Unique continuation for a stationary isotropic lamé system with variable coefficients

We give a simple proof of the unique continuation properties for a stationary isotropic Lamé system with variable coefficients. The key is that by introducing the divergence component of the displacement we transform the Lamé system into an elliptic system with the same principal part.     

Diagonalization of the system of static Lamé equations of isotropic linear elasticity

We find a simplest representation for the general solution to the system of the static Lamé equations of isotropic linear elasticity in the form of a linear combination of the first derivatives of three functions that satisfy three independent harmonic equations. The representation depends on 12 free parameters choosing which it is possible to obtain various representations of the general solution and simplify the boundary value conditions for the solution of boundary value problems as well as the representation of the general solution for dynamic Lamé equations. The system of Lamé equations diagonalizes; i.e., it is reduced to the solution of three independent harmonic equations. The representation implies three conservation laws and some formula for producing new solutions which makes it possible, given a solution, to find new solutions to the static Lamé equations by derivations. In the two-dimensional case of a plane deformation, the so-found solution immediately implies the Kolosov-Muskhelishvili representation for shifts by means of two analytic functions of complex variable. Two examples are given of applications of the proposed method of diagonalization of the two-dimensional elliptic systems.     

Recovering a Lamé kernel in a viscoelastic system

In the equation of viscoelasticity related to a geometric cylinder, we recover the spatial part p of a factorized Lamé kernel depending on two spatial sectional variables. More explicitly, we determine sufficient conditions on the data ensuring the (local) uniqueness of p and its continuous dependence on the data in suitable metrics.     

General decay for a coupled Lamé system of nonlinear viscoelastic equations

In this paper, we consider a coupled Lamé system of nonlinear viscoelastic equations with general source terms. Under some suitable conditions on the initial data and the relaxation functions, we prove an asymptotic stability result of global solution taking into account that the kernel is not necessarily decreasing. This work generalizes and improves earlier results in the literature.     

Maximal regularity for the Lamé system in certain classes of non-smooth domains

The aim of this article is twofold. On the one hand, we study the well-posedness of the Lamé system ${-\mu\Delta-\mu'\nabla{\rm div} }The aim of this article is twofold. On the one hand, we study the well-posedness of the Lamé system -mD-m￠?div{-\mu\Delta-\mu'\nabla{\rm div} } in L ^q (Ω), where Ω is an open subset of \mathbbRⁿ{{\mathbb{R}}^n} satisfying mild regularity assumptions and the Lamé moduli μ, μ′ are such that μ > 0 and μ + μ′ > 0. On the other hand, we prove the analyticity of the semigroup generated by the Lamé operator as well as the maximal regularity property for the time-dependent Lamé system equipped with a homogeneous Dirichlet boundary condition based on off-diagonal estimates.     

Lamé system of elasticity theory in a plane orthotropic medium

The authors develop a functional-theoretic approach to solving boundary-value problems for the Lamé system of elasticity theory. Special attention is paid to the case of a plane orthotropic medium. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 53, Suzdal Conference-2006, Part 1, 2008.     

Wiener Type Regularity of a Boundary Point for the 3D Lamé System

In this paper, we study the 3D Lamé system and establish its weighted positive definiteness for a certain range of elastic constants. By modifying the general theory developed in Maz’ya (J Duke Math 115(3): 479–512, 2002), we then show, under the assumption of weighted positive definiteness, that the divergence of the classical Wiener integral for a boundary point guarantees the continuity of solutions to the Lamé system at this point.     

On the Geometric Properties of the Poisson Kernel for the Lamé Equation

Computational Mathematics and Mathematical Physics - It is shown that the Poisson kernel for the Lamé equation in a disk can be interpreted as a bi-univalent mapping of the projection of an...     

Lamé hypersurfaces and Lagrange multipliers

Starting with a standard calculus problem of finding the rectangle of maximum area, which can be inscribed in an ellipse, we generalizethe result to Lamé curves and hypersurfaces, before investigating the volume enclosed by a generalized Lamé hypersurface. Suggestions of classroom use and a selection of possible classroom exercises are also given.     

Hölder stability and uniqueness for the mean field games system via Carleman estimates

We are concerned with the mathematical study of the mean field games system (MFGS). In the conventional setup, the MFGS is a system of two coupled nonlinear parabolic partial differential equation (PDE)s of the second order in a backward–forward manner, namely, one terminal and one initial condition are prescribed, respectively, for the value function and the population density . In this paper, we show that uniqueness of solutions to the MFGS can be guaranteed if, among all four possible terminal and initial conditions, either only two terminals or only two initial conditions are given. In both cases, Hölder stability estimates are proven. This means that the accuracies of the solutions are estimated in terms of the given data. Moreover, these estimates readily imply uniqueness of corresponding problems for the MFGS. The main mathematical apparatus to establish those results is two new Carleman estimates, which may find application in other contexts associated with coupled parabolic PDEs.     

