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On Some Geometric Inverse Problems for Nonscalar Elliptic Systems

Raul K. C. Araújo, Enrique Fernández-Cara, Diego A. Souza

Abstract

In this paper, we consider several geometric inverse problems for linear elliptic systems. We prove uniqueness and stability results. In particular, we show the way that the observation depends on the perturbations of the domain. In some particular situations, this provides a strategy that could be used to compute approximations to the solution of the inverse problem. In the proofs, we use techniques related to (local) Carleman estimates and differentiation with respect to the domain.

On Some Geometric Inverse Problems for Nonscalar Elliptic Systems

Abstract

In this paper, we consider several geometric inverse problems for linear elliptic systems. We prove uniqueness and stability results. In particular, we show the way that the observation depends on the perturbations of the domain. In some particular situations, this provides a strategy that could be used to compute approximations to the solution of the inverse problem. In the proofs, we use techniques related to (local) Carleman estimates and differentiation with respect to the domain.

Paper Structure

This paper contains 13 sections, 10 theorems, 79 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Unique continuation and uniqueness
  3. A unique continuation property for balls
  4. Unique continuation for general domains
  5. Proof of Theorem 1: uniqueness
  6. Stability
  7. Preliminary results
  8. Proof of Theorem 2: stability
  9. Additional comments and questions
  10. A similar inverse problem with internal observation
  11. A geometric inverse problem for a parabolic system
  12. Additional comments on stability
  13. Reconstruction

Key Result

Theorem 1

Assume that $(\varphi,\psi)\in H^{1/2}(\partial\Omega)\times H^{1/2}(\partial\Omega)$ is nonzero. For $i=0,1$, let $(y^i,z^i)$ be the unique weak solution to an with $D$ replaced by $D^i$ and let $\alpha^i$ and $\beta^i$ be given by the corresponding equalities an1. Then one has the following:

Figures (2)

  • Figure 1: The filled region is the set $D^2\backslash\overline{D}^0$.
  • Figure 2: The deformations of $D$.

Theorems & Definitions (16)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • Theorem 4: Fabre
  • Lemma 2
  • Theorem 5
  • Theorem 6
  • ...and 6 more