Asymptotic Analysis

We extend the classical concept of transformation operators to the one‐dimensional wave equation with spatially nonhomogeneous coefficients containing the first order damping term. The equation governs the vibrations of a damped string. Our results hold in the cases of an infinite, semi‐infinite or a finite string. Transformation operators were introduced in the fifties by I.M. Gelfand, B.M. Levitan and V.A. Marchenko in connection with the inverse scattering problem for the one‐dimensional Schrödinger equation. In the classical case, the transformation operator maps the exponential function (stationary wave function of a free particle) into the so‐called Jost solution of the perturbed Schrödinger equation. In our case, it is natural to introduce two transformation operators, which we call outgoing and incoming transformation operators respectively. (The terminology is motivated by an analog with the Lax–Phillips scattering theory.) The first of them is related to the nonselfadjoint quadratic operator pencil generated by the original problem, and the second one is related to the adjoint pencil. We introduce a pair of asymptotically exponential solutions for each of the pencils and show that our transformation operators map certain exponential type functions to these solutions. Our main results are the proof of the existence of transformation operators (which have the forms of the identity operator plus certain Volterra integral operators) and estimates for their kernels. To obtain these results, we derive a pair of integral equations for the kernels of the transformation operators. These equations are the generalizations of the corresponding classical equation which is valid in the case of a wave equation without damping term. One of possible applications of the method developed in this paper is given in our forthcoming work. In that work, we use the transformation operators to prove the fact that the dynamics generator of a finite string with damping both in the equation and in the boundary conditions is a Riesz spectral operator. The latter result provides a class of nontrivial examples of nonselfadjoint operators which admit an analog of the spectral decomposition. The result also has significant applications in the control theory of distributed parameter systems.

This paper deals with the continuos dependence with respect to the obstacle of the solutions of obstacle problems with L¹(Ω)‐ data associated to non linear differential operators of Leray–Lions type mapping W^1,p₀(Ω) into its dual W^−1,p′(Ω).

This paper is devoted to the study of a semi‐classical NLS equation with a small parameter ε in two space dimensions, with oscillating data that are highly oscillating in one direction only, with the aim of modelling geometric optics with a caustic consisting of a line in $\mathbb{R}^{2}$ . We prove that the phenomena encountered are typically one‐dimensional. In order to describe the results, we introduce spaces that make it possible to define scattering operators with a parameter.

The effect of thin ferromagnetic films is studied in this paper. We shall adopt the Landau–Lifschitz–Gilbert equation as a phenomenological model for the ferromagnetic materials. Asymptotic analysis of the fields and the magnetization vector inside thin ferromagnetic films are performed and their convergences are established. Our results reveal the physical nature of this nonlinear model. They also provide an effective method for overcoming the computational difficulties that arise in thin ferromagnetic coatings.

We consider the linear heat equation with potential in a n‐dimensional thin cilinder Ω_ε=Ω×(0,ε) where Ω is a bounded open smooth set of $\mathbb{R}^{n-1}$ with n≥2 and ε is a small parameter. We study the null controllability problem when the control acts in a cylindrical region ω_ε=ω×(0,ε), where ω⊂Ω is an open and non‐empty subset of Ω. We prove that, under appropriate boundary conditions, for a suitable class of potentials the heat equation is uniformly null controllable as ε→0. We also prove the convergence of the controls to a null control for the n−1‐dimensional heat equation in Ω. Similar results are proved for the semilinear heat equation with globally Lipschitz nonlinearities.

The semi‐classical study of a 1‐dimensional Schrödinger operator near a non‐degenerate maximum of the potential has lead Colin de Verdière and Parisse to prove a microlocal normal form theorem for any 1‐dimensional pseudo‐differential operator with the same kind of singularity. We present here a generalization of this result to pseudo‐differential integrable systems of any finite degree of freedom with a Morse singularity. Our results are based upon Eliasson's classical mechanical study of critical integrable systems.

We study Dirichlet boundary control problem for semilinear parabolic equations, with pointwise state constraints. By penalizing the Dirichlet boundary condition, we define a family of control problems with Robin boundary conditions. We study the asymptotic behavior of solutions of these penalized problems, when the penalty parameter tends to infinity. We also prove that optimality conditions for the Dirichlet boundary control problem can be obtained by passage to the limit in the optimality conditions of the penalized problem.