Carleman estimates for the Zaremba Boundary Condition and Stabilization of Waves

Abstract

In this paper, we shall prove a Carleman estimate for the so-called Zaremba problem. Using some techniques of interpolation and spectral estimates, we deduce a result of stabilization for the wave equation by means of a linear Neumann feedback on the boundary. This extends previous results from the literature: indeed, our logarithmic decay result is obtained while the part where the feedback is applied contacts the boundary zone driven by an homogeneous Dirichlet condition. We also derive a controllability result for the heat equation with the Zaremba boundary condition.

Stabilization of hyperbolic equations with mixed boundary conditions

“…However, to the best of our knowledge, there are no references addressing the stabilization of hyperbolic equations with Neumann-Robin boundary conditions even for C ∞ -regularity of the coefficients, the damping and the boundary. Moreover, as we mentioned before, most of very interesting logarithmic decay results were given in [9,17] and the references therein for the hyperbolic equation under the regularity assumption that the coefficients a jk (•) and a(•) are C ∞ -smooth, and the damping at least is C s (s > 1 2 )-smooth. In this paper, we shall develop an approach based on two global Carleman estimates to prove the boundary stabilization of system (1.4) equipped with boundary condition (1.5), and the internal stabilization of system (1.6) equipped with boundary conditions (1.7), under sharp regularity of the coefficients…”

“…We remark that the logarithmic decay results obtained in [4,16,17] depend on C ∞ -regularity of a jk (•), a(•) and ∂M , while Cornilleau-Robbiano [9] weakened the regularity of the damping a(•) to C s for s > 1 2 , and kept the C ∞ -regularity on the boundary ∂M . Since the classical local Carleman estimate was employed in [9], it seems that the authors do need these strong regularity. Naturally, we want to know whether the logarithmic decay results still hold under some sharp regularity on the coefficients a jk (•), the damping a(•) and the boundary ∂M .…”

“…The aim of this paper is to derive the explicit decay rates of the solutions to hyperbolic equations with a small boundary or internal damping term. This work was motivated by the above mentioned references, especially reference [9] for logarithmic stabilization of wave equations with Zaremba boundary conditions. More precisely, our problem can be formulated as follows.…”

“…Bellassoued [4] established the logarithmic decay results for both the boundary stabilization of the elastic wave equation (∂ 2 t −∆ e )u = 0 with Neumann dissipative term supported on the boundary and the internal stabilization of the elastic wave equation (∂ 2 t − ∆ e + a(x)u)u = 0 with zero Dirichlet boundary condition, where ∆ e = µ∆ g + (µ + λ)∇ g (div •) with real parameters λ, µ satisfying µ > 0, 2λ + µ > 0. Very recently, Cornilleau-Robbiano [9] considered the logarithmic stabilization of the wave equation (∂ 2 t − ∆)u = 0, i.e. the coefficients (a jk ) n×n = I, the identity matrix, with the Zaremba boundary conditions, i.e.…”

“…Assume that ilR ⊂ ρ(A), where ρ(A) is the resolvent set of A. We recall the following characterization on decay rates: (i) The solution of (1.1) decays exponentially if and only if sup{||(A − iβI) −1 || L(X) | β ∈ lR} < ∞ (see [13,22]); (ii) The solution of (1.1) decays polynomially if and only if sup β −l ||(A − iβI) −1 || L(X) ≤ L for some l, L > 0 and all |β| ≥ 1 (see [5,18]); (iii) The solution of (1.1) decays logarithmically if and only if there exists a constant C > 0 such that ∀β ∈ lR, ||(A − iβI) −1 || L(X) ≤ Ce C|β| (see [9,16,17]). We refer to [3,8] and…”