Observability estimate for the wave equation with variable coefficients

TL;DR

This paper establishes improved observability estimates for the wave equation with variable coefficients, considering different observation regions and times, using Carleman estimates to refine previous results.

Contribution

The paper introduces new observability estimates with explicit constants for the wave equation with variable coefficients, reducing the required waiting time and observation region size.

Findings

Improved observability estimates with explicit constants.

Reduced waiting time for observability.

Expanded observation region conditions.

Abstract

This paper is devoted to a study of observability estimate for the wave equation with variable coefficients $(h^{jk}(x))_{n\times n}$ ($n\in\mathbb{N})$. We consider both the observation point lies outside the domain and the observation point lies inside the domain. Based on a Carleman estimate for the ultra-hyperbolic operator and a delicate treatment of observation region, we obtain two observability estimates with explicit observability constants. The key improvements are: (1) we improve the requirement of waiting time $T$; (2) we improve the size of the observation region (see Fingure 1 and Fingure 2 for the case of $(h^{jk}(x))_{n\times n}=I_n)$ .