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Local data inverse problem for the polyharmonic operator with anisotropic perturbations

Sombuddha Bhattacharyya, Pranav Kumar

Abstract

In this article, we study an inverse problem with local data for a linear polyharmonic operator with several lower order tensorial perturbations. We consider our domain to have an inaccessible portion of the boundary where neither the input can be prescribed nor the output can be measured. We prove the unique determination of all the tensorial coefficients of the operator from the knowledge of the Dirichlet and Neumann map on the accessible part of the boundary, under suitable geometric assumptions on the domain.

Local data inverse problem for the polyharmonic operator with anisotropic perturbations

Abstract

In this article, we study an inverse problem with local data for a linear polyharmonic operator with several lower order tensorial perturbations. We consider our domain to have an inaccessible portion of the boundary where neither the input can be prescribed nor the output can be measured. We prove the unique determination of all the tensorial coefficients of the operator from the knowledge of the Dirichlet and Neumann map on the accessible part of the boundary, under suitable geometric assumptions on the domain.
Paper Structure (10 sections, 8 theorems, 120 equations)

This paper contains 10 sections, 8 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Statement of the main result
  3. Preliminary results
  4. Interior Carleman estimates
  5. Construction of Complex Geometric Optics solutions
  6. Momentum ray transforms
  7. Proof of the main result
  8. Construction of special solutions
  9. The integral identity
  10. Recovering the coefficients

Key Result

Theorem 1

Let the coefficients $A^{j}$ for $1 \leq j \leq m$ belong to $C_{c}^{\infty}(\Omega, \mathbb{C}^{n^{j}})$ and $q(x) \in L^{\infty}(\Omega, \mathbb{C})$ in Operator. Suppose Assumption Assumption-1 and $f=\left(f_{0}, f_{1}, \ldots, f_{m-1}\right) \in \prod_{i=0}^{m-1} H^{2 m-2 i-\frac{1}{2}}(\partia where $C>0$ is a constant.

Theorems & Definitions (16)

  • Theorem 1: tanabe2017functional Lemma 5.13
  • Definition 1.1
  • Theorem 2
  • Definition 2.1: kenig2007calderon
  • Proposition 2.1
  • Proposition 2.2
  • Lemma 2.1
  • proof
  • Definition 2.2: Sharafutdinov_book, Chapter 2
  • Definition 2.3: Sharafutdinov_book, Chapter 2
  • ...and 6 more