Observability estimate for the wave equation with variable coefficients

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 $(h^{jk}(x))_{n\times n}=I_n)$ .