Asymptotic Analysis

The paper deals with the dynamic problem for linearly elastic shallow shells with variable thickness. The author shows that the solution of three-dimensional problem converges to the solution of two-dimensional shallow shell equations as the thickness of the shell goes to zero.

We consider scattering by short range perturbations of the semi-classical Laplacian. We prove that when a polynomial bound on the resolvent holds the scattering amplitude is a semi-classical Fourier integral operator associated to the scattering relation near a non-trapped ray. Compared to previous work, we allow the scattering relation to have more general structure.

We consider the stabilization of the wave equation with space variable coefficients in a bounded region with a smooth boundary, subject to Dirichlet boundary conditions on one part of the boundary and linear or nonlinear dissipative boundary conditions of memory type on the remainder part of the boundary. Our stabilization results are mainly based on the use of differential geometry arguments, on the multiplier method and the introduction of suitable Lyapounov functionals.

We study the asymptotic behavior of a control problem for a quadratic cost functional and a linear elliptic equation in a thin domain of wide ε. For each n∈N, we obtain an asymptotic expansion of the control, the direct state, and the adjoint state, giving an approximation of order εⁿ⁺¹. Since this is a singular perturbation problem, one of the main difficulties is the existence of boundary layers, and then the appearance of boundary terms in the asymptotic expansions. We show how this type of problems can be solved by an iterative scheme, which reduces the question to obtain an approximation of order ε. To carry out this procedure, the problem needs to be reformulated in such a way that its structure does not change with the iterates.

With observation restricted to a single component: displacement, velocity, or temperature, we consider observability of the nonscalar thermoelastic system [1−γΔ]w_tt+Δ²w−αΔθ=0,  θ_t−Δθ+αΔw_t=0, coupling heat conduction with a Kirkhoff or Euler–Bernoulli plate model. One does have observability in arbitrarily short time here, but necessarily has blowup of the sensitivity as the observation time T→0 and also as the coupling coefficient α→0. In this paper we are able to examine the asymptotics of this blowup for two situations: global observation (i.e., on all of Ω) and, with significant restrictions, boundary observation. The blowup rates obtained are of optimal order: 𝒪(T^−5/2) for global observation, corresponding to what is known for 3-dimensional systems, and exponential in 𝒪(1/T) for boundary observation, corresponding to what is known for scalar PDE problems. Our methods permit us also to obtain asymptotics as α→0 – a question which can only arise for systems.

We consider the solution of an interface problem posed in a bounded domain coated with a layer of thickness ε and with external boundary conditions of Dirichlet or Neumann type. Our aim is to build a multi-scale expansion as ε goes to 0 for that solution.

After presenting a complete multi-scale expansion in a smooth coated domain, we focus on the case of a corner domain. Singularities appear, obstructing the construction of the expansion terms in the same way as in the smooth case. In order to take these singularities into account, we construct profiles in an infinite coated sectorial domain.

Combining expansions in the smooth case with splittings in regular and singular parts involving the profiles, we construct two families of multi-scale expansions for the solution in the coated domain with corner. We prove optimal estimates for the remainders of the multi-scale expansions.