Neumann map of the above equation determines the Taylor series of $a(x,z)$ at $z=0$ under general assumptions on $a(x,z)$. The determination of the Taylor series can be done in parallel with the detection of an unknown cavity inside the domain or an unknown part of the boundary of the domain. The method relies on the solution of the linearized partial data Calderón problem [FKSU09], and implies the solution of partial data problems for certain semilinear equations $Δu+ a(x,u) = 0$ also proved in [KU19]. The results that we prove are in contrast to the analogous inverse problems for the linear Schrödinger equation. There recovering an unknown cavity (or part of the boundary) and the potential simultaneously are long-standing open problems, and the solution to the Calderón problem with partial data is known only in special cases when $n \geq 3$.

math.AP↗

arXiv2019

Inverse problems for elliptic equations with power type nonlinearities

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 known. Assuming the knowledge of a nonlinear Dirichlet-to-Neumann map, we determine both a potential and a conformal manifold simultaneously in dimension $2$, and a potential on transversally anisotropic manifolds in dimensions $n \geq 3$. In the Euclidean case, we show that one can solve the Calderón problem for certain semilinear equations in a surprisingly simple way without using complex geometrical optics solutions.

math.AP↗

arXiv2018

The Poisson embedding approach to the Calderón problem

We introduce a new approach to the anisotropic Calderón problem, based on a map called Poisson embedding that identifies the points of a Riemannian manifold with distributions on its boundary. We give a new uniqueness result for a large class of Calderón type inverse problems for quasilinear equations in the real analytic case. The approach also leads to a new proof of the result by Lassas and Uhlmann (2001) solving the Calderón problem on real analytic Riemannian manifolds. The proof uses the Poisson embedding to determine the harmonic functions in the manifold up to a harmonic morphism. The method also involves various Runge approximation results for linear elliptic equations.

math.AP↗

arXiv2017

The linearized Calderon problem in transversally anisotropic geometries

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 transform at certain points in the transversal manifold. This leads to recovery of transversal singularities in the linearized problem. The method requires a geometric condition on the transversal manifold related to pairs of intersecting geodesics, but it does not involve the geodesic X-ray transform which has limited earlier results on this problem.

math.AP↗

arXiv2016

The Calderón problem for the conformal Laplacian

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 $\geq 3$ can be determined in this way, giving a positive answer to an earlier conjecture by Lassas and Uhlmann (2001). The proof proceeds as in the standard Calderón problem on a real-analytic Riemannian manifold, but new features appear due to the conformal structure. In particular, we introduce a new coordinate system that replaces harmonic coordinates when determining the conformal class in a neighborhood of the boundary.

math.AP↗

arXiv2016

$n$-harmonic coordinates and the regularity of conformal mappings

This article studies the smoothness of conformal mappings between two Riemannian manifolds whose metric tensors have limited regularity. We show that any bi-Lipschitz conformal mapping or $1$-quasiregular mapping between two manifolds with $C^r$ metric tensors ($r > 1$) is a $C^{r+1}$ conformal (local) diffeomorphism. This result was proved in [12, 27, 33], but we give a new proof of this fact. The proof is based on $n$-harmonic coordinates, a generalization of the standard harmonic coordinates that is particularly suited to studying conformal mappings. We establish the existence of a $p$-harmonic coordinate system for $1 < p < \infty$ on any Riemannian manifold.

math.DG↗

arXiv2016

Equivalence of quasiregular mappings on subRiemannian manifolds via the Popp extension

We show that all the common definitions of quasiregular mappings $f\colon M\to N$ between two equiregular subRiemannian manifolds of homogeneous dimension $Q\geq 2$ are quantitatively equivalent with precise dependences of the quasiregularity constants. As an immediate consequence, we obtain that if $f$ is $1$-quasiregular according to one of the definitions, then it is also $1$-quasiregular according to any other definition. In particular, this recovers a recent theorem of Capogna et al. on the equivalence of $1$-quasiconformal mappings. Our main results answer affirmatively a few open questions from the recent research. The main new ingredient in our proofs is the distortion estimates for particular local extensions of the horizontal metrics. These extensions are named "Popp extensions", and based on these extensions, we introduce a new natural and invariant definition of quasiregularity in the equiregular subRiemannian setting. The analysis on Popp extensions and on the implied distortion is also of independent interest.

math.DG↗

arXiv2015

p-harmonic coordinates for Hölder metrics and applications

We show that on any Riemannian manifold with Hölder continuous metric tensor, there exists a $p$-harmonic coordinate system near any point. When $p = n$ this leads to a useful gauge condition for regularity results in conformal geometry. As applications, we show that any conformal mapping between manifolds having $C^α$ metric tensors is $C^{1+α}$ regular, and that a manifold with $W^{1,n} \cap C^α$ metric tensor and with vanishing Weyl tensor is locally conformally flat if $n \geq 4$. The results extend the works [LS14, LS15] from the case of $C^{1+α}$ metrics to the Hölder continuous case. In an appendix, we also develop some regularity results for overdetermined elliptic systems in divergence form.

math.DG↗