arXiv subjects

Vladimir A. Sharafutdinov

Publications and source records attributed to Vladimir A. Sharafutdinov.

11 recordsLinked to original sources

arXiv2026

Numerical solution of the two-dimensional Calderon problem for domains close to a disk

For a compact Riemannian surface $(M,g)$ with non-empty boundary $Γ$, the Dirichlet-to-Neumann operator (DtN-map) $Λ_g:C^\infty(Γ)\to C^\infty(Γ)$ is defined by $Λ_gf=\left.\frac{\partial u}{\partialν}\right|_Γ$, where $ν$ is the unit outer normal vector to the boundary and $u$ is the solution to the Dirichlet problem $Δ_gu=0,\ u|_Γ=f$. The Calderón problem consists of recovering a Riemannian surface from its DtN-map. It is well known that $(M,g)$ is determined by $Λ_g$ uniquely up to a conformal equivalence. We suggest a method for numerical solution of the Calderón problem. The method works well at least for Riemannian surfaces $(M,g)$ close to $({D},e)$, where ${D}=\{(x,y)\mid x^2+y^2\le1\}$ is the unit disk and $e=dx^2+dy^2$ is the Euclidean metric. Our numerical examples confirm the statement: the DtN-map is very sensitive to small deviations of the shape of a domain.

math.DG↗

arXiv2025

Range characterization of the ray transform on Sobolev spaces of symmetric tensor fields in two dimensions

The ray transform $I_m$ integrates a symmetric $m$ rank tensor field $f$ on $\mathbb{R}^n$ over lines. In the case of $n\ge3$, the range characterization of the operator $I_m$ on weighted Sobolev spaces $H^{s}_t({\mathbb R}^n;S^m{\mathbb R}^n)$ was obtained in [V. Krishnan and V. Sharafutdinov. Range characterization of ray transform on Sobolev spaces of symmetric tensor fields. Inverse Problems and Imaging, 18(6), 1272--1293, 2024]. Here we obtain a range characterization result in higher order weighted Sobolev spaces in two dimensions. Range characterization in the case of $n=2$ is very different from that for $n\ge3$, and this allows us to obtain such a result in higher order weighted Sobolev spaces $H^{r,s}_t(\mathbb{R}^2)$ for any real $r$. Nevertheless, our main tool is again the Reshetnyak formula stating that $\lVert I_mf\rVert_{H^{(r,s+1/2)}_{t+1/2}(T{\mathbb S}^{n-1})}=\lVert f\rVert_{H^{(r,s)}_t({\mathbb R}^n;S^m{\mathbb R}^n)}$ for a solenoidal tensor field $f$.

math.AP↗

arXiv2025

Two-dimensional Calderon problem and flat metrics

For a compact Riemannian manifold $(M,g)$ with boundary $\partial M$, the Diri\-chl\-et-to-Neumann operator $Λ_g:C^\infty(\partial M)\longrightarrow C^\infty(\partial M)$ is defined by $Λ_gf=\left.\frac{\partial u}{\partialν}\right|_{\partial M}$, where $ν$ is the unit outer normal vector to the boundary and $u$ is the solution to the Dirichlet problem $Δ_gu=0,\ u|_{\partial M}=f$. Let $g_\partial$ be the Riemannian metric on $\partial M$ induced by $g$. The Calderon problem is posed as follows: To what extent is $(M,g)$ determined by the data $(\partial M,g_\partial,Λ_g)$? We prove the uniqueness theorem: A compact connected two-dimensional Riemannian manifold $(M,g)$ with non-empty boundary is determined by the data $(\partial M,g_\partial,Λ_g)$ uniquely up to conformal equivalence.

math.DG↗

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 $t^k$, $I_m^kf(x,ξ)=\int_{-\infty}^\infty t^k\langle f(x+tξ),ξ^m\rangle\,\mathrm{d}t$. 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↗

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 $t^k$, $I_m^kf(x,ξ)=\int_{-\infty}^\infty t^k\langle f(x+tξ),ξ^m\rangle\,\mathrm{d}t$. 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

Ray transform on Sobolev spaces of symmetric tensor fields, II: Range characterization

The ray transform $I$ integrates symmetric $m$-tensor field in $\mathbb{R}^n$ over lines. This transform in Sobolev spaces was studied in our earlier work where higher order Reshetnyak formulas (isometry relations) were established. The main focus of the current work is the range characterization. In dimensions $n\geq 3$, the range characterization of the ray transform in Schwartz spaces is well-known; the main ingredient of the characterization is a system of linear differential equations of order $2(m+1)$ which are called John equations. Using the higher order Reshetnyak formulas, the range of the ray transform on Sobolev spaces is characterized in dimensions $n\geq 3$ in this paper.

