Searcharxiv⌕ Search

arXiv subjects

Manas Kar

Publications and source records attributed to Manas Kar.

At least 19 recordsLinked to original sources
arXiv2024

Weighted Divergent Beam Ray Transform: Reconstruction, Unique continuation and Stability

In this article, we establish that any symmetric $m$-tensor field can be recovered pointwise from partial data of the $k$-th weighted divergent ray transform for any $k \in \mathbb{Z}^{+} \cup\{0\}$. Using the unique continuation property of the fractional Laplacian, we further prove the unique continuation of the fractional divergent beam ray transform for both vector fields and symmetric 2-tensor fields. Additionally, we derive explicit reconstruction formulas and stability results for vector fields and symmetric 2-tensor fields in terms of fractional divergent beam ray transform data. Finally, we conclude by proving a unique continuation result for the divergent beam ray transform for functions.

math.AP↗
arXiv2024

Normal operators for momentum ray transforms, II: Saint Venant operator

The momentum ray transform $I_m^k$ integrates a rank $m$ symmetric tensor field $f$ on ${\mathbb R}^n$ over lines with the weight $t^k$, $I_m^kf(x,\xi)=\int_{-\infty}^\infty t^k\langle f(x+t\xi),\xi^m\rangle\,\mathrm{d}t$. Let $N^k_m=(I^k_m)^*I^k_m$ be the normal operator of $I_m^k$. To what extent is a symmetric $m$-tensor field $f$ determined by the data $(N_m^0f,\dots,N_m^rf)$ given for some $0\le r\le m$? The Saint Venant operator $W^r_m$ is a linear differential operator of order $m-r$ with constant coefficients on the space of symmetric $m$-tensor fields. We derive an explicit formula expressing $W^r_mf$ in terms of $(N_m^0f,\dots,N_m^rf)$. The tensor field $W^r_mf$ represents the full local information on $f$ that can be extracted from the data $(N_m^0f,\dots,N_m^rf)$.

math.AP↗
arXiv2024

Normal operators for momentum ray transforms, I: The inversion formula

The momentum ray transform $I_m^k$ integrates a rank $m$ symmetric tensor field $f$ on $\mathbb R^n$ over lines with the weight $t^k$, $I_m^kf(x,\xi)=\int_{-\infty}^\infty t^k\langle f(x+t\xi),\xi^m\rangle\,\mathrm{d}t$. We compute the normal operator $N_m^k=(I_m^k){}^*I_m^k$ and present an inversion formula recovering a rank $m$ tensor field $f$ from the data $(N_m^0f,\dots,N_m^mf)$.

math.AP↗
arXiv2023

Loop group factorization method for the magnetic and thermostatic nonabelian ray transforms

We study the injectivity of the matrix attenuated and nonabelian ray transforms on compact surfaces with boundary for nontrapping $\lambda$-geodesic flows and the general linear group of invertible complex matrices. We generalize the loop group factorization argument of Paternain and Salo to reduce to the setting of the unitary group when $\lambda$ has the vertical Fourier degree at most $2$. This covers the magnetic and thermostatic flows as special cases. Our article settles the general injectivity question of the nonabelian ray transform for simple magnetic flows in combination with an earlier result by Ainsworth. We stress that the injectivity question in the unitary case for simple Gaussian thermostats remains open. Furthermore, we observe that the loop group argument does not apply when $\lambda$ has higher Fourier modes.

math.DG↗
arXiv2023

Electromagnetic waves generated by a dielectric moving at a constant speed

We consider a regular and bounded dielectric body moving at a speed $|V|$, following a constant vector field $V$, with respect to a reference frame. In this frame, the special relativity implies that the Maxwell system is derived through the constitutive equations linking the moving speed $|V|$ and the speed of light $c$ in the background medium (as the vacuum for instance). Based on this model, we derive the well-poseness of the related forward scattering problem in the natural regime where $\frac{|V|}{c}\leq C_{te}$ with an appropriate constant $C_{te} <1$ that we estimate. In particular, we show the invertibility of the related Lippmann-Schwinger system in this regime and state the corresponding Born series in terms of the ratio $\frac{|V|}{c}$. As an application, we state and show the unique identifiability of the inverse problem of detecting the dielectric body, without knowing the moving speed $|V|$ or $V$, by illuminating it with incident electromagnetic waves propagating at the speed $c$. Such identifiability result makes sense in the regime under consideration.

