326 research outputs found
Carleman estimates, observability inequalities and null controllability for interior degenerate non smooth parabolic equations
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
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
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
In this paper we establish some new Carleman estimates for the
Baouendi-Grushin operators , 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
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
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 class of the coefficient in
front of .The approach applies in particular to the (possibly degenerate
or singular) heat equation with a(x)\textgreater{}0
for a.e. and , or to the heat equation with
inverse square potential with
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