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Quantitative strong unique continuation for elliptic operators -- application to an inverse spectral problem

Mourad Choulli

TL;DR

The work delivers a quantitative form of strong unique continuation for elliptic operators with unbounded lower-order terms, leveraging the three-ball and doubling inequalities. It extends Koch–Tataru-type results to a broad coefficient regime and derives uniform bounds for eigenfunctions, enabling a rigidity result for inverse spectral problems. Specifically, it proves that two Dirichlet–Laplace–Beltrami operators are gauge equivalent whenever their metrics coincide near the boundary and their boundary spectral data agree on a positive-measure subset. This advances stability and identifiability in inverse boundary/spectral problems for elliptic operators on manifolds, with explicit bounds and interpolation inequalities derived from a unified quantitative framework.

Abstract

Based on the three-ball inequality and the doubling inequality established in [23], we quantify the strong unique continuation established by Koch and Tataru [21] for elliptic operators with unbounded lower-order coefficients. We also derive a uniform quantitative strong unique continuation for eigenfunctions that we use to prove that two Dirichlet-Laplace-Beltrami operators are gauge equivalent whenever their corresponding metrics coincide in the vicinity of the boundary and their boundary spectral data coincide on a subset of positive measure.

Quantitative strong unique continuation for elliptic operators -- application to an inverse spectral problem

TL;DR

The work delivers a quantitative form of strong unique continuation for elliptic operators with unbounded lower-order terms, leveraging the three-ball and doubling inequalities. It extends Koch–Tataru-type results to a broad coefficient regime and derives uniform bounds for eigenfunctions, enabling a rigidity result for inverse spectral problems. Specifically, it proves that two Dirichlet–Laplace–Beltrami operators are gauge equivalent whenever their metrics coincide near the boundary and their boundary spectral data agree on a positive-measure subset. This advances stability and identifiability in inverse boundary/spectral problems for elliptic operators on manifolds, with explicit bounds and interpolation inequalities derived from a unified quantitative framework.

Abstract

Based on the three-ball inequality and the doubling inequality established in [23], we quantify the strong unique continuation established by Koch and Tataru [21] for elliptic operators with unbounded lower-order coefficients. We also derive a uniform quantitative strong unique continuation for eigenfunctions that we use to prove that two Dirichlet-Laplace-Beltrami operators are gauge equivalent whenever their corresponding metrics coincide in the vicinity of the boundary and their boundary spectral data coincide on a subset of positive measure.
Paper Structure (7 sections, 6 theorems, 59 equations)

This paper contains 7 sections, 6 theorems, 59 equations.

Table of Contents

  1. Quantitative strong unique continuation
  2. Statement of the main theorem
  3. Quantitative unique continuation from a set of positive measure
  4. Strong unique continuation for eigenfunctions
  5. Determining a metric tensor from a partial spectral boundary data
  6. Proof of Theorem \ref{['theorem0']}
  7. Geometric lemma

Key Result

Theorem 1.1

There exist $\rho_\ast \in \mathscr{C}$, $0<\rho_\ast \le \rho_0$, $\tau\in \mathscr{C}$ with $\tau<1/4$ so that, for all $0<\rho <\rho_\ast$, we find $C<1$, $c>1$ and $\mathfrak{a}>1$ in $\mathscr{C}_\rho$ with the property that if $u\in \mathscr{S}_0$ then where $\mathfrak{b}_r=\ln (\tau\rho/r)+\mathfrak{a}$.

Theorems & Definitions (9)

  • Theorem 1.1
  • Corollary 1.1
  • Proposition 1.1
  • Theorem 1.2
  • proof
  • Theorem 2.1
  • proof : Proof of Theorem \ref{['theorem0']}
  • Lemma A.1
  • proof