Asymptotic Analysis

We consider a class of modified Schrödinger operators where the semiclassical Laplacian is perturbed with h-dependent interface conditions occurring at the boundaries of the potential’s support. Under positivity assumptions on the potential, we show that this modification produces a small perturbation on the dynamics as h → 0 , independently from the time scale. In the case of a time dependent potential, this yields uniform-in-h stability estimates for products of instantaneous propagators. Then, following a standard approach, the non-autonomous dynamical system is defined as a limit of stepwise propagators and its small-h expansion is provided under suitable regularity assumptions on the potential’s variations.

We study the stochastic diffusive limit of a kinetic radiative transfer equation, which is non linear, involving a small parameter and perturbed by a smooth random term. Under an appropriate scaling for the small parameter, using a generalization of the perturbed test-functions method, we show the convergence in law to a stochastic non linear fluid limit.

We consider the Cauchy problem in R n for some types of damped wave equations. We derive asymptotic profiles of solutions with weighted L 1 , 1 ( R n ) initial data by employing a simple method introduced in [Math. Meth. Appl. Sci. 27 (2004), 865–889].

We consider the initial value problem for a system of cubic nonlinear Schrödinger equations with different masses in one space dimension. Under a suitable structural condition on the nonlinearity, we will show that the small amplitude solution exists globally and decays at the rate O ( t − 1 / 2 ( log t ) − 1 / 2 ) in L ∞ as t tends to infinity, if the system satisfies certain mass relations.

In this paper we study the asymptotic behaviour as ε → 0 of the spectrum of the elliptic operator A ε = − 1 b ε div ( a ε ∇ ) posed in a bounded domain Ω ⊂ R n ( n ⩾ 2 ) subject to Dirichlet boundary conditions on ∂ Ω . When ε → 0 both coefficients a ε and b ε become high contrast in a small neighborhood of a hyperplane Γ intersecting Ω. We prove that the spectrum of A ε converges to the spectrum of an operator acting in L 2 ( Ω ) ⊕ L 2 ( Γ ) and generated by the operation − Δ in Ω ∖ Γ , Dirichlet boundary conditions on ∂ Ω and certain interface conditions on Γ containing the spectral parameter in a nonlinear manner. The eigenvalues of this operator may accumulate at a finite point. Then we study the same problem, when Ω is an infinite straight strip (“waveguide”) and Γ is parallel to its boundary. We show that A ε has at least one gap in the spectrum when ε is small enough and describe the asymptotic behaviour of this gap as ε → 0 . The proofs are based on methods of homogenization theory.

We consider a system of reaction–diffusion equations in a bounded interval of the real line, with emphasis on the metastable dynamics, whereby the time-dependent solution approaches the steady state in an asymptotically exponentially long time interval as the viscosity coefficient ε > 0 goes to zero. To rigorous describe such behavior, we analyze the dynamics of layered solutions localized far from the stable configurations of the system, and we derive an ODE for the position of the internal interfaces.

We consider an optimal control problem associated to Dirichlet boundary value problem for linear elliptic equations on a bounded domain Ω. We take the matrix-valued coefficients A = A sym + A skew of such system as a control in L 1 ( Ω ; S sym N ) ⊕ L 2 p ( Ω ; S skew N ) . One of the important features of the class of admissible controls is the fact that the matrices A ( x ) are unbounded on Ω and eigenvalues of the symmetric parts A sym may vanish in Ω. In spite of the fact that the equations of this type can exhibit non-uniqueness of weak solutions, the corresponding OCP is well-possed and admits at least one solution. At the same time, optimal solutions to such problem can inherit a singular character of the matrices A skew . We indicate two types of optimal solutions to the above problem and show that one of them can not be attained by optimal solutions of regularized problems for coercive elliptic equations with bounded coefficients, using the Steklov smoothing of matrix-valued controls A.