Observability inequalities and measurable sets

Source 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.