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.where D is a subset of Ω × (0, T ), andwhere J is a subset of ∂Ω × (0, T ). Such apriori estimates are called observability inequalities.In the case that D = ω × (0, T ) and J = Γ × (0, T ) with ω and Γ accordingly open and nonempty subsets of Ω and ∂Ω, both inequalities (1.2) and (1.3) (where ∂Ω is smooth) were essentially first established, via the Lebeau-Robbiano spectral inequalities in [28] (See also [29,34,16]). These two estimates were set up to the linear parabolic equations (where ∂Ω is of class C 2 ), based on the Carleman inequality provided in [18]. In the case when D = ω × (0, T ) and J = Γ × (0, T ) 1991 Mathematics Subject Classification. Primary: 35B37.

We prove the interior and boundary null-controllability of some parabolic evolutions with controls acting over measurable sets.1991 Mathematics Subject Classification. Primary: 35B37.

<p style='text-indent:20px;'>We consider a heat equation and a wave equation in one spatial dimension. This article deals with the inverse problem of determining the size of the spatial interval from some extra boundary information on the solution. Under several different circumstances, we prove uniqueness, non-uniqueness and some size estimates. Moreover, we numerically solve the inverse problems and compute accurate approximations of the size. This is illustrated with several satisfactory numerical experiments.</p>

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