Poincaré mapping for a time-delay impulsive system

The state estimation problem in the system of time-delay impulsive differential equation formulated in mathematical biology is considered. The discrete mapping (the Poincaré mapping) completely describing the evolution of the state of a hybrid observer from impulse to impulse is obtained; this allows one to study the properties of the original system based on its discrete dynamics.     

Homogenization of the Neumann problem for the Lamé equations of linear elasticity in domains with a periodic system of channels of small length

The problem of homogenization is considered for the solutions of the Neumann problem for the Lamé system of plane elasticity in two-dimensional domains with channels that have the form of rectilinear cylinders of length ε ^q (ε is a small positive parameter, q = const > 0) and radius a _ɛ. The bases of the channels form an ε-periodic structure on the hyperplane {x ∈ ℝ²: x ₁ = 0} and their number is equal to N _ɛ= O(ɛ⁻¹) as ε → 0. Under the limit condition lim on the parameters characterizing the geometry of the domain, the weak H ¹-limit of the generalized solution of this problem is found. __________ Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 25, pp. 310–322, 2005.     

Transport solutions of the Lamé equations and shock elastic waves

The Lamé system describing the dynamics of an isotropic elastic medium affected by a steady transport load moving at subsonic, transonic, or supersonic speed is considered. Its fundamental and generalized solutions in a moving frame of reference tied to the transport load are analyzed. Shock waves arising in the medium at supersonic speeds are studied. Conditions on the jump in the stress, displacement rate, and energy across the shock front are obtained using distribution theory. Numerical results concerning the dynamics of an elastic medium influenced by concentrated transport loads moving at sub-, tran- and supersonic speeds are presented.     

Existence of martingale and stationary suitable weak solutions for a stochastic Navier–Stokes system

The existence of suitable weak solutions of 3D Navier–Stokes equations, driven by a random body force, is proved. These solutions satisfy a local balance of energy. Existence of statistically stationary solutions is also proved.     

Multigrid methods for saddle point problems: Stokes and Lamé systems

We develop new multigrid methods for a class of saddle point problems that include the Stokes system in fluid flow and the Lamé system in linear elasticity as special cases. The new smoothers in the multigrid methods involve optimal preconditioners for the discrete Laplace operator. We prove uniform convergence of the \(W\) -cycle algorithm in the energy norm and present numerical results for \(W\) -cycle and \(V\) -cycle algorithms.     

Asymptotics of eigenvalues of the two-dimensional Dirichlet boundary-value problem for the Lamé operator in a domain with a small hole

The Dirichlet boundary-value problem for the eigenvalues of the Lamé operator in a twodimensional bounded domain with a small hole is studied. The asymptotics of the eigenvalue of this boundary-value problem is constructed and justified up to the power of the parameter defining the diameter of the hole.     

Distribution of zeros of the Hermite-Padé polynomials for a system of three functions,and the Nuttall condenser

The well-known approach of J. Nuttall to the derivation of strong asymptotic formulas for the Hermite-Padé polynomials for a set of m multivalued functions is based on the conjecture that there exists a canonical (in the sense of decomposition into sheets) m-sheeted Riemann surface possessing certain properties. In this paper, for m = 3, we introduce a notion of an abstract Nuttall condenser and describe a procedure for constructing (based on this condenser) a three-sheeted Riemann surface $\Re _3$ that has a canonical decomposition. We consider a system of three functions $\mathfrak{f}_1 ,\mathfrak{f}_2 ,\mathfrak{f}_3$ that are rational on the constructed Riemann surface and satisfy the independence condition det . In the case of m = 3, we refine the main theorem from Nuttall’s paper of 1981. In particular, we show that in this case the complement ?? \ B of the open (possibly, disconnected) set B ? ?? introduced in Nuttall’s paper consists of a finite number of analytic arcs. We also propose a new conjecture concerning strong asymptotic formulas for the Padé approximants.     

Harnack estimate for the H~k-flow

We obtain the Harnack estimate of the solution to H~k-flow in Euclidean space R~(n 1),for k>0.By using this estimate,we get some corollaries about the translation soliton.