Carleman inequalities for the two-dimensional heat equation in singular domains

Null controllability for the semilinear heat equation in a non-smooth domain

“…In [9], the author has studied approximate controllability with globally Lipschitz nonlinearities using the fixedpoint method. We also cite [5], in which the authors proved exact controllability to trajectories where the non-linearity checks condition (2). They have used Carleman's estimates and Kakutani's fixed point theorem.…”

“…In [3], the authors were interested by the exact controllability to trajectories with discontinuous diffusion coefficients. In [2], the authors have established a null-controllability result for the linear heat equation in polygonal or cracked domains of R 2 . They were able to justify Carleman's estimate by building a suitable weight function.…”

“…We prove a global Carleman inequality for the linear problem with a potential. As in [2], the loss of regularity due to the geometry of the domain prevents us from doing some integrations by parts, so we use a density result. In case 2, we have circumvented the tip of the crack and worked in a sub-domain with a Lipschitzian boundary.…”

A Carleman estimate for the two dimensional heat equation with mixed boundary conditions

“…where f is given in L 2 (Ω) and ∂u ∂ν is the normal derivative of u. It is well known, see [8], that the solution of (1) is not in H 2 (Ω), and more precisely the solution, according to [7] is given by:…”

“…To the best of our knowledge, very few results on Carleman estimates in the presence of singularities have been established. We cite [2] for the Laplace equation for a domain with a corner, [1] for the heat equation in a singular domain and [3] for the wave equation with mixed conditions using microlocal approach. Our methodology here is in a similar spirit of [1,[4][5][6].…”