Asymptotic Analysis

This note is concerned with the damped wave equation ε²∂_ttu+∂_tu−Δu+f(u)=g depending on a small parameter ε and with the corresponding parabolic equation ∂_tu−Δu+f(u)=g obtained in the singular limit ε→0. The existence of a family ℳ_ε of exponential attractors which is Hölder continuous with respect to ε is proved.

In this paper, we are concerned with inverse problems of determining spatially varying two Lamé coefficients and the mass density by a finite number of boundary observations. Our main results are Lipschitz stability estimates for the inverse problems under suitable conditions on initial values and boundary values. In particular, if we take suitable quadratic functions as initial displacements, then we can prove the Lipschitz stability with the minimum times of observations.

We consider some elliptic boundary value problems in a self-similar ramified domain of $\mathbb{R}^{2}$ with a fractal boundary with Laplace's equation and nonhomogeneous Neumann boundary conditions. The Hausdorff dimension of the fractal boundary is greater than one. The goal is twofold: first rigorously define the boundary value problems, second approximate the restriction of the solutions to subdomains obtained by stopping the geometric construction after a finite number of steps. For the first task, a key step is the definition of a trace operator. For the second task, a multiscale strategy based on transparent boundary conditions and on a wavelet expansion of the Neumann datum is proposed, following an idea contained in a previous work by the same authors. Error estimates are given and numerical results are presented.

The question studied concerns the behaviour, as α→∞, of the functions defined by the series with general terms a_k=k^−ν(k+α)⁻¹ and a_k=k^−ν(k−α)⁻¹, k=1,2,… , where the parameter $\alpha\not=k$ and the value ν>0. The asymptotic expansions of such functions in terms of powers of α⁻¹ when α→∞ are found (a logarithmical factor is also present in the only term when ν is integer). It is shown that the coefficients of the asymptotic expansions obtained are determined by values of the Riemann zeta-function ζ(z) on the sequence of points ν−m, where m=0,1,… and m≠ν−1. The Mellin transform technique, the Cauchy theorem on residues, and the recurrence formulae, connecting the series in question for the values of parameter ν and ν+1, are employed when deriving the asymptotic expansions.

We consider the semiclassical asymptotic behaviour of the number of eigenvalues smaller than E for elliptic operators in $L^{2}(\mathbb{R}^{d})$ . We describe a method of obtaining remainder estimates related to the volume of the region of the phase space in which the principal symbol takes values belonging to the intervals [E′;E′+h], where E′ is close to E. This method allows us to derive sharp remainder estimates O(h^1−d) for a class of symbols with critical points and non-smooth coefficients.