Skip to main content
SpringerLink
Log in

Elliptic equations in hilbert space and associated spectral problems

  • Published:
Journal of Soviet Mathematics Aims and scope Submit manuscript

Abstract

Problems are formulated for abstract higher-order elliptic equations on the semiaxis and on a finite interval and general theorems for the Fredholm solvability and exact solvability of these equations given emission conditions to infinity are proved. A classification of the real spectrum of the pencil associated with the equation is presented, and possible rules for rigorous selection of the segment of its eigenelements and associated elements formulated. Completeness, minimality, and the basis property of the fundamental solutions of the equation in the solution space, along with the properties of the derivative chains of the eigenelements and associated elements of the pencil that correspond to problems on the semiaxis and on a finite interval are studied.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Literature cited

  1. M. S. Agranovich, “Summability of series in the root vectors of non-self-adjoint elliptic operators,” Funkts. Anal. Prilozhen.,10, No. 3, 1–12 (1976).

    Google Scholar 

  2. M. S. Agranovich, “On series in root vectors of nearly self-adjoint operators,” Funkts. Anal. Prilozhen.,11, No. 4, 65–67 (1977).

    Google Scholar 

  3. M. S. Agranovich, “Spectral properties of diffraction problems,” in N. N. Voitovich, B. Z. Katsenelenbaum, and A. N. Sivov, Generalized Method of Natural Vibrations in Diffraction Theory [in Russian], Nauka, Moscow (1977), pp. 288–362.

    Google Scholar 

  4. M. S. Agranovich and M. I. Vishik, “Elliptical problems with a parameter and the general form of hyperbolic problems,” Usp. Mat. Nauk,19, No. 3, 53–161 (1964).

    Google Scholar 

  5. T. Ya. Azizov and I. S. Iokhvidov, Foundations of the Theory of Linear Operators in Spaces with Indefinite Metric [in Russian], Nauka, Moscow (1986).

    Google Scholar 

  6. Yu. M. Berezanskii, Decompositions in Eigenfunctions of Self-Adjoint Operators [in Russian], Naukova Dumka, Kiev (1965).

    Google Scholar 

  7. B. R. Vainberg, “Principles of emission, limiting absorption, and limiting amplitude in the general theory of partial differential equations,” Usp. Mat. Nauk,21, No. 3, 115–194 (1966).

    Google Scholar 

  8. V. N. Vizitei and A. S. Markus, “Convergence of multiple decompositions in a system of eigenelements and adjoint vectors of an operator pencil,” Mat. Sb.,66, No. 2, 287–320 (1965).

    Google Scholar 

  9. I. I. Vorovich, “Certain mathematical problems of plate and shell theory,” Proceedings of the 2nd All-Union Conference on Theoretical and Applied Mechanics [in Russian], Nauka, Moscow (1966), No. 3, 116–136.

    Google Scholar 

  10. I. I. Vorovich and V. A. Babeshko, Mixed Dynamic Problems of the Theory of Elasticity of Nonclassical Domains [in Russian], Nauka, Moscow (1979).

    Google Scholar 

  11. I. I. Vorovich and V. E. Koval'chuk, On the basis properties of a system of homogeneous solutions (a problem of elasticity theory for the rectangle),” Prikl. Mat. Mekh.,31, No. 5, 861–869 (1967).

    Google Scholar 

  12. V. V. Vlasov, “Abbreviated minimality of a segment of the system of root vectors of the Keldysh pencil,” Dokl. Akad. Nauk SSSR,263, No. 6, 1289–1293 (1982).

    Google Scholar 

  13. M. G. Gasymov, “Theory of regular-type evolutionary equations,” Dokl. Akad. Nauk SSSR,200, No. 1, 13–16 (1971).

    Google Scholar 

  14. M. G. Gasymov, “Theory of polynomial operator pencils,” Dokl. Akad. Nauk SSSR,199, No. 4, 747–750 (1971).

    Google Scholar 

  15. M. G. Gasymov, “On the multiple completeness of a portion of the eigenelements and associated vectors of polynomial operator pencils,” Izv. Akad. Nauk Arm SSR, Mat.,6, No. 2–3, 131–147 (1971).

