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Uspekhi Mat. Nauk, 1999, Volume 54, Issue 5(329), Pages 77–142 (Mi umn204)  

This article is cited in 106 scientific papers (total in 106 papers)

The polynomial property of self-adjoint elliptic boundary-value problems and an algebraic description of their attributes

S. A. Nazarov

Saint-Petersburg State University

Abstract: We describe a wide class of boundary-value problems for which the application of elliptic theory can be reduced to elementary algebraic operations and which is characterized by the following polynomial property: the sesquilinear form corresponding to the problem degenerates only on some finite-dimensional linear space $\mathscr P$ of vector polynomials. Under this condition the boundary-value problem is elliptic, and its kernel and cokernel can be expressed in terms of $\mathscr P$. For domains with piecewise-smooth boundary or infinite ends (conic, cylindrical, or periodic), we also present fragments of asymptotic formulae for the solutions, give specific versions of general conditional theorems on the Fredholm property (in particular, by modifying the ordinary weighted norms), and compute the index of the operator corresponding to the boundary-value problem. The polynomial property is also helpful for asymptotic analysis of boundary-value problems in thin domains and junctions of such domains. Namely, simple manipulations with $\mathscr P$ permit one to find the size of the system obtained by dimension reduction as well as the orders of the differential operators occurring in that system and provide complete information on the boundary layer structure. The results are illustrated by examples from elasticity and hydromechanics.


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English version:
Russian Mathematical Surveys, 1999, 54:5, 947–1014

Bibliographic databases:

MSC: Primary 35J40, 35J55; Secondary 35B40, 35C20, 47A53, 35Q30, 47A56, 73R05, 47B15
Received: 15.04.1999

Citation: S. A. Nazarov, “The polynomial property of self-adjoint elliptic boundary-value problems and an algebraic description of their attributes”, Uspekhi Mat. Nauk, 54:5(329) (1999), 77–142; Russian Math. Surveys, 54:5 (1999), 947–1014

Citation in format AMSBIB
\by S.~A.~Nazarov
\paper The polynomial property of self-adjoint elliptic boundary-value problems and an algebraic description of their attributes
\jour Uspekhi Mat. Nauk
\yr 1999
\vol 54
\issue 5(329)
\pages 77--142
\jour Russian Math. Surveys
\yr 1999
\vol 54
\issue 5
\pages 947--1014

