Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Increasing stability for the inverse source problems in electrodynamics

Suliang Si

Abstract

We are concerned with increasing stability in the inverse source problems for the time-dependent Maxwell equations in R^3 , where the source term is compactly supported in both time and spatial variables. By using the Fourier transform, sharp bounds of the analytic continuation and the Huygens principle, increasing stability estimates of the L^2 -norm of the source function are obtained. The main goal of this paper is to understand increasing stability for the Maxwell equations in the time domain.

Increasing stability for the inverse source problems in electrodynamics

Abstract

We are concerned with increasing stability in the inverse source problems for the time-dependent Maxwell equations in R^3 , where the source term is compactly supported in both time and spatial variables. By using the Fourier transform, sharp bounds of the analytic continuation and the Huygens principle, increasing stability estimates of the L^2 -norm of the source function are obtained. The main goal of this paper is to understand increasing stability for the Maxwell equations in the time domain.
Paper Structure (9 sections, 6 theorems, 129 equations, 2 figures)

This paper contains 9 sections, 6 theorems, 129 equations, 2 figures.

Table of Contents

  1. Introduction
  2. statement of the problem
  3. Motivations
  4. Known results
  5. Main results
  6. Preliminaries and Proof of Theorem \ref{['TH1']}
  7. Proof of Theorem \ref{['TH3']}
  8. Proof of Theorem \ref{['TH4']}
  9. Acknowledgment

Key Result

Theorem 2.1

Let the condition af hold and let $T>2R+T_0$. Assume that $g(t)$ is given and $||\textbf{f}||_{H^1({\mathbb R}^3)^3}\leq M$ where $M>1$ is a constant. If $\nabla\cdot \textbf{f}(x)=0$, then there exists a constant $C>0$ depending on $\delta$, $T_0$, $n$ and $R$ such that where $\epsilon=(\int_0^T\int_{\partial B_R}(|T(\textbf{E}\times\nu)|^2+|\textbf{E}\times\nu|^2 )ds(\textbf{x})dt)^{\frac{1}{2}

Figures (2)

  • Figure 1: $E_s$ is the shadow area, $\zeta^2=|\xi|^2$.
  • Figure 2: $E_s$ is the shadow area, $\tilde{\xi}^2=\xi_1^2+\xi_2^2$.

Theorems & Definitions (8)

  • Theorem 2.1
  • Remark 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Lemma 5.1