# Observability inequalities and measurable sets

### Jone Apraiz

Universidad del Pais Vasco, Donostia-San Sebastian, Spain### Luis Escauriaza

Universidad del Pais Vasco, Bilbao, Spain### Gengsheng Wang

Wuhan University, Wuhan, Hubei, China### C. Zhang

Wuhan University, Wuhan, Hubei, China

## Abstract

This paper presents two observability inequalities for the heat equation over $\Omega\times (0,T)$. In the first one, the observation is from a subset of positive measure in $\Omega\times (0,T)$, while in the second, the observation is from a subset of positive surface measure on $\partial\Omega\times (0,T)$. It also proves the Lebeau-Robbiano spectral inequality when $\Omega$ is a bounded Lipschitz and locally star-shaped domain. Some applications for the above-mentioned observability inequalities are provided.

## Cite this article

Jone Apraiz, Luis Escauriaza, Gengsheng Wang, C. Zhang, Observability inequalities and measurable sets. J. Eur. Math. Soc. 16 (2014), no. 11, pp. 2433–2475

DOI 10.4171/JEMS/490