We find new quantitative estimates on the space-time analyticity of solutions to linear parabolic equations with time-independent coefficients and apply them to obtain observability inequalities for its solutions over measurable sets.1991 Mathematics Subject Classification. Primary: 35B37.Theorem 2. Assume that J ⊂ ∂Ω × (0, T ) is a measurable set with positive surface measure in ∂Ω × (0, T ). Then, there is N = N (Ω, J, T, δ) such that the inequality

We find new quantitative estimates on the space-time analyticity of solutions to linear parabolic equations with analytic coefficients near the initial time. We apply the estimates to obtain observability inequalities and null-controllability of parabolic evolutions over measurable sets. 2 (Ω × [0, 1]) Schauder estimates hold [5, Theorem 6]; i.e., there is K > 0 such 1991 Mathematics Subject Classification. Primary: 35B37.

Abstract. In this note we prove an end-point regularity result on the L p integrability of the second derivatives of solutions to non-divergence form uniformly elliptic equations whose second derivatives are a priori only known to be integrable. The main assumption on the elliptic operator is the Dini continuity of the coefficients. We provide a counterexample showing that the Dini condition is somehow optimal. We also give a counterexample related to the BM O regularity of second derivatives of solutions to elliptic equations. These results are analogous to corresponding results for divergence form elliptic equations in [4,16].

This paper deals with the numerical approximation of null controls for the wave equation posed in a bounded domain of R n. The goal is to compute approximations of controls that drive the solution from a prescribed initial state to zero at a large enough controllability time. In [Cindea & Münch, A mixed formulation for the direct approximation of the control of minimal L 2-norm for linear type wave equations], we have introduced a space-time variational approach ensuring strong convergent approximations with respect to the discretization parameter. The method, which relies on generalized observability inequality, requires H 2-finite element approximation both in time and space. Following a similar approach, we present and analyze a variational method still leading to strong convergent results but using simpler H 1-approximation. The main point is to preliminary restate the second order wave equation into a first order system and then prove an appropriate observability inequality.

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