    Google Scholar 

  16. M. G. Gasymov, “Solvability of boundary-value problems for a class of differential operator equations,” Dokl. Akad. Nauk SSSR,235, No. 3, 505–508 (1977).

    Google Scholar 

  17. A. M. Gomilko, Certain Problems in the Spectral Theory of Operators and Quadratic Operator Pencils and Applications [in Russian]. Dissertation for the Degree of Candidate in Physics and Mathematics, Moscow (1982).

  18. V. I. Gorbachuk and M. A. Gorbachuk, Boundary-Value Problems for Differential Operator Equations [in Russian], Naukova Dumka, Kiev (1984).

    Google Scholar 

  19. I. Ts. Gokhberg and M. G. Krein, “Systems of integral equations on the half-line with kernels that are functions of a difference of independent variables,” Usp. Mat. Nauk,13, No. 3, 3–72 (1958).

    Google Scholar 

  20. I. Ts. Gokhberg and M. G. Krein, Introduction to the Theory of Linear Non-Self-Adjoint Operators in Hilbert Space [in Russian], Nauka, Moscow (1965).

    Google Scholar 

  21. N. Dunford and J. T. Schwartz, Linear Operators [Russian translation], Vol. 3, Mir, Moscow (1974).

    Google Scholar 

  22. A. A. Dezin, General Problems of the Theory of Boundary-Value Problems [in Russian], Nauka, Moscow (1980).

    Google Scholar 

  23. Yu. A. Dubinskii, “Certain differential operator equations of arbitrary order,” Mat. Sb.,90, No. 1, 3–22 (1973).

    Google Scholar 

  24. I. D. Evzerov, “Domains of definition of fractional degrees of ordinary differential operators in spaces,” Mat. Zametki,21, No. 4, 509–518 (1977).

    Google Scholar 

  25. A. S. Zil'bergleit and Yu. I. Kopilevich, Spectral Theory of Regular Waveguides [in Russian], Leningrad (1983).

  26. T. Kato, Theory of Perturbations of Linear Operators [Russian translation], Mir, Moscow (1972).

    Google Scholar 

  27. V. E. Katsel'son, “Conditions on the basis properties of a system of root vectors of certain operator classes,” Candidate's Dissertation, Mathematics and Physics, Kharkov (1967).

  28. V. E. Katsnel'son, “On the convergence and summability of series in root vectors of operator classes,” Funkts. Anal. Prilozhen.,1, No. 2, 39–51 (1967).

    Google Scholar 

  29. M. V. Keldysh, “Eigenvalues and eigenfunctions of certain classes of non-self-adjoint equations,” Dokl. Akad. Nauk SSSR,77, No. 1, 11–14 (1951).

    Google Scholar 

  30. M. V. Keldysh, “On the completeness of the eigenfunctions of certain classes of non-self-adjoint operators,” Usp. Mat. Nauk,26, No. 4, 15–41 (1971).

    Google Scholar 

  31. M. V. Keldysh and V. B. Lidskii, “Problems of the spectral theory of non-self-adjoint operators,” Proceedings of the 4th All-Union Mathematical Congress [in Russian], Izd-vo Akad. Nauk SSSR, Leningrad (1963), 101–120.

    Google Scholar 

  32. A. A. Kirillov and A. D. Gvishiani, Theorems and Problems in Analysis [in Russian], Nauka, Moscow (1979).

    Google Scholar 

  33. V. E. Koval'chuk, “On the behavior of the solution of the first principal problem of elasticity theory for a linear rectangular plate,” Prikl. Mat. Mekh.,33, No. 3, 511–518 (1969).

    Google Scholar 

  34. V. A. Kondrat'ev, “Boundary-value problems for elliptic equations in domains with conical and corner points,” Trudy MMO,16, 209–292 (1967).

