Consider the Lamé operator $\mathcal{L}(\mathbf{ u} ) :=μΔ\mathbf{u}+(λ+μ) \nabla(\nabla \cdot \mathbf{ u} )$ that arises in the theory of linear elasticity. This paper studies the geometric properties of the (generalized) Lamé eigenfunction $\mathbf{u}$, namely $-\mathcal{L}(\mathbf{ u} )=κ\mathbf{ u}$ with $κ\in\mathbb{R}_+$ and $\mathbf{ u}\in L^2(Ω)^2$, $Ω\subset\mathbb{R}^2$. We introduce the so-called homogeneous line segments of $\mathbf{u}$ in $Ω$, on which $\mathbf{u}$, its traction or their combination via an impedance parameter is vanishing. We give a comprehensive study on characterizing the presence of one or two such line segments and its implication to the uniqueness of $\mathbf{u}$. The results can be regarded as generalizing the classical Holmgren's uniqueness principle for the Lamé operator in two aspects. We establish the results by analyzing the development of analytic microlocal singularities of $\mathbf{u}$ with the presence of the aforesaid line segments. Finally, we apply the results to the inverse elastic problems in establishing two novel unique identifiability results. It is shown that a generalized impedance obstacle as well as its boundary impedance can be determined by using at most four far-field patterns. Unique determination by a minimal number of far-field patterns is a longstanding problem in inverse elastic scattering theory.
In our earlier paper [9], it is proved that a homogeneous rigid, traction or impedance condition on one or two intersecting line segments together with a certain zero point-value condition implies that the solution to the Lamé system must be identically zero, which is referred to as the generalized Holmgren principle (GHP). The GHP enables us to solve a longstanding inverse scattering problem of determining a polygonal elastic obstacle of general impedance type by at most a few far-field measurements. In this paper, we include all the possible physical boundary conditions from linear elasticit…
Dynamic Response of a Finite Circular Plate on an Elastic Half-Space Using the Truncated Lamb Kernel
Lane Department of Computer Science and Electrical Engineering, West Virginia University（西弗吉尼亚大学计算机科学与电气工程学院）；School of Mathematical and Data Sciences, West Virginia University（西弗吉尼亚大学数学与数据科学学院）
We consider the inverse boundary value problem for the system of equations describing elastic waves in isotropic media on a bounded domain in $\mathbb{R}^3$ via a finite-time Laplace transform. The data is the dynamical Dirichlet-to-Neumann map. More precisely, using the full symbol of the transformed Dirichlet-to-Neumann map viewed as a semiclassical pseudodifferential operator, we give an explicit reconstruction of both Lamé parameters and the density, as well as their derivatives, at the boundary. We also show how this boundary reconstruction leads to a decomposition of incoming and outgoin…
We first formulate an inverse problem for a linear fractional Lamé system. We determine the Lamé parameters from exterior partial measurements of the Dirichlet-to-Neumann map. We further study an inverse obstacle problem as well as an inverse problem for a nonlinear fractional Lamé system. Our arguments are based on the unique continuation property for the fractional operator as well as the associated Runge approximation property.
We consider an isotropic elastic medium occupying a bounded domain D whose density and Lamé parameters are piecewise smooth. In the elastic wave initial value inverse problem, we are given the solution operator for the elastic wave equation, but only outside the domain D and only for initial data supported outside D, and we study the recovery of the density and Lamé parameters. For known density, results have recently been obtained using the scattering control method to recover wave speeds. Here, we extend this result to include the recovery of the density in addition to the Lamé parameters un…
Consider an isotropic elastic medium $Ω\subset \mathbb{R}^3$ whose Lamé parameters are piecewise smooth. In the elastic wave initial value inverse problem, we are given the solution operator for the elastic wave equation, but only outside $Ω$ and only for initial data supported outside $Ω$. Using the recently introduced scattering control series in the acoustic case, we prove that piecewise smooth Lamé parameters are uniquely determined by this map under certain geometric conditions. We also show the extent that multiple scattering in the interior may be suppressed and eliminated with access t…
In this paper we study Steklov eigenvalues for the Lamé operator which arise in the theory of linear elasticity. In this eigenproblem the spectral parameter appears in a Robin boundary condition, linking the traction and the displacement. To establish the existence of a countable spectrum for this problem, we present an extension of Korn's inequality. We also show that a proposed conforming Galerkin scheme provides convergent approximations to the true eigenvalues. A standard finite element method is used to conduct numerical experiments on 2D and 3D domains to support our theoretical findings…
We study the propagation of transient waves under the action of a vertical step point load on the surface of a half-space filled by a block medium. The block medium is modeled by a square lattice of masses connected by springs in the directions of the axes x, y, and in the diagonal directions. The problem is solved by two methods. Analytically, we obtain asymptotic solutions in the vicinity of the Rayleigh wave at large time intervals. Numerically, we obtain a solution for any finite time interval. We compare these solutions with each other and with the solution to the Lamb problem for an elas…
In this paper we show that the shear modulus $μ$ of an isotropic elastic body can be stably recovered by the knowledge of one single displacement field whose boundary data can be assigned independently of the unknown elasticity tensor.
