This paper presents two observability inequalities for the heat equation over Ω× (0,T) . In the first one, the observation is from a subset of positive measure in Ω× (0,T) , while in the second, the observation is from a subset of positive surface measure on ∂Ω× (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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Apraiz et al. (2014) studied this question.
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