    Google Scholar 

  35. V. A. Kondrat'ev and O. A. Oleinik, “Boundary-value problems for partial differential equations in nonsmooth domains,” Usp. Mat. Nauk,38, No. 3, 3–76 (1983).

    Google Scholar 

  36. L. S. Koplienko and B. A. Plamenevskii, “On the emission principle for periodic problems,” Diff. Uravn.,19, No. 10, 1713–1723 (1983).

    Google Scholar 

  37. A. G. Kostyuchenko and M. B. Orazov, “On certain properties of the roots of a self-adjoint quadratic pencil,” Funkts. Anal. Prilozhen.,9, No. 4, 28–40 (1975).

    Google Scholar 

  38. A. G. Kostyuchenko and M. B. Orazov, “Vibration of an elastic semicylinder and associated self-adjoint quadratic pencils,” Trudy Seminara im. I. G. Petrovskogo,6, 97–146, Izd-vo Mosk. Univ., Moscow (1981).

    Google Scholar 

  39. A. G. Kostyuchenko and A. A. Shkalikov, “Self-adjoint quadratic operator pencils and elliptic problems,” Funkts. Anal. Prilozhen.,17, No. 2, 38–61 (1983).

    Google Scholar 

  40. A. G. Kostyuchenko and A. A. Shkalikov, “Theory of self-adjoint quadratic operator pencils,” Vestn. Mosk. Univ. Ser. Mat., Mekh., No. 6, 40–51 (1983).

    Google Scholar 

  41. M. A. Krasnosel'skii, P. P. Zabreiko, E. I. Pustyl'nik, and P. E. Sobolevskii, Integral Operators in Spaces of Summable Functions [in Russian], Nauka, Moscow (1966).

    Google Scholar 

  42. M. G. Krein, Fundamentals of the Theory of λ-Zones of Stability of a Canonical System of Linear Differential Equations with Periodic Coefficients (in memory of A. A. Andronov), Gostekhizdat, Moscow (1955), 423–498.

    Google Scholar 

  43. M. G. Krein and G. K. Langer, “Certain mathematical principles of the theory of damped vibrations of continua,” Proceedings of the International Symposium on the Application of the Theory of Functions of a Complex Variable to Continuum Mechanics, Nauka, Moscow (1965), 283–322.

    Google Scholar 

  44. M. G. Krein and G. Ya. Lyubarskii, “Analytic properties of multipliers of periodic canonical positive-type differential systems,” Izv. Akad. Nauk SSSR, Ser. Mat.,26, No. 4, 549–572 (1962).

    Google Scholar 

  45. S. G. Krein and M. I. Khazan, “Differential equations in Banach space,” Mat. Anal.,21, 130–263 (1983) (Itogi Nauki).

    Google Scholar 

  46. O. A. Ladyzhenskaya, Boundary-Value Problems of Mathematical Physics [in Russian], Nauka, Moscow (1973).

    Google Scholar 

  47. V. B. Lidskii, “On the summability of series in principal vectors of non-self-adjoint operators,” Trudy MMO,11, 3–35 (1962).

    Google Scholar 

  48. V. B. Lidskii, “Decomposition in a Fourier series in the principal vectors of a non-self-adjoint elliptic operator,” Mat. Sb.,57, No. 2, 137–150 (1962).

    Google Scholar 

  49. J.-L. Lions and E. Magenes, Nonhomogeneous Boundary-Value Problems and Applications [Russian translation], Mir, Moscow (1971).

    Google Scholar 

  50. B. Ya. Levin, Distribution of the Roots of Entire Functions [in Russian], Gostekhizdat, Moscow (1956).

    Google Scholar 

  51. V. G. Maz'ya and B. A. Plamenevskii, “On the asymptotic behavior of the solutions of differential equations in Hilbert space,” Izv. Akad. Nauk SSSR, Ser. Mat.,36, No. 5, 1080–1133 (1972).

    Google Scholar 

  52. V. G. Maz'ya and B. A. Plamenevskii, “On boundary-value problems for second-order elliptic equations in a domain with edges,” Vestn. Leningr. Univ., Ser. Mat., No. 1, 102–108 (1975).

