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Stable determination of the potential for the Helmholtz equation in the high frequency limit from boundary measurements

Mourad Choulli, Hiroshi Takase

TL;DR

The paper addresses the stable identification of a potential in the Helmholtz equation $(-\Delta+q-\lambda)u=0$ from boundary measurements taken on a portion of the boundary in the high-frequency limit. It develops a framework based on global quantitative uniqueness of continuation, quantitative Runge approximation, and complex geometric optics (CGO) solutions to link boundary data differences to interior potential differences, establishing a triple-logarithmic stability bound with explicit $\lambda$-dependent constants via a stability function $\Phi_c$. The main results include a triple-log stability for partial Dirichlet-to-Neumann maps and an analogous bound for the interior impedance problem via the Robin-to-Dirichlet map, both with carefully tracked frequency dependence through $\mathbf{e}_\lambda$, $\mathbf{b}_\lambda$, and the function $\Phi_c$. These results clarify the ill-posedness scaling at high frequency and provide rigorous, explicit stability estimates that are robust to the high-frequency regime and boundary-measurement limitations.

Abstract

We establish a triple logarithmic stability estimate of determining the potential in a Helmholtz equation from a partial Dirichlet-to-Neumann map in the high frequency limit. This estimate is proved under the assumption that the potential is known near the boundary of a domain when the dimension is greater than or equal to $3$. In addition, we show a triple logarithmic stability for an interior impedance problem.

Stable determination of the potential for the Helmholtz equation in the high frequency limit from boundary measurements

TL;DR

The paper addresses the stable identification of a potential in the Helmholtz equation from boundary measurements taken on a portion of the boundary in the high-frequency limit. It develops a framework based on global quantitative uniqueness of continuation, quantitative Runge approximation, and complex geometric optics (CGO) solutions to link boundary data differences to interior potential differences, establishing a triple-logarithmic stability bound with explicit -dependent constants via a stability function . The main results include a triple-log stability for partial Dirichlet-to-Neumann maps and an analogous bound for the interior impedance problem via the Robin-to-Dirichlet map, both with carefully tracked frequency dependence through , , and the function . These results clarify the ill-posedness scaling at high frequency and provide rigorous, explicit stability estimates that are robust to the high-frequency regime and boundary-measurement limitations.

Abstract

We establish a triple logarithmic stability estimate of determining the potential in a Helmholtz equation from a partial Dirichlet-to-Neumann map in the high frequency limit. This estimate is proved under the assumption that the potential is known near the boundary of a domain when the dimension is greater than or equal to . In addition, we show a triple logarithmic stability for an interior impedance problem.
Paper Structure (15 sections, 17 theorems, 214 equations)

This paper contains 15 sections, 17 theorems, 214 equations.

Table of Contents

  1. Introduction and main result
  2. Global quantitative uniqueness of continuation
  3. Quantitative Runge approximation
  4. Integral inequality
  5. Proof of Theorem \ref{['triple_log_stability']}
  6. The interior impedance problem
  7. Statement of the result
  8. Global quantitative uniqueness of continuation
  9. Quantitative Runge approximation
  10. Integral inequality
  11. Proof of Theorem \ref{['triple_log_stability_2']}
  12. A priori estimates
  13. Local quantitative uniqueness of continuation
  14. Complex geometric optics solutions
  15. Boundedness of a multiplication operator

Key Result

Theorem 1.1

Let $n\ge 3$ be an integer. Assume $g=\mathbf{I}$ and set $\varsigma=(n,\Omega,\Omega_0,\kappa)$. There exist constants $C=C(\varsigma,\Gamma,\Sigma)>0$, $c=c(\varsigma,\Gamma)>0$ and $\theta=\theta(\varsigma,\Sigma)\in(0,1)$ such that for all $\lambda\in\rho(A_{q_0})$ so that $\lambda\ge 1$ and for

Theorems & Definitions (33)

  • Theorem 1.1
  • Theorem 2.1
  • proof
  • Corollary 2.2
  • proof
  • Corollary 2.3
  • proof
  • Theorem 3.1
  • proof
  • Proposition 4.1
  • ...and 23 more