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

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.

DOI: https://doi.org/10.4213/rm204

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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

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
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Related articles on Google Scholar: Russian articles, English articles

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
2. Nazarov, SA, “Artificial boundary conditions for elliptic systems in domains with conical outlets to infinity”, Doklady Mathematics, 63:2 (2001), 277
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
4. Sokołowski J., Żochowski A., “Optimality conditions for simultaneous topology and shape optimization”, SIAM J. Control Optim., 42:4 (2003), 1198–1221
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
6. Nazarov S.A., Sokołowski J., “Asymptotic analysis of shape functionals”, J. Math. Pures Appl. (9), 82:2 (2003), 125–196
7. Nardinocchi P., Teresi L., Tiero A., “Constitutive identification of affine rods”, Mech. Res. Commun., 30:1 (2003), 61–68
8. S. A. Nazarov, “Elliptic Boundary Value Problems in Hybrid Domains”, Funct. Anal. Appl., 38:4 (2004), 283–297
9. Kulikov A.A., Nazarov S.A., “Griffith formula for a crack in a piezoelectric body”, Dokl. Math., 49:12 (2004), 768–771
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
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
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
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
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
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
16. A. A. Kulikov, S. A. Nazarov, “Cracks in piezoelectric and electroconductive bodies”, J. Appl. Industr. Math., 1:2 (2007), 201–216
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
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
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
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
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
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
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
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
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
26. S. A. Nazarov, “Dirichlet problem in an angular domain with rapidly oscillating boundary: Modeling of the problem and asymptotics of the solution”, St. Petersburg Math. J., 19:2 (2008), 297–326
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
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
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
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
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
32. J. Appl. Industr. Math., 4:1 (2010), 99–116
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
34. S. A. Nazarov, “Concentration of trapped modes in problems of the linearized theory of water waves”, Sb. Math., 199:12 (2008), 1783–1807
35. Kulikov, A, “On Airy functions and stresses in nonisotropic heterogeneous 2d-elasticity”, Zamm-Zeitschrift fur Angewandte Mathematik und Mechanik, 88:12 (2008), 955
36. S. A. Nazarov, “Trapped modes in a cylindrical elastic waveguide with a damping gasket”, Comput. Math. Math. Phys., 48:5 (2008), 816–833
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
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
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
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
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
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
43. S. A. Nazarov, “The Eshelby theorem and the problem on optimal patch”, St. Petersburg Math. J., 21:5 (2010), 791–818
44. Cardone, G, “A criterion for the existence of the essential spectrum for beak-shaped elastic bodies”, Journal de Mathematiques Pures et Appliquees, 92:6 (2009), 628
45. Cardone, G, “Gaps in the essential spectrum of periodic elastic waveguides”, Zamm-Zeitschrift fur Angewandte Mathematik und Mechanik, 89:9 (2009), 729
46. Cardone, G, “Korn's inequality for periodic solids and convergence rate of homogenization”, Applicable Analysis, 88:6 (2009), 847
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
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
49. Nazarov, SA, “Neumann LAPLACIAN ON A DOMAIN WITH TANGENTIAL COMPONENTS IN THE BOUNDARY”, Annales Academiae Scientiarum Fennicae-Mathematica, 34:1 (2009), 131
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
51. S. A. Nazarov, “An example of multiple gaps in the spectrum of a periodic waveguide”, Sb. Math., 201:4 (2010), 569–594
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
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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
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
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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
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
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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
78. S. A. Nazarov, “Elastic waves trapped by a homogeneous anisotropic semicylinder”, Sb. Math., 204:11 (2013), 1639–1670
79. S. A. Nazarov, “The Mandelstam Energy Radiation Conditions and the Umov–Poynting Vector in Elastic Waveguides”, J Math Sci, 2013
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
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
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
83. S. A. Nazarov, “Asymptotics of eigen-oscillations of a massive elastic body with a thin baffle”, Izv. Math., 77:1 (2013), 87–142
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
85. G. Leugering, S. A. Nazarov, “The Eshelby Theorem and its Variants for Piezoelectric Media”, Arch Rational Mech Anal, 2014
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
87. Nazarov S.A. Specovius-Neugebauer M. Steigemann M., “Crack Propagation in Anisotropic Composite Structures”, Asymptotic Anal., 86:3-4 (2014), 123–153
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
89. Cardone G. Nazarov S.A. Taskinen J., “Spectra of Open Waveguides in Periodic Media”, 269, no. 8, 2015, 2328–2364
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
91. S. A. Nazarov, “Discrete spectrum of cranked quantum and elastic waveguides”, Comput. Math. Math. Phys., 56:5 (2016), 864–880
92. Nazarov S.A., Ruotsalainen K.M., Silvola M., “Trapped Modes in Piezoelectric and Elastic Waveguides”, J. Elast., 124:2 (2016), 193–223
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
94. Kozlov V. Nazarov S.A., “on the Spectrum of An Elastic Solid With Cusps”, Adv. Differ. Equat., 21:9-10 (2016), 887–944
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
96. Nazarov S.A., Ruotsalainen K.M., “A Rigorous Interpretation of Approximate Computations of Embedded Eigenfrequencies of Water Waves”, Z. Anal. ihre. Anwend., 35:2 (2016), 211–242
97. S. A. Nazarov, “The asymptotic behaviour of the scattering matrix in a neighbourhood of the endpoints of a spectral gap”, Sb. Math., 208:1 (2017), 103–156
98. Nazarov S.A. Slutskij A.S., “A Folded Plate Clamped Along One Side Only”, C. R. Mec., 345:12 (2017), 903–907
99. Bakharev F.L. Taskinen J., “Bands in the Spectrum of a Periodic Elastic Waveguide”, Z. Angew. Math. Phys., 68:5 (2017), 102
100. Pettersson I., Piatnitski A., “Stationary Convection-Diffusion Equation in An Infinite Cylinder”, J. Differ. Equ., 264:7 (2018), 4456–4487
101. Gomez D. Nazarov S.A. Perez M.E., “Homogenization of Winkler-Steklov Spectral Conditions in Three-Dimensional Linear Elasticity”, Z. Angew. Math. Phys., 69:2 (2018), 35
102. Suslina T.A., “Spectral Approach to Homogenization of Elliptic Operators in a Perforated Space”, Rev. Math. Phys., 30:8, SI (2018), 1840016
103. S. A. Nazarov, “The asymptotics of natural oscillations of a long two-dimensional Kirchhoff plate with variable cross-section”, Sb. Math., 209:9 (2018), 1287–1336
104. S. A. Nazarov, “Breakdown of cycles and the possibility of opening spectral gaps in a square lattice of thin acoustic waveguides”, Izv. Math., 82:6 (2018), 1148–1195
105. S. A. Nazarov, “Asymptotics of the deflection of a cruciform junction of two narrow Kirchhoff plates”, Comput. Math. Math. Phys., 58:7 (2018), 1150–1171
106. Nazarov S.A. Slutskii A.S., “Asymptotics of Natural Oscillations of Elastic Junctions With Readily Movable Elements”, Mech. Sol., 53:1 (2018), 101–115
107. Chesnel L., Nazarov S.A., “Non Reflection and Perfect Reflection Via Fano Resonance in Waveguides”, Commun. Math. Sci., 16:7 (2018), 1779–1800
108. F. L. Bakharev, S. A. Nazarov, “Asimptotika sobstvennykh chisel dlinnykh plastin Kirkhgofa s zaschemlennymi krayami”, Matem. sb., 210:4 (2019), 3–26
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