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    This publication is cited in the following articles:
    1. Nazarov S.A., “Weighted spaces with detached asymptotics in application to the Navier–Stokes equations”, Advances in Mathematical Fluid Mechanics, 2000, 159–191  crossref  mathscinet  isi
    2. Nazarov, SA, “Artificial boundary conditions for elliptic systems in domains with conical outlets to infinity”, Doklady Mathematics, 63:2 (2001), 277  zmath  isi  elib  scopus  scopus
    3. Maz'ya, VG, “Point estimates for Green's matrix to boundary value problems for second order elliptic systems in a polyhedral cone”, Zeitschrift fur Angewandte Mathematik und Mechanik, 82:5 (2002), 291  crossref  mathscinet  zmath  isi  scopus  scopus
    4. Sokołowski J., Żochowski A., “Optimality conditions for simultaneous topology and shape optimization”, SIAM J. Control Optim., 42:4 (2003), 1198–1221  crossref  mathscinet  zmath  isi
    5. Maz'ya V.G., Roßmann J., “Weighted $L_p$ estimates of solutions to boundary value problems for second order elliptic systems in polyhedral domains”, ZAMM Z. Angew. Math. Mech., 83:7 (2003), 435–467  crossref  mathscinet  zmath  isi
    6. Nazarov S.A., Sokołowski J., “Asymptotic analysis of shape functionals”, J. Math. Pures Appl. (9), 82:2 (2003), 125–196  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    7. Nardinocchi P., Teresi L., Tiero A., “Constitutive identification of affine rods”, Mech. Res. Commun., 30:1 (2003), 61–68  crossref  zmath  isi  scopus  scopus
    8. S. A. Nazarov, “Elliptic Boundary Value Problems in Hybrid Domains”, Funct. Anal. Appl., 38:4 (2004), 283–297  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    9. Kulikov A.A., Nazarov S.A., “Griffith formula for a crack in a piezoelectric body”, Dokl. Math., 49:12 (2004), 768–771  mathnet  crossref  mathscinet  adsnasa  isi  elib  scopus
    10. Nazarov S.A., Slutskii A.S., “Branching periodicity: homogenization of the Dirichlet problem for an elliptic system”, Dokl. Math., 70:1 (2004), 628–631  mathnet  mathscinet  mathscinet  zmath  isi  elib  scopus
    11. Nazarov S.A., Specovius-Neugebauer M., “Artificial boundary conditions for Petrovsky systems of second order in exterior domains and in other domains of conical type”, Math. Methods Appl. Sci., 27:13 (2004), 1507–1544  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    12. S. A. Nazarov, “Estimates for the accuracy of modelling boundary-value problems at the junction of domains with different limit dimensions”, Izv. Math., 68:6 (2004), 1179–1215  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    13. S. A. Nazarov, “Estimates for second order derivatives of eigenvectors in thin anisotropic plates with variable thickness”, J. Math. Sci. (N. Y.), 132:1 (2006), 91–102  mathnet  crossref  mathscinet  zmath  elib  elib
    14. S. A. Nazarov, M. Specovius-Neugebauer, “Artificial boundary conditions for external boundary problem with a cylindrical inhomogeneity”, Comput. Math. Math. Phys., 44:12 (2004), 2087–2103  mathnet  mathscinet  zmath
    15. S. A. Nazarov, A. S. Slutskij, “Homogenization of an Elliptic System as the Cells of Periodicity are Refined in One Direction”, Math. Notes, 78:6 (2005), 814–826  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    16. A. A. Kulikov, S. A. Nazarov, “Cracks in piezoelectric and electroconductive bodies”, J. Appl. Industr. Math., 1:2 (2007), 201–216  mathnet  crossref  mathscinet  elib
    17. S. A. Nazarov, A. S. Slutskij, “Averaging of an elliptic system under condensing perforation of a domain”, St. Petersburg Math. J., 17:6 (2006), 989–1014  mathnet  crossref  mathscinet  zmath  elib
    18. De Maio, U, “Asymptotic approximation for the solution to the Robin problem in a thick multi-level junction”, Mathematical Models & Methods in Applied Sciences, 15:12 (2005), 1897  crossref  mathscinet  zmath  isi  scopus  scopus
