Observability Inequalities and Measurable Sets

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

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