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A generalized Helmholtz-type decomposition of symmetric tensor fields and applications to ray transforms

Antti Kykkänen, Rohit Kumar Mishra, Suman Kumar Sahoo

Abstract

We study a solenoidal-potential type decomposition of a symmetric $m$-tensor field in $\Rb^2$, and its implications to injectivity questions for the momentum and elastic ray transforms. For symmetric tensor fields, a general decomposition with a restriction on the dimension and order of the decomposition was proved in~\cite{Rohit_Suman}. We extend the result to dimension $2$ under a mean-zero assumption. We use the decomposition in $2$ dimensions to prove the injectivity of the momentum and elastic ray transforms. We also prove a connection between the two integral transforms for $2$-tensors.

A generalized Helmholtz-type decomposition of symmetric tensor fields and applications to ray transforms

Abstract

We study a solenoidal-potential type decomposition of a symmetric -tensor field in , and its implications to injectivity questions for the momentum and elastic ray transforms. For symmetric tensor fields, a general decomposition with a restriction on the dimension and order of the decomposition was proved in~\cite{Rohit_Suman}. We extend the result to dimension under a mean-zero assumption. We use the decomposition in dimensions to prove the injectivity of the momentum and elastic ray transforms. We also prove a connection between the two integral transforms for -tensors.
Paper Structure (7 sections, 11 theorems, 89 equations)

This paper contains 7 sections, 11 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Definitions and notation
  3. A decomposition result for symmetric $2$-tensor fields in
  4. Applications to momentum ray transforms
  5. Applications to Elastic ray transform
  6. Connections between two types of transform
  7. Acknowledgments

Key Result

Lemma 2

Given a tensor field $f\in C^{\infty}(\mathbb{R}^2;S^2)$, there exist unique $g\in C^{\infty}(\mathbb{R}^2\setminus\{0\};S^2)$ and $v\in C^{\infty}(\mathbb{R}^2\setminus\{0\})$ such that the following decomposition of $f$ holds: and $g$ satisfies $j^2_{x} g(x) =0$$(x \in \mathbb{R}^2\setminus\{0\})$. For $x \in \mathbb{R}^2\setminus\{0\}$, the fields $g$ and $v$ are explicitly expressed in terms

Theorems & Definitions (22)

  • Definition 1: $k$-solenoidal and $k$-potential tensor fields
  • Lemma 2: Rohit_Suman and Sharafutdinov_book
  • Theorem 3
  • Remark 4
  • proof : Proof of Theorem \ref{['th:new_decomposition']}
  • Definition 5
  • Theorem 6
  • proof
  • Definition 7
  • Theorem 8
  • ...and 12 more