    19. Nazarov, SA, “A crack at the interface of anisotropic bodies. Singularities of the elastic fields and a criterion for fracture when the crack surfaces are in contact”, Pmm Journal of Applied Mathematics and Mechanics, 69:3 (2005), 473  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    20. S. A. Nazarov, “Homogenization of elliptic systems with periodic coefficients: Weighted $L^p$ and $L^\infty$ estimates for asymptotic remainders”, St. Petersburg Math. J., 18:2 (2007), 269–304  mathnet  crossref  mathscinet  zmath  elib
    21. S. A. Nazarov, Ya. Taskinen, “Asymptotics of a solution to the Neumann problem in a thin domain with the sharp edge”, J. Math. Sci. (N. Y.), 142:6 (2007), 2630–2644  mathnet  crossref  mathscinet  zmath  elib  elib
    22. D'Apice, C, “Asymptotic approximation of the solution to the robin problem in a thick multistructure”, International Journal For Multiscale Computational Engineering, 4:5–6 (2006), 545  crossref  isi  elib  scopus  scopus
    23. Costabel, M, “Analysis of crack singularities in an aging elastic material”, ESAIM-Mathematical Modelling and Numerical Analysis-Modelisation Mathematique et Analyse Numerique, 40:3 (2006), 553  crossref  mathscinet  zmath  isi  elib
    24. S. A. Nazarov, “On the concentration of the point spectrum on the continuous one in problems of the linearized theory of water-waves”, J. Math. Sci. (N. Y.), 152:5 (2008), 674–689  mathnet  crossref  elib
    25. D. Gomez, S. A. Nazarov, M. E. Perez, “The formal asymptotics of eigenmodes for oscillating elastic spatial body with concentrated masses”, J. Math. Sci. (N. Y.), 148:5 (2008), 650–674  mathnet  crossref  mathscinet  zmath  elib  elib
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    27. M. S. Agranovich, “To the Theory of the Dirichlet and Neumann Problems for Strongly Elliptic Systems in Lipschitz Domains”, Funct. Anal. Appl., 41:4 (2007), 247–263  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    28. Nazarov, SA, “A criterion of the continuous spectrum for elasticity and other self-adjoint systems on sharp peak-shaped domains”, Comptes Rendus Mecanique, 335:12 (2007), 751  crossref  zmath  adsnasa  isi  elib  scopus  scopus
    29. D'Apice, C, “Asymptotic analysis of a perturbed parabolic problem in a thick junction of type 3 : 2 : 2”, Networks and Heterogeneous Media, 2:2 (2007), 255  crossref  mathscinet  zmath  isi
    30. Blazy, S, “Artificial boundary conditions of pressure type for viscous flows in a system of pipes”, Journal of Mathematical Fluid Mechanics, 9:1 (2007), 1  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    31. T. Durante, T. A. Mel'nik, “Asymptotic analysis of a parabolic problem in a thick two-level junction”, Zhurn. matem. fiz., anal., geom., 3:3 (2007), 313–341  mathnet  mathscinet  zmath  elib
    32. J. Appl. Industr. Math., 4:1 (2010), 99–116  mathnet  crossref  mathscinet
    33. S. A. Nazarov, M. Specovius-Neugebauer, “Singularities at the tip of a crack on the interface of piezoelectric bodies”, J. Math. Sci. (N. Y.), 159:4 (2009), 524–540  mathnet  crossref  zmath  elib  elib
    34. S. A. Nazarov, “Concentration of trapped modes in problems of the linearized theory of water waves”, Sb. Math., 199:12 (2008), 1783–1807  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    35. Kulikov, A, “On Airy functions and stresses in nonisotropic heterogeneous 2d-elasticity”, Zamm-Zeitschrift fur Angewandte Mathematik und Mechanik, 88:12 (2008), 955  crossref  mathscinet  zmath  isi  elib
    36. S. A. Nazarov, “Trapped modes in a cylindrical elastic waveguide with a damping gasket”, Comput. Math. Math. Phys., 48:5 (2008), 816–833  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    37. Mel'nyk, TA, “Homogenization of a boundary-value problem with a nonlinear boundary condition in a thick junction of type 3 : 2 : 1”, Mathematical Methods in the Applied Sciences, 31:9 (2008), 1005  crossref  mathscinet  zmath  adsnasa  isi
