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 TMF, 1999, Volume 119, Number 2, Pages 295–307 (Mi tmf740)

Two physical applications of the Laplace operator perturbed on a null set

I. Yu. Popov, D. A. Zubok

St. Petersburg State University of Information Technologies, Mechanics and Optics

Abstract: Two physical applications of the Laplace operator perturbed on a set of zero measure are suggested. The approach is based on the theory of self-adjoint extensions of symmetrical operators. The first application is a solvable model of scattering of a plane wave by a perturbed thin cylinder. “Nonlocal” extensions are described. The model parameters can be chosen such that the model solution is an approximation of the corresponding “realistic” solution. The second application is the description of the time evolution of a one-dimensional quasi-Chaplygin medium, which can be reduced using a hodograph transform to the ill-posed problem of the Laplace operator perturbed on a set of codimension two in $\mathbf R^3$. Stability and instability conditions are obtained.

DOI: https://doi.org/10.4213/tmf740

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English version:
Theoretical and Mathematical Physics, 1999, 119:2, 629–639

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Revised: 02.11.1998

Citation: I. Yu. Popov, D. A. Zubok, “Two physical applications of the Laplace operator perturbed on a null set”, TMF, 119:2 (1999), 295–307; Theoret. and Math. Phys., 119:2 (1999), 629–639

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. B. E. Kanguzhin, D. B. Nurakhmetov, N. E. Tokmagambetov, “Laplace operator with $\delta$-like potentials”, Russian Math. (Iz. VUZ), 58:2 (2014), 6–12
2. Nalzhupbayeva G., “Formulas For the Eigenvalues of the Iterated Laplacian With Singular Potentials”, International Conference Functional Analysis in Interdisciplinary Applications (FAIA2017), AIP Conference Proceedings, 1880, eds. Kalmenov T., Sadybekov M., Amer Inst Physics, 2017, UNSP 050005
3. Nalzhupbayeva G., “Remark on the Eigenvalues of the M-Laplacian in a Punctured Domain”, Complex Anal. Oper. Theory, 12:3 (2018), 599–606
4. Nalzhupbayeva G., “Spectral Properties of One Elliptic Operator in a Punctured Domain”, AIP Conference Proceedings, 1997, eds. Ashyralyev A., Lukashov A., Sadybekov M., Amer Inst Physics, 2018, UNSP 020083-1
5. Nalzhupbayeva G., “Spectral Properties of the Iterated Laplacian With a Potential in a Punctured Domain”, Filomat, 32:8 (2018), 2897–2900
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