    Google Scholar 

  53. A. S. Markus, “Decomposition in root vectors of a weakly perturbated self-adjoint operator,” Dokl. Akad. Nauk SSSR,142, No. 3, 538–541 (1962).

    Google Scholar 

  54. A. S. Markus, “Certain signs for the completeness of a system of root vectors of a linear operator in Banach space,” Mat. Sb.,70, No. 4, 526–561 (1966).

    Google Scholar 

  55. A. S. Markus, “Spectral theory of polynomial pencils in Banach space,” Sib. Mat. Zh.,8, No. 6, 1346–1369 (1967).

    Google Scholar 

  56. A. S. Markus, Introduction to the Spectral Theory of Polynomial Operator Pencils [in Russian], Shtiintsa, Kishinev (1986).

    Google Scholar 

  57. A. S. Markus and V. I. Matsaev, “Convergence of decompositions in eigenvectors of a nearly self-adjoint operator,” Lineinye Operatory i Integral'nye Uravneniya. Mat. Issled., No. 61, 104–129 (1981) (Shtiintsa, Kishinev).

    Google Scholar 

  58. A. S. Markus and V. I. Matsaev, “Theorems comparing the spectra of linear operators and spectral asymptotic expressions,” Trudy MMO,45, 133–181 (1982).

    Google Scholar 

  59. V. I. Matsaev, “A method of estimating the resolvent of non-self-adjoint operators,” Dokl. Akad. Nauk SSSR,154, No. 5, 1034–1037 (1964).

    Google Scholar 

  60. V. I. Matsaev, “Certain theorems on the completeness of the root vectors of completely continuous operators,” Dokl. Akad. Nauk SSSR,155, No. 2, 273–276 (1964).

    Google Scholar 

  61. S. Mizohata, Theory of Partial Differential Equations [Russian translation], Mir, Moscow (1977).

    Google Scholar 

  62. S. S. Mirzoev, “Multiple completeness of root vectors of polynomial operator pencils corresponding to boundary-value problems on the semiaxis,” Funkts. Anal. Prilozhen.,17, No. 2, 84–85 (1983).

    Google Scholar 

  63. S. S. Mirzoev, “Conditions for well-formed solvability of boundary-value problems for differential operator equations,” Dokl. Akad. Nauk SSSR,273, No. 2, 292–295 (1983).

    Google Scholar 

  64. M. A. Naimark, Linear Differential Operators [in Russian], Nauka, Moscow (1968).

    Google Scholar 

  65. M. B. Orazov, “On the completeness of elementary solutions for certain operator equations on the semiaxis and an interval,” Dokl. Akad. Nauk SSSR,245, No. 4, 788–792 (1979).

    Google Scholar 

  66. L. S. Pontryagin, “Hermitian operators in a space with indefinite metric,” Izv. Akad. Nauk SSSR,8, 243–280 (1944).

    Google Scholar 

  67. G. V. Radzievskii, “A method of proving the completeness of the root vectors of operator functions,” Dokl. Akad. Nauk SSSR,214, No. 2, 291–294 (1974).

    Google Scholar 

  68. G. V. Radzievskii, “Basis property of derivative chains,” Izv. Akad. Nauk SSSR, Ser. Mat.,39, No. 5, 1182–1218 (1975).

    Google Scholar 

  69. G. V. Radzievskii, Quadratic Operator Pencils [in Russian], Preprint 76-24 of the Institute of Mathematics, Kiev (1976).

  70. G. V. Radzievskii, “Completeness problem for root vectors in the spectral theory of operator functions,” Usp. Mat. Nauk,37, No. 2, 81–145 (1982).

    Google Scholar 

  71. G. V. Radzievskii, “A method of proving minimality and basis property of a segment of root vectors,” Funkts. Anal. Prilozhen.,17, No. 1, 24–30 (1983).

    Google Scholar 

  72. G. V. Radzievskii, “Quadratic operator pencil (equivalence of segment of root vectors),” Preprint 84-32 of Institute of Mathematics, Kiev (1984).