    38. S. A. Nazarov, “Asymptotics of solutions and modelling the problems of elasticity theory in domains with rapidly oscillating boundaries”, Izv. Math., 72:3 (2008), 509–564  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    39. S. A. Nazarov, “The Essential Spectrum of Boundary Value Problems for Systems of Differential Equations in a Bounded Domain with a Cusp”, Funct. Anal. Appl., 43:1 (2009), 44–54  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    40. S. A. Nazarov, “Opening a gap in the essential spectrum of the elasticity problem in a periodic semi-layer”, St. Petersburg Math. J., 21:2 (2010), 281–307  mathnet  crossref  mathscinet  zmath  isi
    41. S. A. Nazarov, “A Gap in the Essential Spectrum of the Neumann Problem for an Elliptic System in a Periodic Domain”, Funct. Anal. Appl., 43:3 (2009), 239–241  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    42. G. Cardone, A. Corbo Esposito, S. A. Nazarov, “Homogenization of the mixed boundary value problem for a formally self-adjoint system in a periodically perforated domain”, St. Petersburg Math. J., 21:4 (2010), 601–634  mathnet  crossref  mathscinet  zmath  isi
    43. S. A. Nazarov, “The Eshelby theorem and the problem on optimal patch”, St. Petersburg Math. J., 21:5 (2010), 791–818  mathnet  crossref  mathscinet  zmath  isi
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    45. Cardone, G, “Gaps in the essential spectrum of periodic elastic waveguides”, Zamm-Zeitschrift fur Angewandte Mathematik und Mechanik, 89:9 (2009), 729  crossref  mathscinet  zmath  isi  elib
    46. Cardone, G, “Korn's inequality for periodic solids and convergence rate of homogenization”, Applicable Analysis, 88:6 (2009), 847  crossref  mathscinet  zmath  isi  elib
    47. Pankratova, I, “ON THE BEHAVIOUR AT INFINITY OF SOLUTIONS TO STATIONARY CONVECTION-DIFFUSION EQUATION IN A CYLINDER”, Discrete and Continuous Dynamical Systems-Series B, 11:4 (2009), 935  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    48. Cardone, G, “Asymptotics of solutions of the Neumann problem in a domain with closely posed components of the boundary”, Asymptotic Analysis, 62:1–2 (2009), 41  mathscinet  zmath  isi  elib  scopus  scopus
    49. Nazarov, SA, “Neumann LAPLACIAN ON A DOMAIN WITH TANGENTIAL COMPONENTS IN THE BOUNDARY”, Annales Academiae Scientiarum Fennicae-Mathematica, 34:1 (2009), 131  mathscinet  zmath  isi  elib  scopus  scopus
    50. Durante T., Kardone D., Nazarov S.A., “Modelirovanie sochlenenii plastin i sterzhnei posredstvom samosopryazhennykh rasshirenii”, Vestn. Sankt-Peterburgskogo un-ta. Ser. 1: Matem., Mekh., Astronom., 2009, no. 2, 3–14  mathscinet  zmath  elib
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    52. S. A. Nazarov, A. S. Slutskii, “Homogenization of a mixed boundary-value problem in a domain with anisotropic fractal perforation”, Izv. Math., 74:2 (2010), 379–409  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    53. S. A. Nazarov, “Opening of a Gap in the Continuous Spectrum of a Periodically Perturbed Waveguide”, Math. Notes, 87:5 (2010), 738–756  mathnet  crossref  crossref  mathscinet  isi  elib
    54. Sergey A. Nazarov, Iryna L. Pankratova, Andrey L. Piatnitski, “Homogenization of the Spectral Problem for Periodic Elliptic Operators with Sign-Changing Density Function”, Arch Rational Mech Anal, 2010  crossref  mathscinet  isi
    55. V. A. Kozlov, S. A. Nazarov, “The spectrum asymptotics for the Dirichlet problem in the case of the biharmonic operator in a domain with highly indented boundary”, St. Petersburg Math. J., 22:6 (2011), 941–983  mathnet  crossref  mathscinet  zmath  isi
    56. S. A. Nazarov, “On the asymptotics of an eigenvalue of a waveguide with thin shielding obstacle and Wood's anomalies”, J. Math. Sci. (N. Y.), 178:3 (2011), 292–312  mathnet  crossref
    57. Nazarov S.A., “Trapped modes in a T-shaped waveguide”, Acoustical Physics, 56:6 (2010), 1004–1015  crossref  adsnasa  isi  scopus  scopus