We investigate Lamé systems in periodically perforated domains, and establish quantitative homogenization results in the setting where the domain is clamped at the boundary of the holes. Our method is based on layer potentials and it provides a unified proof for various regimes of hole-cell ratios (the ratio between the size of the holes and the size of the periodic cells), and, more importantly, it yields natural correctors that facilitate error estimates. A key ingredient is the asymptotic analysis for the rescaled cell problems, and this is studied by exploring the convergence of the period…
We consider the inverse problem of determining the Lamé parameters and the density of a three-dimensional elastic body from the local time-harmonic Dirichlet-to-Neumann map. We prove uniqueness and Lipschitz stability of this inverse problem when the Lamé parameters and the density are assumed to be piecewise constant on a given domain partition.
For a bounded domain $Ω\subset {\Bbb R}^n$ with smooth boundary, we explicitly calculate the first two coefficients of the asymptotic expansion of the trace of the strongly continuous semigroup associated with the Navier-Lamé operator on $Ω$ as $t\to 0^+$. These coefficients (i.e., spectral invariants) provide precise information for the volume of the elastic body $Ω$ and the surface area of the boundary $\partial Ω$ in terms of the spectrum of the Navier-Lamé problem. As an application, we show that an $n$-dimensional ball is uniquely determined by its Navier-Lamé spectrum among all bounded e…
It is known that the Neumann--Poincaré operator for the Lamé system of linear elasticity is polynomially compact and, as a consequence, that its spectrum consists of three non-empty sequences of eigenvalues accumulating to certain numbers determined by Lamé parameters, if the boundary of the domain where the operator is defined is $C^\infty$-smooth. We extend this result to less smooth boundaries, namely, $C^{1, α}$-smooth boundaries for some $α> 0$. The results are obtained by proving certain identities for surface Riesz transforms, which are singular integral operators of nonconvolution type…
We consider the inverse problem of identifying an unknown inclusion contained in an elastic body by the Dirichlet-to-Neumann map. The body is made by linearly elastic, homogeneous and isotropic material. The Lamé moduli of the inclusion are constant and different from those of the surrounding material. Under mild a-priori regularity assumptions on the unknown defect, we establish a logarithmic stability estimate. For the proof, we extend the approach used for electrical and thermal conductors in a novel way. Main tools are propagation of smallness arguments based on three-spheres inequality fo…
We study an inverse problem for fractional elasticity. In analogy to the classical problem of linear elasticity, we consider the unique recovery of the Lamé parameters associated to a linear, isotropic fractional elasticity operator from fractional Dirichlet-to-Neumann data. In our analysis we make use of a fractional matrix Schrödinger equation via a generalization of the so-called Liouville reduction, a technique classically used in the study of the scalar conductivity equation. We conclude that unique recovery is possible if the Lamé parameters agree and are constant in the exterior, and th…
We study boundary determination for an inverse problem associated to the time-harmonic Maxwell equations and another associated to the isotropic elasticity system. We identify the electromagnetic parameters and the Lamé moduli for these two systems from the corresponding boundary measurements. In a first step we reconstruct Lipschitz magnetic permeability, electric permittivity and conductivity on the surface from the ideal boundary measurements. Then, we study inverse problems for Maxwell equations and the isotropic elasticity system assuming that the data contains measurement errors. For bot…
We consider a multilayer hyperbolic-parabolic PDE system which constitutes a coupling of 3D thermal - 2D elastic - 3D elastic dynamics, in which the boundary interface coupling between 3D fluid and 3D structure is realized via a 2D elastic equation. Our main result here is one of strong decay for the given multilayered - heat system. That is, the solution to this composite PDE system is stabilized asymptotically to the zero state. Our proof of strong stability takes place in the "frequency domain" and ultimately appeals to the pointwise resolvent condition introduced by Tomilov [45]. This very…
