Topological obstructions to the diagonalisation of pseudodifferential systems

Capoferri, Matteo; Rozenblum, Grigori; Saveliev, Nikolai; Vassiliev, Dmitri

doi:10.1090/bproc/147

Cited by 3 publications

(2 citation statements)

References 27 publications

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“…In the current paper, we assume that there are no such obstructions, so that one can perform a global diagonalization. Necessary and sufficient conditions for the existence of topological obstructions will be examined extensively in a separate paper [8].…”

Section: Statement Of the Problemmentioning

confidence: 99%

Diagonalization of elliptic systems via pseudodifferential projections

Capoferri

2022

Journal of Differential Equations

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Section: Statement Of the Problemmentioning

confidence: 99%

Diagonalization of elliptic systems via pseudodifferential projections

Capoferri

2022

Journal of Differential Equations

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“…We should mention that an alternative approach to the elastic NP spectral problem can, probably, be based upon recent results on diagonalizing matrix pseudodifferential operators, see [13,14]; the important initial step, the global diagonalization of the principal symbol, is possible according to the results of [15]. We plan to explore this approach in the future.…”

Section: Introductionmentioning

confidence: 99%

The discrete spectrum of the Neumann-Poincaré operator in 3D elasticity

Rozenblum

2023

J. Pseudo-Differ. Oper. Appl.

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For the Neumann-Poincaré (double layer potential) operator in the three-dimensional elasticity we establish asymptotic formulas for eigenvalues converging to the points of the essential spectrum and discuss geometric and mechanical meaning of coefficients in these formulas. In particular, we establish that for any body, there are infinitely many eigenvalues converging from above to each point of the essential spectrum. On the other hand, if there is a point where the boundary is concave (in particular, if the body contains cavities) then for each point of the essential spectrum there exists a sequence of eigenvalues converging to this point from below. The reasoning is based upon the representation of the Neumann-Poincaré operator as a zero order pseudodifferential operator on the boundary and the earlier results by the author on the eigenvalue asymptotics for polynomially compact pseudodifferential operators.

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A decomposition theorem of surface vector fields and spectral structure of the Neumann-Poincaré operator in elasticity

Fukushima,

Ji,

Kang

2023

Trans. Amer. Math. Soc.

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We prove that the space of vector fields on the boundary of a bounded domain with the Lipschitz boundary in three dimensions is decomposed into three subspaces: elements of the first one extend to inside the domain as divergence-free and rotation-free vector fields, the second one to the outside as divergence-free and rotation-free vector fields, and the third one to both the inside and the outside as divergence-free harmonic vector fields. We then show that each subspace in the decomposition is infinite-dimensional. We also prove under a mild regularity assumption on the boundary that the decomposition is almost direct in the sense that any intersection of two subspaces is finite-dimensional. We actually prove that the dimension of intersection is bounded by the first Betti number of the boundary. In particular, if the boundary is simply connected, then the decomposition is direct. We apply this decomposition theorem to investigate spectral properties of the Neumann-Poincaré operator in elasticity, whose cubic polynomial is known to be compact. We prove that each linear factor of the cubic polynomial is compact on each subspace of decomposition separately and those subspaces characterize eigenspaces of the Neumann-Poincaré operator. We then prove all the results for three dimensions, decomposition of surface vector fields and spectral structure, are extended to higher dimensions. We also prove analogous but different results in two dimensions.

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Topological obstructions to the diagonalisation of pseudodifferential systems

Cited by 3 publications

References 27 publications

Diagonalization of elliptic systems via pseudodifferential projections

Diagonalization of elliptic systems via pseudodifferential projections

The discrete spectrum of the Neumann-Poincaré operator in 3D elasticity

A decomposition theorem of surface vector fields and spectral structure of the Neumann-Poincaré operator in elasticity

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