    58. Nazarov S.A., “Gap in the essential spectrum of an elliptic formally self-adjoint system of differential equations”, Differential Equations, 46:5 (2010), 730–741  crossref  mathscinet  zmath  isi  elib
    59. Cardone G., Nazarov S.A., Sokolowski J., “Asymptotic Analysis, Polarization Matrices, and Topological Derivatives for Piezoelectric Materials With Small Voids”, SIAM Journal on Control and Optimization, 48:6 (2010), 3925–3961  crossref  mathscinet  zmath  isi
    60. S. A. Nazarov, “Formation of gaps in the spectrum of the problem of waves on the surface of a periodic channel”, Comput. Math. Math. Phys., 50:6 (2010), 1038–1054  mathnet  crossref  mathscinet  adsnasa  isi  elib
    61. Cardone G., Durante T., Nazarov S.A., “Localization Effect for Eigenfunctions of the Mixed Boundary Value Problem in a Thin Cylinder With Distorted Ends”, SIAM J Math Anal, 42:6 (2010), 2581–2609  crossref  mathscinet  zmath  isi  elib
    62. S. A. Nazarov, A. V. Shanin, “Calculation of characteristics of trapped modes in T-shaped waveguides”, Comput. Math. Math. Phys., 51:1 (2011), 96–110  mathnet  crossref  mathscinet  isi  elib
    63. S. A. Nazarov, “Asymptotic expansions of eigenvalues in the continuous spectrum of a regularly perturbed quantum waveguide”, Theoret. and Math. Phys., 167:2 (2011), 606–627  mathnet  crossref  crossref  mathscinet  adsnasa  isi
    64. S. A. Nazarov, “Asymptotics of trapped modes and eigenvalues below the continuous spectrum of a waveguide with a thin shielding obstacle”, St. Petersburg Math. J., 23:3 (2012), 571–601  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    65. S. A. Nazarov, “On the spectrum of the Laplace operator on the infinite Dirichlet ladder”, St. Petersburg Math. J., 23:6 (2012), 1023–1045  mathnet  crossref  mathscinet  isi  elib  elib
    66. Nazarov S.A., Taskinen J., “Radiation Conditions At the TOP of a Rotational Cusp in the Theory of Water-Waves”, M2AN Math Model Numer Anal, 45:5 (2011), 947–979  crossref  mathscinet  zmath  isi
    67. J. H. Videman, V. Chiado' Piat, S. A. Nazarov, “Asymptotics of frequency of a surface wave trapped by a slightly inclined barrier in a liquid layer”, J. Math. Sci. (N. Y.), 185:4 (2012), 536–553  mathnet  crossref  mathscinet
    68. G. Cardone, S. A. Nazarov, K. Ruotsalainen, “Asymptotic behaviour of an eigenvalue in the continuous spectrum of a narrowed waveguide”, Sb. Math., 203:2 (2012), 153–182  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    69. S. A. Nazarov, “Notes to the proof of a weighted Korn inequality for an elastic body with peak-shaped cusps”, J Math Sci, 2012  crossref  mathscinet
    70. G. Leugering, S. Nazarov, F. Schury, M. Stingl, “The Eshelby Theorem and Application to the Optimization of an Elastic Patch”, SIAM J. Appl. Math, 72:2 (2012), 512  crossref  mathscinet  zmath  isi  elib
    71. S. A. Nazarov, “Enforced stability of an eigenvalue in the continuous spectrum of a waveguide with an obstacle”, Comput. Math. Math. Phys., 52:3 (2012), 448–464  mathnet  crossref  zmath  isi  elib  elib
    72. S. A. Nazarov, J. Taskinen, “Structure of the spectrum of the periodic family of identical cells connected through apertures of reducing sizes”, J. Math. Sci. (N. Y.), 194:1 (2013), 72–82  mathnet  crossref  mathscinet
    73. Nazarov S.A., “Asymptotics of the Reflection Coefficient at Critical Frequencies in a Narrowing Waveguide”, Russ. J. Math. Phys., 19:2 (2012), 216–233  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    74. Kozlov V., Nazarov S., “On the Hadamard Formula for Second Order Systems in Non-Smooth Domains”, Commun. Partial Differ. Equ., 37:5 (2012), 901–933  crossref  mathscinet  zmath  isi  elib