In the region between close-to-touching hard inclusions, the stress may be arbitrarily large as the inclusions get closer. The stress is represented by the gradient of a solution to the Lamé system of linear elasticity. We consider the problem of characterizing the gradient blow-up of the solution in the narrow region between two inclusions and estimating its magnitude. We introduce singular functions which are constructed in terms of nuclei of strain and hence are solutions of the Lamé system, and then show that the singular behavior of the gradient in the narrow region can be precisely captu…
This paper is concerned with the elastic near cloaking for the Lamé system in three-dimensions using the notion of elastic scattering coefficients (ESC). Accordingly, the ESC of arbitrary three-dimensional objects are designed and some of their properties are discussed using elements of the elastic layer potential theory. Then, near-cloaking structures, coined as ESC-vanishing-structures, are constructed for the elastic cloaking at a fixed frequency or a band of frequencies. These multi-layered structures are designed so that their first few significant ESC vanish ahead of transformation-elast…
We consider the inverse problem of determining the Lamé moduli for a piecewise constant elasticity tensor ${\mathbb C}= \sum_{j} {\mathbb C}_j χ_{D_j}$, where $\{D_j\}$ is a known finite partition of the body $Ω$, from the Dirichlet-to-Neumann map. We prove that Lipschitz stability estimates can be derived under $C^{1,α}$ regularity assumptions on the interfaces.
We establish a decoupling result for the $P$ and $S$ waves of linear, isotropic elasticity, in the setting of twice-differentiable Lamé parameters. Precisely, we show that the $P\leftrightarrow S$ components of the wave propagation operator are regularizing of order one on $L^2$ data, by establishing the diagonalization of the elastic system modulo a $L^2$-bounded operator. Effecting the diagonalization in the setting of twice-differentiable coefficients depends upon the symbol of the conjugation operator having a particular structure.
We consider a heterogeneous elastic structure which is stratified in some direction. We derive the limit problem under the assumption that the Lamé coefficients and their inverses weakly* converge to Radon measures. Our method applies also to linear second-order elliptic systems of partial differential equations and in particular, for the case $d=1$, this addresses the previously open problem of determining the asymptotic behaviour in this context for the general anisotropic heat equation.
Several recent papers have focused their attention in proving the correct analogue to the Lieb-Thirring inequalities for non self-adjoint operators and in finding bounds on the distribution of their eigenvalues in the complex plane. This paper provides some improvement in the state of the art in this topic. Precisely, we address the question of finding quantitative bounds on the discrete spectrum of the perturbed Lamé operator of elasticity $-Δ^\ast + V$ in terms of $L^p$-norms of the potential. Original results within the self-adjoint framework are provided too.
This paper is concerned with the theoretical study of plasmonic resonances for linear elasticity governed by the Lamé system in $\mathbb{R}^3$, and their application for cloaking due to anomalous localized resonances. We derive a very general and novel class of elastic structures that can induce plasmonic resonances. It is shown that if either one of the two convexity conditions on the Lamé parameters is broken, then we can construct certain plasmon structures that induce resonances. This significantly extends the relevant existing studies in the literature where the violation of both convexit…
We study increasing stability in the inverse source problems for the Helmholtz equation and the classical Lame system from (minimal) boundary data at multiple wave numbers. By using the Fourier transform with respect to wave numbers, explicit bounds for analytic continuation of the data to larger wave numbers, the Hyugens' principle, and sharp bounds in the corresponding dynamical initial boundary value problems, increasing (with growing wave numbers interval) stability estimates for source terms are obtained.
