Analyticity of the resolvent for elastic waves in a perturbed isotropic half space

In this article the analytic and asymptotic properties of the resolvent for elastic waves in a three dimensional domain perturbed from the isotropic half space R ³₊ are studied. In this case, the asymptotic expansion of the resolvent at the origin has logarithmic terms. This property is totally different from the case of the whole 3‐dimensional space. The existence of the surface waves like the Rayleigh waves makes this difference. As an application of the asymptotic properties of the resolvent, the rate of the local energy decay estimates for the dynamical equations is obtained. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Existence and uniqueness of a rayleigh surface wave propagating along the free boundary of a transversely isotropic elastic half space

In this article the existence and the uniqueness of a Rayleigh surface wave propagating along the free boundary of a transversely isotropic elastic half space are proved by a spectral method.     

Study of generalized eigenfunctions of a perturbed isotropic elastic half‐space

We consider the self‐adjoint operator governing the propagation of elastic waves in a perturbed isotropic half‐space (perturbation with compact support of a homogeneous isotropic half‐space) with a free boundary condition. We propose a method to obtain, numerical values included, a complete set of generalized eigenfunctions that diagonalize this operator. The first step gives an explicit representation of these functions using a perturbative method. The unbounded boundary is a new difficulty compared with the method used by Wilcox [25], who set the problem in the complement of bounded open set. The second step is based on a boundary integral equations method which allows us to compute these functions. For this, we need to determine explicitly the Green's function of (A₀ – ω²), where A₀ is the self‐adjoint operator describing elastic waves in a homogeneous isotropic half‐space. Copyright © 2000 John Wiley & Sons, Ltd.     

Interference transverse sh waves in a transversally,isotropic, inhomogeneous half space

In the paper Love waves in a transversally, Isotropic, inhomogeneous half space are investigated in the case where the plane of isotropy does not coincide with the boundary plane. The dependence on the angle between them is established to the solution obtained.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 104, pp. 228–234, 1981.     

Polarization-spectral anomalies of surface waves in an almost layered elastic half space

The propagation of Love and Rayleigh waves in an isotropic elastic half space whose properties vary little in a horizontal direction is considered. Expressions are obtained for the admixed components of the displacements.Translated from Zapiski Naukchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 165, pp. 102–114.     

Scattering of electromagnetic plane waves by a spheroid uniformly moving in free space

We investigate the scattering process, generated by a plane electromagnetic field that is incident upon a moving perfectly conducting spheroid. An accurate treatment of the electromagnetic waves interaction with scatterers in uniform motion is based on the special relativity principle. In the object's frame the incident wave is assumed to have a wavelength which is much larger than the characteristic dimension of the scatterer and thus the low‐frequency approximation method is applicable to the scattering problem. For the near electromagnetic field we obtain the zeroth‐order low‐frequency coefficients, while in the far field we calculate the leading terms for the scattering amplitude and scattering cross‐section. Finally, using the inverse Lorentz transform, we obtain the same approximations in the observer's frame. Copyright © 2006 John Wiley & Sons, Ltd.     

The limiting absorption principle in the problem of a transparent wedge

The problem of diffraction of a plane wave by a transparent wedge is considered (the wave numbers inside the wedge and outside it are different). Absorption in a medium is assumed to occur. The existence of a solution satisfying the limiting absorption principle is proved. The existence theorem and the fact that the limit passage can be performed are proved for the corresponding system of equations. Bibliography: 3 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 332, 2006, pp. 138–148.     

Variational analysis of a perturbed free boundary problem

We consider maps u:X→Y of Riemannian manifolds which are locally energy minimizing subject to a constraint of the form Im(u) where M denotes a smooth bounded domain contained in a regular ball B of the target manifold. In general local minima are regular up to a set of vanishing Hausdorff measure, here we show for star-shaped obstacles M singular points can be exluded.     

An application of the limiting absorption principle to a mixed boundary value problem in an infinite strip

We present a complete proof of the existence and uniqueness of solutions of a mixed boundary value problem for the homogeneous Laplace equation in an unbounded parallel strip based on the principle of limiting absorption.     