  73. G. V. Radzievskii, “Minimality, basis property, and completeness of segment of root vectors of a quadratic operator pencil,” Dokl. Akad. Nauk SSSR,283, No. 1, 53–57 (1985).

    Google Scholar 

  74. G. V. Radzievskii and S. V. Ashurov, “Polynomial operator pencil (minimality of segment of root vectors),” Preprint IM-85-71, Kiev (1985).

  75. A. G. Sveshnikov, “On the emission principle,” Dokl. Akad. Nauk SSSR,73, No. 5, 917–920 (1950).

    Google Scholar 

  76. Ya. D. Tamarkin, On Certain General Problems of the Theory of Ordinary Differential Equations and on the Decomposition of Arbitrary Functions in Series [in Russian], Prague (1917).

  77. E. Titchmarsh, Theory of Functions [Russian translation], Nauka, Moscow (1980).

    Google Scholar 

  78. H. Tribel, Theory of Interpolation. Functional Spaces. Differential Operators [Russian translation], Mir, Moscow (1980).

    Google Scholar 

  79. Yu. A. Ustinov and Yu. I. Yudovich, “Completeness of the system of elementary solutions of the biharmonic equation in a semiband,” Prikl. Mat. Mekh.,37, No. 4, 706–714 (1973).

    Google Scholar 

  80. G. Fichera, Existence Theorems in Elasticity Theory [Russian translation], Mir, Moscow (1974).

    Google Scholar 

  81. E. Hille and R. Phillips, Functional Analysis and Semigroups [Russian translation], IL, Moscow (1962).

    Google Scholar 

  82. A. A. Shkalikov, “Weakly perturbed operator pencils,” Abstracts of Papers Read to the Republic Symposium on Differential Equations [in Russian], Izd-vo Turkm. Un-ta, Ashkhabad (1978), 132–133.

    Google Scholar 

  83. A. A. Shkalikov, “Basis property of the eigenvectors of quadratic operator pencils,” Mat. Zametki,30, No. 3, 371–385 (1981).

    Google Scholar 

  84. A. A. Shkalikov, “Boundary-value problems for ordinary differential operators with a parameter under boundary conditions,” Funkts. Anal. Prilozhen.,16, No. 4, 92–93 (1982).

    Google Scholar 

  85. A. A. Shkalikov, “Decomposition in eigenfunctions in the two-dimensional problem of elasticity theory,” in: Nonclassical Problems of Equations of Mathematical Physics [in Russian], Novosibirsk (1982), 171–174.

  86. A. A. Shkalikov, “On bounds of meromorphic functions and summation of series in root vectors of non-self-adjoint operators,” Dokl. Akad. Nauk SSSR,268, No. 6, 1310–1314 (1983).

    Google Scholar 

  87. A. A. Shkalikov, “Boundary-value problems for ordinary differential equations with a parameter under boundary conditions,” Trudy Seminara im. I. G. Petrovskogo, No. 9, 190–229 (1983) (Izd-vo Mosk. Un-ta, Moscow).

    Google Scholar 

  88. A. A. Shkalikov, “Certain topics in the theory of polynomial operator pencils,” Usp. Mat. Nauk,37, No. 4, 98 (1982).

    Google Scholar 

  89. A. A. Shkalikov, “Differential operator equations on the semiaxis and associated spectral problems for self-adjoint operator pencils,” Dokl. Akad. Nauk SSSR,276, No. 2, 309–314 (1984).

    Google Scholar 

  90. A. A. Shkalikov, “Tauberian-type theorems on the distribution of the zeros of holomorphic functions,” Mat. Sb.,123, No. 3, 317–347 (1984).

    Google Scholar 

  91. A. A. Shkalikov, “Differential operator equations on the semiaxis and associated spectral problems for polynomial operator pencils,” Usp. Mat. Nauk,39, No. 4, 106 (1984).