    75. S. A. Nazarov, “Scheme for interpretation of approximately computed eigenvalues embedded in a continuous spectrum”, Comput. Math. Math. Phys., 53:6 (2013), 702–720  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    76. G. A. Chechkin, T. A. Mel'nyk, “Spatial-skin effect for eigenvibrations of a thick cascade junction with ‘heavy’ concentrated masses”, Math. Meth. Appl. Sci, 2013, n/a  crossref  mathscinet  isi  scopus  scopus
    77. Fedor Bakharev, Sergey Nazarov, Guido Sweers, “A sufficient condition for a discrete spectrum of the Kirchhoff plate with an infinite peak”, Math. Mech. Compl. Sys, 1:2 (2013), 233  crossref  zmath
    78. S. A. Nazarov, “Elastic waves trapped by a homogeneous anisotropic semicylinder”, Sb. Math., 204:11 (2013), 1639–1670  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    79. S. A. Nazarov, “The Mandelstam Energy Radiation Conditions and the Umov–Poynting Vector in Elastic Waveguides”, J Math Sci, 2013  crossref  mathscinet
    80. S. A. Nazarov, “Enforced Stability of a Simple Eigenvalue in the Continuous Spectrum of a Waveguide”, Funct. Anal. Appl., 47:3 (2013), 195–209  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    81. Nazarov S.A., Taskinen J., “Spectral Anomalies of the Robin Laplacian in Non-Lipschitz Domains”, J. Math. Sci.-Univ. Tokyo, 20:1 (2013), 27–90  mathscinet  zmath  isi  scopus  scopus
    82. Dhia A. -S. Bonnet-Ben, Nazarov S.A., “Obstacles in Acoustic Waveguides Becoming “Invisible” at Given Frequencies”, Acoust. Phys., 59:6 (2013), 633–639  crossref  adsnasa  isi
    83. S. A. Nazarov, “Asymptotics of eigen-oscillations of a massive elastic body with a thin baffle”, Izv. Math., 77:1 (2013), 87–142  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    84. S. A. Nazarov, “Bounded solutions in a $\mathrm{T}$-shaped waveguide and the spectral properties of the Dirichlet ladder”, Comput. Math. Math. Phys., 54:8 (2014), 1261–1279  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    85. G. Leugering, S. A. Nazarov, “The Eshelby Theorem and its Variants for Piezoelectric Media”, Arch Rational Mech Anal, 2014  crossref  mathscinet  isi  scopus  scopus
    86. Leugering G., Nazarov S.A., Slutskij A.S., “Asymptotic Analysis of 3-D Thin Piezoelectric Rods”, ZAMM-Z. Angew. Math. Mech., 94:6 (2014), 529–550  crossref  mathscinet  zmath  isi
    87. Nazarov S.A. Specovius-Neugebauer M. Steigemann M., “Crack Propagation in Anisotropic Composite Structures”, Asymptotic Anal., 86:3-4 (2014), 123–153  crossref  mathscinet  zmath  isi  scopus  scopus
    88. S. A. Nazarov, “Asymptotics of the eigenvalues of boundary value problems for the Laplace operator in a three-dimensional domain with a thin closed tube”, Trans. Moscow Math. Soc., 76:1 (2015), 1–53  mathnet  crossref  elib
    89. Cardone G. Nazarov S.A. Taskinen J., “Spectra of Open Waveguides in Periodic Media”, 269, no. 8, 2015, 2328–2364  crossref  mathscinet  zmath  isi
    90. Nazarov S.A., “Near-threshold effects of the scattering of waves in a distorted elastic two-dimensional waveguide”, Pmm-J. Appl. Math. Mech., 79:4 (2015), 374–387  crossref  mathscinet  isi  scopus
    91. S. A. Nazarov, “Discrete spectrum of cranked quantum and elastic waveguides”, Comput. Math. Math. Phys., 56:5 (2016), 864–880  mathnet  crossref  crossref  isi  elib
    92. Nazarov S.A., Ruotsalainen K.M., Silvola M., “Trapped Modes in Piezoelectric and Elastic Waveguides”, J. Elast., 124:2 (2016), 193–223  crossref  mathscinet  zmath  isi  elib  scopus
    93. Buttazzo G. Cardone G. Nazarov S.A., “Thin Elastic Plates Supported Over Small Areas. i: Korn'S Inequalities and Boundary Layers”, J. Convex Anal., 23:2 (2016), 347–386  mathscinet  zmath  isi
    94. Kozlov V. Nazarov S.A., “on the Spectrum of An Elastic Solid With Cusps”, Adv. Differ. Equat., 21:9-10 (2016), 887–944  mathscinet  zmath  isi  elib
    95. Chesnel L., Nazarov S.A., “Team organization may help swarms of flies to become invisible in closed waveguides”, Inverse Probl. Imaging, 10:4 (2016), 977–1006  crossref  mathscinet  zmath  isi  elib  scopus
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