Scattering of elastic waves through a heterogeneous medium

In this paper we study the elastic wave diffraction in R ³ through a heterogeneous medium, with periodic structure, which occupies a bounded domain. We show that, as the period tends to zero, the solution tends, in some sense, to the solution corresponding to the diffraction by an obstacle made of the classical “homogenized medium”. An analogous result is also proved for the scattering frequencies and the associated scattering functions.     

Weighted polyharmonic equation with Navier boundary conditions in a half space

We study positive solutions of the following polyharmonic equation with Hardy weights associated to Navier boundary conditions on a half space:?????(-?)~mu(x)=u~p(x)/|x|~s,in R_+~n,u(x)=-?u(x)=…=(-?)~(m-1)u(x)=0,on ?R_+~n,(0.1)where m is any positive integer satisfying 02mn.We first prove that the positive solutions of(0.1)are super polyharmonic,i.e.,(-?)~iu0,i=0,1,...,m-1.(0.2) For α=2m,applying this important property,we establish the equivalence between (0.1) and the integral equation u(x)=c_n∫R_+~n(1/|x-y|~(n-α)-1/|x~*-y|~(n-α))u~p(y)/|y|~sdy,(0.3) where x~*=(x1,...,x_(n-1),-x_n) is the reflection of the point x about the plane R~(n-1).Then,we use the method of moving planes in integral forms to derive rotational symmetry and monotonicity for the positive solution of(0.3),in whichαcan be any real number between 0 and n.By some Pohozaev type identities in integral forms,we prove a Liouville type theorem—the non-existence of positive solutions for(0.1).     

Spectral analysis of a perturbed multistratified isotropic elastic strip: new method

We study the self‐adjoint operator (𝒟(A), A) associated with an elastic isotropic and multistratified strip Ω = {(x₁, x₂) ∈ ℝ²; 0 < x₂ < L}, which means that there exists a constant a > 0 such that the density ρ and Lamé coefficients λ and μ are, for (−1)^kx₁ ⩾ a, k = 1, 2, respectively, equal to functions ρ _k, λ _k and μ _k, depending only on x₂. Thanks to [4] the properties of the free operators A^k, k = 1, 2, associated with ρ _k, λ _k and μ _k, are well‐known. We study A by considering it as a ‘compact perturbation’ of the pair (A¹, A²). The difficulty is: if ψ ∈ C(ℝ²) and u ∈ D(A) then ψ u does not necessarily belong to D(A). It has already been encountered in other studies concerning elasticity (cf. [10,18]). Adapting the techniques used there to overcome this difficulty imposes restrictive conditions on λ _k and μ _k. The purpose of this paper is to propose a new method, which removes definitively this difficulty and enables us without restrictive conditions on λ _k and μ _k to prove a limiting absorption principle for A. Copyright © 1999 John Wiley & Sons, Ltd.     

The boundary layer analysis for stokes equations on a half space

The time dependent Stokes equations on a half space are considered. We decompose the solution of these equations in an inviscid solution: a boundary layer solution and a correction. Bounds on these solutions are given, in the appropriate Sobolev spaces, in terms of the norms of the initial and boundary data. The correction is shown to be of the same order of magnitude of the square root of the viscosity.     

Investigation of a problem of an elastic half space subjected to stochastic temperature distribution at the boundary

The present work investigates the responses of stochastic type temperature distribution applied at the boundary of an elastic medium in the context of thermoelasticity without energy dissipation. We consider an one dimensional problem of half space and assume that the bounding surface of the half space is traction free and is subjected to two types of time dependent temperature distributions which are of stochastic types. In order to compare the results predicted by stochastic temperature distributions with the results of deterministic type temperature distribution, the stochastic type temperature distributions applied at the boundary are taken in such a way that they reduce to the cases of deterministic types as special cases. Integral transform technique along with stochastic calculus is used to solve the problem. The approximated solutions for physical fields like, stress, temperature, displacement etc. are derived for very small values of time where stochastic type boundary conditions are taken to be of white noise type. The problem is further illustrated with graphical representation of numerical solutions of the problem for a particular case. A detailed comparison of the results of stochastic temperature, displacement and stress distributions inside the half space with the corresponding results of deterministic distributions is presented and special features of the effects of stochastic type boundary conditions are highlighted.