math.AP↗

arXiv2023

Steady flows of ideal incompressible fluid

A new important relation between fluid mechanics and differential geometry is established. We study smooth steady solutions to the Euler equations with the additional property: the velocity vector is orthogonal to the gradient of the pressure at any point. Such solutions are called Gavrilov flows. Local structure of a Gavrilov flow is described in terms of geometry of isobaric hypersurfaces. In the 3D case, we obtain a system of PDEs for an axisymmetric Gavrilov flow and find consistency conditions for the system. Two numerical examples of axisymmetric Gavrilov flows are presented: with pressure function periodic in the axial direction, and with isobaric surfaces diffeomorphic to the torus.

math-ph↗

arXiv2021

Ray transform on Sobolev spaces of symmetric tensor fields, I: Higher order Reshetnyak formulas

For an integer $r\ge0$, we prove the $r$th order Reshetnyak formula for the ray transform of rank $m$ symmetric tensor fields on $\mathbb{R}^n$. Certain differential operators $A^{(m,r,l)}\ (0\le l\le r)$ on the sphere $\mathbb{S}^{n-1}$ are main ingredients of the formula. The operators are defined by an algorithm that can be applied for any $r$ although the volume of calculations grows fast with $r$. The algorithm is realized for small values of $r$ and Reshetnyak formulas of orders $0,1,2$ are presented in an explicit form.

math.AP↗

arXiv2020

Killing tensor fields of third rank on a two-dimensional Riemannian torus

A rank $m$ symmetric tensor field on a Riemannian manifold is called a Killing field if the symmetric part of its covariant derivative is equal to zero. Such a field determines the first integral of the geodesic flow which is a degree $m$ homogeneous polynomial in velocities. There exist global isothermal coordinates on a two-dimensional Riemannian torus such that the metric is of the form $ds^2=λ(z)|dz|^2$ in the coordinates. The torus admits a third rank Killing tensor field if and only if the function $λ$ satisfies the equation $\Re\big(\frac{\partial}{\partial z}\big(λ(cΔ^{-1}λ_{zz}+a)\big)\big)=0$ with some complex constants $a$ and $c\neq0$. The latter equation is equivalent to some system of quadratic equations relating Fourier coefficients of the function $λ$. If the functions $λ$ and $λ+λ_0$ satisfy the equation for a real constant $λ_0\neq0$, then there exists a non-zero Killing vector field on the torus.

math.DG↗

arXiv2019

Momentum Ray Transforms, II: Range Characterization In the Schwartz space

The momentum ray transform $I^k$ integrates a rank $m$ symmetric tensor field $f$ over lines of ${\R}^n$ with the weight $t^k$: $ (I^k\!f)(x,ξ)=\int_{-\infty}^\infty t^kłf(x+tξ),ξ^m\r\,dt. $ We give the range characterization for the operator $f\mapsto(I^0\!f,I^1\!f,\dots, I^m\!f)$ on the Schwartz space of rank $m$ smooth fast decaying tensor fields. In dimensions $n\ge3$, the range is characterized by certain differential equations of order $2(m+1)$ which generalize the classical John equations. In the two-dimensional case, the range is characterized by certain integral conditions which generalize the classical Gelfand -- Helgason -- Ludwig conditions.

math.AP↗

arXiv2011

On conformal Killing symmetric tensor fields on Riemannian manifolds

A vector field on a Riemannian manifold is called conformal Killing if it generates one-parameter group of conformal transformations. The class of conformal Killing symmetric tensor fields of an arbitrary rank is a natural generalization of the class of conformal Killing vector fields, and appears in different geometric and physical problems. A symmetric tensor field is a trace-free field if the contraction of the field with the metric tensor is identically equal to zero. On a Riemannian manifold of dimension at least three, the space of trace-free conformal Killing symmetric tensor fields of arbitrary rank is of a finite dimension. On a two-dimensional manifold, the space can be of infinite dimension. Nevertheless, on a connected manifold of any dimension, a trace-free conformal Killing tensor field is uniquely determined by its $C^\infty$-jet at any point. We prove the statement: On a connected manifold, a trace-free conformal Killing tensor field is identically equal to zero if it vanishes on some hypersurface. This statement is a basis of the theorem on decomposition of a symmetric tensor field on a compact Riemannian manifold with boundary to a sum of three fields such that the first summand is a trace-free and divergence-free field, the second summand is a potential field with a trace-free potential vanishing on the boundary, and the last summand is a multiple of the metric tensor. The classical theorem by Bochner - Yano states the absence of conformal Killing vector fields on a closed manifold of negative Ricci curvature. We generalize the latter statements to arbitrary rank tensor fields but under the stronger hypothesis: the sectional curvature is assumed to be negative. There is no nontrivial trace-free conformal Killing symmetric tensor field of any rank on a closed negatively curved Riemannian manifold.

math.DG↗