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

Abstract

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 wellposedness 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.

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

“…As an immediate consequence, one can prove, using a standard technique (e.g., see [27,Section 7.4]), null controllability for the linear degenerate/singular problem (1.1). Clearly, this result generalizes the result obtained in [7]: in fact, if we consider Neumann boundary conditions and if λ = 0 (that is, if we consider the purely degenerate case), we obtain exactly the result of [7] in the case of a problem in non divergence form.…”

“…From now on, we make the following assumptions on a, b and λ: Observe that the assumption λ = 0 is not restrictive since the case λ = 0 is considered in [7] and in [23].…”

“…In this case we will proceed as in [7], using Theorem 4.1 in the case of Dirichlet boundary conditions and a technique based on cut off functions.…”

“…On the functions a, b and on the constant λ we make the following assumptions: Remark 6. Since we require identity (5.3) far from x 0 , once a is given, it is easy to find g, h, g 0 and h 0 with the desired properties (see [7,Remark 4] for some examples). Now, we associate to problem (1.1) the homogeneous adjoint problem…”

“…where k and α are positive constants. To the best of our knowledge, [6], [7], [22], [23] and [25] are the first papers deal with well posedness and Carleman estimates (and, consequently, null controllability) for operators (in divergence and in non divergence form with Dirichlet or Neumann boundary conditions) with purely degeneracy (i.e. λ = 0) at the interior of the space domain.…”