An improved spectral inequality for sums of eigenfunctions

TL;DR

The paper addresses bounding the $H^s$-norm of a finite sum of eigenfunctions by its norm on a strictly smaller open set, improving upon the universal $e^{C\sqrt{\lambda}}$ rate. It introduces frequency numbers $\Lambda_n(\tau)$ depending on the coefficients $a_k$ and proves monotonicity properties, enabling non-uniform spectral inequalities: $\|\psi\|_{L^2(\Omega)} \le C_1 e^{C_2 \sqrt{\Lambda_1(\tau_0)}} \|\psi\|_{L^2(\omega)}$ and $|\psi|_{H^s(\Omega)} \le C_1 e^{C_2 \sqrt{\Lambda_{1+s}(\tau_s)}} |\psi|_{H^s(\omega)}$. These results generalize Lebeau–Jerison and are shown to be asymptotically tight via a counterexample, with numerical tests on a disk illustrating tighter bounds for rapidly decaying coefficient sequences. The approach links spectral inequalities to quantitative unique continuation and observability, yielding non-uniform bounds that reflect the actual frequency content of the sum. The work provides a framework for sharper observability estimates in PDEs with variable coefficients and variable spectral composition.

Abstract

We establish a new spectral inequality for the quantified estimation of the $H^s$-norm, $s\ge 0$ of a finite linear combination of eigenfunctions in a domain in terms of its $H^s$-norm in a strictly open subset of the whole domain. The corresponding upper bound depends exponentially on the square root of the frequency number associated to the linear combination.

Figures (4)

• Figure 1: The intersection for the case of the Example with $p=1.2$ between $\frac{C}{(1-\theta)\tau^2}$ and $\Lambda_\eta(\tau)$, $\eta=1+s$ gives the optimal time $\tau_s$. We show the value of $\tau_0$.
• Figure 2: Comparison between $\sqrt{\lambda}$ and $\sqrt{\Lambda_1}$ for different choices of the constant $\theta$. Example with $p=1.0$.
• Figure 3: Comparison between $\sqrt{\lambda}$ and $\sqrt{\Lambda_1}$ for different choices of the constant $\theta$. Example with $p=1.2$.
• Figure 4: Comparison between $\sqrt{\lambda}$ and $\sqrt{\Lambda_1}$ for different choices of the constant $\theta$. Example with $p=1.3$.

Theorems & Definitions (13)

• Theorem 1: Lebeau-Jerison
• Theorem 2: Lebeau-Jerison
• proof
• Definition 3
• Proposition 4
• proof
• Remark 5
• Theorem 6
• proof
• Remark 7