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Lipschitz stability for an elliptic inverse problem with two measurements

Mourad Choulli, Hiroshi Takase

TL;DR

The paper addresses recovering unknown boundary values for solutions to elliptic equations outside a bounded obstacle from two boundary measurements on a surrounding Lipschitz boundary. It develops a Carleman framework with a second large parameter and a carefully constructed weight to obtain conditional Lipschitz stability for the unknown boundary data within an admissible set, for both exterior and interior problems, and extends the approach to the parabolic case. The main contributions are the first global Lipschitz stability results under an a priori constraint on the boundary data, plus quantitative uniqueness-type results and a parabolic extension. The findings have potential implications for inverse boundary value problems where only boundary data are accessible, and they provide a rigorous stability benchmark via Carleman estimates and weighted energy methods.

Abstract

We consider the problem of determining the unknown boundary values of a solution of an elliptic equation outside a bounded open set $B$ from the knowledge of the values of this solution on a boundary of an arbitrary Lipschitz bounded domain surrounding $B$. We obtain for this inverse problem Lipschitz stability for an admissible class of unknown boundary functions. Our analysis applies as well to an interior problem. We also give an extension to the parabolic case.

Lipschitz stability for an elliptic inverse problem with two measurements

TL;DR

The paper addresses recovering unknown boundary values for solutions to elliptic equations outside a bounded obstacle from two boundary measurements on a surrounding Lipschitz boundary. It develops a Carleman framework with a second large parameter and a carefully constructed weight to obtain conditional Lipschitz stability for the unknown boundary data within an admissible set, for both exterior and interior problems, and extends the approach to the parabolic case. The main contributions are the first global Lipschitz stability results under an a priori constraint on the boundary data, plus quantitative uniqueness-type results and a parabolic extension. The findings have potential implications for inverse boundary value problems where only boundary data are accessible, and they provide a rigorous stability benchmark via Carleman estimates and weighted energy methods.

Abstract

We consider the problem of determining the unknown boundary values of a solution of an elliptic equation outside a bounded open set from the knowledge of the values of this solution on a boundary of an arbitrary Lipschitz bounded domain surrounding . We obtain for this inverse problem Lipschitz stability for an admissible class of unknown boundary functions. Our analysis applies as well to an interior problem. We also give an extension to the parabolic case.
Paper Structure (7 sections, 10 theorems, 63 equations)

This paper contains 7 sections, 10 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Carleman type inequality
  3. Proofs of main results
  4. Extension to the parabolic case
  5. Comments for quantitative uniqueness of continuation
  6. Elliptic case
  7. Parabolic case

Key Result

Theorem 1.1

Let $B\subset\mathbb{R}^n$ be a $C^2$ bounded open set, $\Omega\Supset B$ be a Lipschitz bounded domain such that $\Omega\setminus\overline{B}$ is connected and set $\zeta_0=(g,p,B,\Omega,\beta/\alpha)$. Then there exists $C=C(\zeta_0)>0$ such that for any $\mathfrak{a}\in \mathcal{A}$ and $u(\mathf where $\nu$ denotes the outer unit normal to $\Gamma$.

Theorems & Definitions (17)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • proof : Proof of Theorem \ref{['exLipschitz']}
  • proof : Proof of Theorem \ref{['inLipschitz']}
  • ...and 7 more