    Google Scholar 

  92. A. A. Shkalikov, “On the minimality of derivative chains corresponding to a segment of the eigenelements and associated elements of self-adjoint operator pencils,” Vestn. Mosk. Univ. Ser. Mat., Mekh., No. 6, 10–19 (1985).

    Google Scholar 

  93. A. A. Shkalikov, “On the principles of selection and the properties of the segment of eigenelements and associated elements of operator pencils,” Vestn. Mosk. Univ. Ser. Mat. Mekh., No. 4, 16–25 (1988).

    Google Scholar 

  94. A. A. Shkalikov, “On the minimality and completeness of systems constructed from the segment of eigenelements and associated elements of quadratic operator pencils,” Dokl. Akad. Nauk SSSR,285, No. 6, 1100–1106 (1985).

    Google Scholar 

  95. S. Agmon, “On the eigenfunctions and eigenvalues of general elliptic boundary-value problems,” Commun. Pure Appl. Math.,15, 119–147 (1962).

    Google Scholar 

  96. S. Agmon, Lectures on Elliptic Boundary-Value Problems, New York (1965).

  97. S. Agmon and S. Nirenberg, “Properties of solutions of ordinary differential equations in Banach space,” Commun. Pure Appl. Math.,16, 121–239 (1963).

    Google Scholar 

  98. G. D. Birkhoff, “Boundary-value and expansion problems of ordinary differential equations,” Trans. Am. Math. Soc.,9, 373–395 (1908).

    Google Scholar 

  99. G. D. Birkhoff, “On the asymptotic character of the solution of certain linear differential equations containing a parameter,” Trans. Am. Math. Soc.,9, 219–231 (1908).

    Google Scholar 

  100. F. Brouder, “On the eigenfunctions and eigenvalues of the general elliptic differential operators,” Proc. Nat. Acad. Sci. USA,39, 433–439 (1953).

    Google Scholar 

  101. F. Brouder, “On the spectral theory of strongly elliptic differential operators,” Proc. Nat. Acad. Sci. USA,45, 1423–1431 (1959).

    Google Scholar 

  102. F. Brouder, “On the spectral theory of elliptic differential operators,” Math. Ann.,142, 22–130 (1961).

    Google Scholar 

  103. T. Carleman, “Zur Theorie der linearen integralgleichungen,” Math. Z.,9, 196–217 (1921).

    Google Scholar 

  104. I. Gohberg [Gokhberg], P. Lancaster, and L. Rodman, “Spectral analysis of self-associated matrix polynomials,” Ann. Math. Ser. 2,112, No. 1, 33–71 (1980).

    Google Scholar 

  105. P. Grisvard, “Characterisation de quelques espaces d'interpolation,” Arch. Rat. Mech. Anal.,25, 40–63 (1967).

    Google Scholar 

  106. P. Grisvard, Boundary-Value Problems in Nonsmooth Domains, University of Nice (1981).

  107. H. Langer, “Factorization of operator pencils,” Acta Scient. Math. Szeged,38, 83–96 (1976).

    Google Scholar 

  108. P. D. Lax, “Phragmén-Lindelöf theorem in harmonic analysis and its application to some questions in the theory of elliptic equations,” Commun. Pure Appl. Math.,10, 361–389 (1957).

    Google Scholar 

  109. R. Seeley, “Interpolation in Lp with boundary conditions,” Stud. Math.,44, 47–60 (1972).

    Google Scholar 

  110. A. A. Shkalikov, “Estimates of meromorphic functions and summability theorems,” Pac. J. Math.,103, No. 1, 569–582 (1982).

    Google Scholar 

  111. S. Ya. Yakubov, Linear Differential Operator Equations and Applications [in Russian], Inst. Mat. Akad. Nauk Azer. SSR, Baku (1985).

    Google Scholar 

Download references

Authors
  1. A. A. Shkalikov
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 14, pp. 140–224, 1989.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Shkalikov, A.A. Elliptic equations in hilbert space and associated spectral problems. J Math Sci 51, 2399–2467 (1990). https://doi.org/10.1007/BF01097162

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01097162

Keywords

Search

Navigation