Observability Inequalities and Measurable Sets

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