math.AP↗
arXiv2023

Broken ray transform for twisted geodesics on surfaces with a reflecting obstacle

We prove a uniqueness result for the broken ray transform acting on the sums of functions and $1$-forms on surfaces in the presence of an external force and a reflecting obstacle. We assume that the considered twisted geodesic flows have nonpositive curvature. The broken rays are generated from the twisted geodesic flows by the law of reflection on the boundary of a suitably convex obstacle. Our work generalizes recent results for the broken geodesic ray transform on surfaces to more general families of curves including the magnetic flows and Gaussian thermostats.

math.DG↗
arXiv2022

Determining coefficients for a fractional $p$-Laplace equation from exterior measurements

We consider an inverse problem of determining the coefficients of a fractional $p\,$-Laplace equation in the exterior domain. Assuming suitable local regularity of the coefficients in the exterior domain, we offer an explicit reconstruction formula in the region where the exterior measurements are performed. This formula is then used to establish a global uniqueness result for real-analytic coefficents. In addition, we also derive a stability estimate for the unique determination of the coefficients in the exterior measurement set.

math.AP↗
arXiv2022

The fractional $p\,$-biharmonic systems: optimal Poincar\'e constants, unique continuation and inverse problems

This article investigates nonlocal, fully nonlinear generalizations of the classical biharmonic operator $(-\Delta)^2$. These fractional $p$-biharmonic operators appear naturally in the variational characterization of the optimal fractional Poincar\'e constants in Bessel potential spaces. We study the following basic questions for anisotropic fractional $p$-biharmonic systems: existence and uniqueness of weak solutions to the associated interior source and exterior value problems, unique continuation properties (UCP), monotonicity relations, and inverse problems for the exterior Dirichlet-to-Neumann maps. Furthermore, we show the UCP for the fractional Laplacian in all Bessel potential spaces $H^{t,p}$ for any $t\in \mathbb{R}$, $1 \leq p < \infty$ and $s \in \mathbb{R}_+ \setminus \mathbb{N}$: If $u\in H^{t,p}(\mathbb{R}^n)$ satisfies $(-\Delta)^su=u=0$ in a nonempty open set $V$, then $u\equiv 0$ in $\mathbb{R}^n$. This property of the fractional Laplacian is then used to obtain a UCP for the fractional $p$-biharmonic systems and plays a central role in the analysis of the associated inverse problems. Our proofs use variational methods and the Caffarelli-Silvestre extension.

math.AP↗
arXiv2020

Reconstructing unknown inclusions for the biharmonic equation

Herein, we study an inverse problem for detecting unknown obstacles by the enclosure method using the Dirichlet--to--Neumann map for measurements. We justify the method for an penetrable obstacle case involving a biharmonic equation. We use complex geometrical optics solutions with a logarithmic phase to reconstruct some non--convex parts of the obstacle.

math.AP↗
arXiv2017

Superconductive and insulating inclusions for linear and non-linear conductivity equations

We detect an inclusion with infinite conductivity from boundary measurements represented by the Dirichlet-to-Neumann map for the conductivity equation. We use both the enclosure method and the probe method. We use the enclosure method to prove partial results when the underlying equation is the quasilinear $p$-Laplace equation. Further, we rigorously treat the forward problem for the partial differential equation $\operatorname{div}(σ\lvert\nabla u\rvert^{p-2}\nabla u)=0$ where the measurable conductivity $σ\colonΩ\to[0,\infty]$ is zero or infinity in large sets and $1<p<\infty$.

math.AP↗
arXiv2017

Monotonicity and enclosure methods for the p-Laplace equation

We show that the convex hull of a monotone perturbation of a homogeneous background conductivity in the $p$-conductivity equation is determined by knowledge of the nonlinear Dirichlet-Neumann operator. We give two independent proofs, one of which is based on the monotonicity method and the other on the enclosure method. Our results are constructive and require no jump or smoothness properties on the conductivity perturbation or its support.

