326 research outputs found

    Carleman estimates, observability inequalities and null controllability for interior degenerate non smooth parabolic equations

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    We show Carleman estimates, observability inequalities and null controllability results for parabolic equations with non smooth coefficients degenerating at an interior point.Comment: Accepted in Memoirs of the American Mathematical Societ

    Carleman estimates for parabolic equations with interior degeneracy and Neumann boundary conditions

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    We consider a parabolic problem with degeneracy in the interior of the spatial domain and Neumann boundary conditions. In particular, we will focus on the well-posedness of the problem and on Carleman estimates for the associated adjoint problem. The novelty of the present paper is that for the first time it is considered a problem with an interior degeneracy and Neumann boundary conditions so that no previous result can be adapted to this situation. As a consequence new observability inequalities are established.Comment: Accepted in J. Anal. Math. arXiv admin note: text overlap with arXiv:1508.0401

    Interior degenerate/singular parabolic equations in nondivergence form: well-posedness and Carleman estimates

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    We consider non smooth general degenerate/singular parabolic equations in non divergence form with degeneracy and singularity occurring in the interior of the spatial domain, in presence of Dirichlet or Neumann boundary conditions. In particular, we consider well posedness of the problem and then we prove Carleman estimates for the associated adjoint problem.Comment: Accepted in Journal of Differential Equations. arXiv admin note: text overlap with arXiv:1507.0778

    Carleman estimates for Baouendi-Grushin operators with applications to quantitative uniqueness and strong unique continuation

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    In this paper we establish some new L2−L2L^{2}-L^{2} Carleman estimates for the Baouendi-Grushin operators Bγ\mathscr{B}_\gamma, in (1.1) below. We apply such estimates to obtain: (i) an extension of the Bourgain-Kenig quantitative unique continuation; (ii) the strong unique continuation property for some degenerate sublinear equations.Comment: revised version of the file, several references have been adde

    Inverse coefficient problem for Grushin-type parabolic operators

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    The approach to Lipschitz stability for uniformly parabolic equations introduced by Imanuvilov and Yamamoto in 1998 based on Carleman estimates, seems hard to apply to the case of Grushin-type operators studied in this paper. Indeed, such estimates are still missing for parabolic operators degenerating in the interior of the space domain. Nevertheless, we are able to prove Lipschitz stability results for inverse coefficient problems for such operators, with locally distributed measurements in arbitrary space dimension. For this purpose, we follow a strategy that combines Fourier decomposition and Carleman inequalities for certain heat equations with nonsmooth coefficients (solved by the Fourier modes)

    Null controllability of one-dimensional parabolic equations by the flatness approach

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    We consider linear one-dimensional parabolic equations with space dependent coefficients that are only measurable and that may be degenerate or singular.Considering generalized Robin-Neumann boundary conditions at both extremities, we prove the null controllability with one boundary control by following the flatness approach, which providesexplicitly the control and the associated trajectory as series. Both the control and the trajectory have a Gevrey regularity in time related to the LpL^p class of the coefficient in front of u_tu\_t.The approach applies in particular to the (possibly degenerate or singular) heat equation (a(x)u_x)_x−u_t=0(a(x)u\_x)\_x-u\_t=0 with a(x)\textgreater{}0 for a.e. x∈(0,1)x\in (0,1) and a+1/a∈L1(0,1)a+1/a \in L^1(0,1), or to the heat equation with inverse square potential u_xx+(μ/∣x∣2)u−u_t=0u\_{xx}+(\mu / |x|^2)u-u\_t=0with μ≥1/4\mu\ge 1/4
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