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Partial data inverse problems for the nonlinear magnetic Schrödinger equation

Ru-Yu Lai, Gunther Uhlmann, Lili Yan

Abstract

In this paper, we study the partial data inverse problem for nonlinear magnetic Schrödinger equations. We show that the knowledge of the Dirichlet-to-Neumann map, measured on an arbitrary part of the boundary, determines the time-dependent linear coefficients, electric and magnetic potentials, and nonlinear coefficients, provided that the divergence of the magnetic potential is given. Additionally, we also investigate both the forward and inverse problems for the linear magnetic Schrödinger equation with a time-dependent leading term. In particular, all coefficients are uniquely recovered from boundary data.

Partial data inverse problems for the nonlinear magnetic Schrödinger equation

Abstract

In this paper, we study the partial data inverse problem for nonlinear magnetic Schrödinger equations. We show that the knowledge of the Dirichlet-to-Neumann map, measured on an arbitrary part of the boundary, determines the time-dependent linear coefficients, electric and magnetic potentials, and nonlinear coefficients, provided that the divergence of the magnetic potential is given. Additionally, we also investigate both the forward and inverse problems for the linear magnetic Schrödinger equation with a time-dependent leading term. In particular, all coefficients are uniquely recovered from boundary data.

Paper Structure

This paper contains 20 sections, 20 theorems, 275 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Outline
  4. Preliminary results and the forward problem
  5. Function spaces and notations
  6. Well-posedness for the linear magnetic Schrödinger equation (MSE)
  7. Well-posedness for the nonlinear MSE
  8. Geometric optics solutions
  9. GO solutions with non-concentrated amplitudes
  10. GO solutions with concentrated amplitudes
  11. Recovery of the magnetic and electric potentials
  12. An integral identity
  13. Recovery of $A$ and $q$
  14. Recovery of the nonlinear coefficient
  15. Recovery of $B_{\sigma\beta}$ with $\sigma+\beta=2$
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $\Omega$ be a bounded and simply connected domain in $\mathbb{R}^n$ with smooth boundary $\partial\Omega$, $n\geq 2$. Suppose that $A_j \in C^\infty(\overline{Q};\mathbb{R}^n)$ and $q_j\in C^\infty(\overline{Q};\mathbb{R})$ satisfy $A_j=0$ and $q_j=0$ in $(0,T)\times\mathcal{O}$ for $j=1,\,2$, a for all integers $\sigma$ and $\beta$ such that $1\le\sigma$, $0\le\beta$ and $2\le\sigma+\beta\le

Theorems & Definitions (44)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • proof : Proof of Theorem \ref{['thm_linear_wellposedness']} (a)
  • proof : Proof of Theorem \ref{['thm_linear_wellposedness']} (b)
  • Remark 2.1
  • Remark 2.2
  • Proposition 2.1
  • proof
  • Remark 2.3
  • ...and 34 more