math.AP↗
arXiv2016

Inverse problems for $p$-Laplace type equations under monotonicity assumptions

We consider inverse problems for $p$-Laplace type equations under monotonicity assumptions. In two dimensions, we show that any two conductivities satisfying $σ_1 \geq σ_2$ and having the same nonlinear Dirichlet-to-Neumann map must be identical. The proof is based on a monotonicity inequality and the unique continuation principle for $p$-Laplace type equations. In higher dimensions, where unique continuation is not known, we obtain a similar result for conductivities close to constant.

math.AP↗
arXiv2015

Quantitative uniqueness estimates for $p$-Laplace type equations in the plane

In this article our main concern is to prove the quantitative unique estimates for the $p$-Laplace equation, $1 \max\{p,2\}$ or $q=p>2$, if $\|u\|_{L^\infty(\mathbb{R}^2)}\leq C_0$, then $u$ satisfies the following asymptotic estimates at $R\gg 1$ \[ \inf_{|z_0|=R}\sup_{|z-z_0|<1} |u(z)| \geq e^{-CR^{1-\frac{2}{q}}\log R}, \] where $C$ depends only on $p$, $q$, $\tilde{M}$ and $C_0$. When $q=\max\{p,2\}$ and $p\in (1,2]$, under similar assumptions, we have \[ \inf_{|z_0|=R} \sup_{|z-z_0|<1} |u(z)| \geq R^{-C}, \] where $C$ depends only on $p$, $\tilde{M}$ and $C_0$. As an immediate consequence, we obtain the strong unique continuation principle (SUCP) for nontrivial solutions of this equation. We also prove the SUCP for the weighted $p$-Laplace equation with a locally positive locally Lipschitz weight.

math.AP↗
arXiv2015

Enclosure method for the p-Laplace equation

We study the enclosure method for the p-Calderón problem, which is a nonlinear generalization of the inverse conductivity problem due to Calderón that involves the p-Laplace equation. The method allows one to reconstruct the convex hull of an inclusion in the nonlinear model by using exponentially growing solutions introduced by Wolff. We justify this method for the penetrable obstacle case, where the inclusion is modelled as a jump in the conductivity. The result is based on a monotonicity inequality and the properties of the Wolff solutions.

math.AP↗
arXiv2014

An $H^{s,p}(\curl;Ω)$ estimate for the Maxwell system

We derive an $H_{0}^{s,p}(\curl;Ω)$ estimate for the solutions of the Maxwell type equations modeled with anisotropic and $W^{s, \infty}(Ω)$-regular coefficients. Here, we obtain the regularity of the solutions for the integrability and smoothness indices $(p, s)$ in a plan domain characterized by the apriori lower/upper bounds of $a$ and the apriori upper bound of its H{ö}lder semi-norm of order $s$. The proof relies on a perturbation argument generalizing Gr{ö}ger's $L^p$-type estimate, known for the elliptic problems, to the Maxwell system.

math.AP↗
arXiv2013

Reconstruction of interfaces from the elastic farfield measurements using CGO solutions

In this work, we are concerned with the inverse scattering by interfaces for the linearized and isotropic elastic model at a fixed frequency. First, we derive complex geometrical optic solutions with linear or spherical phases having a computable dominant part and an $H^α$-decaying remainder term with $α<3$, where $H^α$ is the classical Sobolev space. Second, based on these properties, we estimate the convex hull as well as non convex parts of the interface using the farfields of only one of the two reflected body waves (pressure waves or shear waves) as measurements. The results are given for both the impenetrable obstacles, with traction boundary conditions, and the penetrable obstacles. In the analysis, we require the surfaces of the obstacles to be Lipschitz regular and, for the penetrable obstacles, the Lamé coefficients to be measurable and bounded with the usual jump conditions across the interface.

math.AP↗
arXiv2013

On the inverse elastic scattering by interfaces using one type of scattered waves

We deal with the problem of the linearized and isotropic elastic inverse scattering by interfaces. We prove that the scattered $P$-parts or $S$-parts of the far field pattern, corresponding to all the incident plane waves of pressure or shear types, uniquely determine the obstacles for both the penetrable and impenetrable obstacles. In addition, we state a reconstruction procedure. In the analysis, we assume only the Lipschitz regularity of the interfaces and, for the penetrable case, the Lam{é} coefficients to be measurable and bounded, inside the obstacles, with the usual jumps across these interfaces.

math.AP↗