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Observability from a measurable set for functions in a Gevrey class

Igor Kukavica, Linfeng Li

Abstract

We provide an observability inequality in terms of a measurable set for general Gevrey regular functions. As an application, we establish an observability estimate from a measurable set for sums of Laplace eigenfunctions in a compact and connected boundaryless Riemannian manifold that belongs to the Gevrey class. The estimate has an explicit dependence on the maximal eigenvalue.

Observability from a measurable set for functions in a Gevrey class

Abstract

We provide an observability inequality in terms of a measurable set for general Gevrey regular functions. As an application, we establish an observability estimate from a measurable set for sums of Laplace eigenfunctions in a compact and connected boundaryless Riemannian manifold that belongs to the Gevrey class. The estimate has an explicit dependence on the maximal eigenvalue.

Paper Structure

This paper contains 3 sections, 5 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Proof of the main results

Key Result

Theorem 2.1

Suppose that $\Omega$ is a $C^1$ domain. Let $M,\delta, \kappa,r_0>0$ be constants, let $\sigma\geq 1$, and assume that $f\in C^\infty (\Omega)$ satisfies EQ110 and EQ130. Then, for any measurable set $E\subset \Omega$ with positive measure, we have where $C=C(M,\delta,\kappa, r_0, \sigma, |\Omega|/|E|, \Omega)>0$ is a constant.

Theorems & Definitions (8)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Lemma 3.1
  • proof : Proof of Theorem \ref{['T01']}
  • Lemma 3.2: LM2
  • proof : Proof of Theorem \ref{['T02']}
  • proof : Proof of Theorem \ref{['T03']}