This paper is devoted to providing quantitative bounds on the location of eigenvalues, both discrete and embedded, of non self-adjoint Lamé operators of elasticity $-Δ^\ast + V$ in terms of suitable norms of the potential $V$. In particular, this allows to get sufficient conditions on the size of the potential such that the point spectrum of the perturbed operator remains empty. In three dimensions we show full spectral stability under suitable form-subordinated perturbations: we prove that the spectrum is purely continuous and coincides with the non negative semi-axis as in the free case.
We discuss the determination of the Lamé parameters of an elastic material by the means of boundary measurements. We will combine previous results of Eskin-Ralston and Isakov to prove inverse results in the case of bounded domains with partial data. Moreover, we generalise these results to infinite cylinders.
We consider the problem of elastic diffraction tomography, which consists in reconstructing elastic properties (i.e. mass density and elastic Lamé parameters) of a weakly scattering medium from full-field data of scattered waves outside the medium. Elastic diffraction tomography refers to the elastic inverse scattering problem after linearization using a firstorder Born approximation. In this paper, we prove the Fourier diffraction theorem, which relates the 2D Fourier transform of scattered waves with the Fourier transform of the scatterer in the 3D spatial Fourier domain. Elastic wave mode s…
We prove dispersive estimate for the elastic wave equation by which we extend the known Strichartz estimates for the classical wave equation to those for the elastic wave equation. In particular, the endpoint Strichartz estimates are deduced. For the purpose we diagonalize the symbols of the Lamé operator and its semigroup, which also gives an alternative and simpler proofs of the previous results on perturbed elastic wave equations. Furthermore, we obtain uniform Sobolev inequalities for the elastic wave operator.
We deal with the problem of the linearized and isotropic elastic inverse scattering by interfaces. We prove that the scattered $P$-parts or $S$-parts of the far field pattern, corresponding to all the incident plane waves of pressure or shear types, uniquely determine the obstacles for both the penetrable and impenetrable obstacles. In addition, we state a reconstruction procedure. In the analysis, we assume only the Lipschitz regularity of the interfaces and, for the penetrable case, the Lam{é} coefficients to be measurable and bounded, inside the obstacles, with the usual jumps across these …
For a compact connected Riemannian $n$-manifold $(Ω,g)$ with smooth boundary, we explicitly calculate the first two coefficients $a_0$ and $a_1$ of the asymptotic expansion of $\sum_{k=1}^\infty e^{-t τ_k^\mp}= a_0t^{-n/2} \mp a_1 t^{-(n-1)/2}+a_2^\mp t^{-(n-2)/2} +\cdots+ a_m^\mp t^{-(n-m)/2} +O(t^{-(n-m-1)/2})$ as $t\to 0^+$, where $τ^-_k$ (respectively, $τ^+_k$) is the $k$-th Navier-Lamé eigenvalue on $Ω$ with Dirichlet (respectively, Neumann) boundary condition. These two coefficients provide precise information for the volume of the elastic body $Ω$ and the surface area of the boundary $\…
We study some particular cases of the $n$-well problem in two-dimensional linear elasticity. Assuming that every well in $\mathcal{U}\subset\mathbb{R}^{2\times 2}_\text{sym}$ belong to the same two-dimensional affine subspace, we characterize the symmetric lamination convex hull $L^e(\mathcal{U})$ for any number of wells in terms of the symmetric lamination convex hull of all three-well subsets contained in $\mathcal{U}$. For a family of four-well sets where two pairs of wells are rank-one compatible, we show that the symmetric lamination convex and quasiconvex hulls coincide, but are strictly…
We investigate the problem of recovering the initial displacement f for a solution u of a linear, isotropic, non-homogeneous elastic wave equation, given measurements of u on [0,T] x \partial Ω, where Ω\subset\R^3 is some bounded domain containing the support of f. For the acoustic wave equation, this problem is known as thermoacoustic tomography (TAT), and has been well-studied; for the elastic wave equation, the situation is somewhat more subtle, and we give sufficient conditions on the Lamé parameters to ensure that recovery is possible.
