• Corpus ID: 251320541

Two-term spectral asymptotics in linear elasticity on a Riemannian manifold

@inproceedings{Liu2022TwotermSA,
  title={Two-term spectral asymptotics in linear elasticity on a Riemannian manifold},
  author={Genqiang Liu},
  year={2022}
}
. In this note, by explaining two key methods that were employed in [Liu-21] and by giving some remarks, we show that the proof of Theorem 1.1 in [Liu-21] is a rigorous proof based on theory of strongly continuous semigroups and pseudodifferential operators. All remarks and comments to paper [Liu-21], which were given by Matteo Capoferri, Leonid Friedlander, Michael Levitin and Dmitri Vassiliev in [CaFrLeVa-22], are incorrect. The so-called “numerical counter-examples” in [CaFrLeVa-22] are… 
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References

SHOWING 1-10 OF 25 REFERENCES

Providence

The geometric invariants for the spectrum of the Stokes operator

For a bounded domain $$\Omega \subset {\mathbb {R}}^n$$ Ω ⊂ R n with smooth boundary, we explicitly calculate the first two coefficients of the asymptotic expansion for the integral of the trace of…

Geometric Invariants of Spectrum of the Navier–Lamé Operator

  • Genqiang Liu
  • Mathematics
    The Journal of Geometric Analysis
  • 2020
For a compact connected Riemannian n-manifold (Ω,g)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy}…

Determining the viscosity from the boundary information for incompressible fluid

For the Stokes equations in a compact connected Riemannian $n$-manifold $(\Omega,g)$ with smooth boundary $\partial \Omega$, we give an equivalent new system of elliptic equations with $(n+1)$…

Determining anisotropic real-analytic metric from boundary electromagnetic information

For a compact Riemannian manifold $(M, g)$ with smooth boundary $\partial M$, we explicitly give local representation and full symbol expression for the boundary electromagnetic admittance map…

Determination of isometric real-analytic metric and spectral invariants for elastic Dirichlet-to-Neumann map on Riemannian manifolds

For a compact Riemannian manifold $(\Omega, g)$ with smooth boundary $\partial \Omega$, we explicitly give local representation and full symbol expression for the elastic Dirichlet-to-Neumann map…

Second term of the spectral asymptotic expansion of the Laplace - Beltrami operator on manifolds with boundary

A weld head control and guidance system utilizing non-contact proximity sensors which generate a signal based on the electrical conductivity of a workpiece. The system includes electrical controls…

The Asymptotic Distribution of Eigenvalues of Partial Differential Operators

Main results Oscillatory integrals Construction of the wave group Singularities of the wave group Proof of main results Mechanical applications Appendix A. Spectral problem on the half-line Appendix…

Non-homogeneous boundary value problems and applications

7 Scalar and Vector Ultra-Distributions.- 1. Scalar-Valued Functions of Class Mk.- 1.1 The Sequences {Mk}.- 1.2 The Space $${D_{{M_k}}}\left( H \right)$$.- 1.3 The Spaces $${D_{{M_k}}}\left( H…

Curvature and the Eigenvalues of the Laplacian

A famous formula of H. Weyl [17] states that if D is a bounded region of R d with a piecewise smooth boundary B, and if 0 > γ1 ≥ γ2 ≥ γ3 ≥ etc. ↓−∞ is the spectrum of the problem …