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Elliptic Functional Differential Equations with Contractions and Extensions of Independent Variables of the Unknown Function

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References

  1. M. S. Agranovich and M. I. Vishik, “Elliptic problems with parameter and parabolic problems of general form,” Russ. Math. Surv., 9, No. 3, 53–157 (1964).

    Article  MATH  Google Scholar 

  2. V. A. Ambartsumyan, “On the theory of brightness fluctuations in Milky Way,” Dokl. Akad. Nauk SSSR, 44, 244–247 (1944).

    MathSciNet  Google Scholar 

  3. A.B. Antonevich, “The index and the normal solvability of a general elliptic boundary-value problem with a finite group of translations on the boundary,” Differ. Uravn., 8, 309–317 (1972).

    MATH  Google Scholar 

  4. A. B. Antonevich, “Elliptic pseudodifferential operators with a finite group of translations,” Izv. Akad. Nauk SSSR. Ser. Mat., 37, 663–675 (1973).

    MathSciNet  Google Scholar 

  5. A.B. Antonevich, “Strongly nonlocal boundary value problems for elliptic equations,” Izv. Math. USSR, 34, No. 1, 1–21 (1990).

    Article  MathSciNet  Google Scholar 

  6. A. B. Antonevich and A.V. Lebedev, “On the Noethericity of a functional-partial differential operator that contains a linear transformation of the argument,” Differ. Uravn., 18, 987–996 (1982).

    MathSciNet  MATH  Google Scholar 

  7. A. Antonevich and A. Lebedev, Functional Differential Equations. I. C -theory, Longman, Harlow (1994).

  8. P. Auscher, S. Hofmann, A. McIntosh, and P. Tchamitchian, “The Kato square root problem for higher order elliptic operators and systems on ℝn ,J. Evol. Equ., 1, No. 4, 361–385 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  9. A. Axelsson, S. Keith, and A. McIntosh, “The Kato square root problem for mixed boundary value problems,” J. London Math. Soc. (2), 74, 113–130 (2006).

  10. I. Babuška and R. Výborný, “Continuous dependence of the eigenvalues on the domain,” Czech. Math. J., 15, 169–178 (1965).

    MathSciNet  MATH  Google Scholar 

  11. R. Bellman and K. Cooke, Theory of Differential-Difference Equations, Academic Press, New York (1957).

    MATH  Google Scholar 

  12. A. V. Bitsadze and A.A. Samarskii, “On some simple generalizations of linear elliptic boundary problems,” Sov. Math., Dokl., 10, 398–400 (1969).

  13. L. V. Borodulina and L. E. Rossovskii, “Solvability of elliptic functional differential equations with compressions of the arguments in weighted spaces,” J. Math. Sci. (N.Y.), 143, No. 4, 3205–3216 (2007).

  14. V. I. Burenkov, Sobolev Spaces on Domains, Teubner, Stuttgart (1998).

    Book  MATH  Google Scholar 

  15. V. I. Burenkov and E.B. Davies, “Spectral stability of the Neumann Laplacian,” J. Differ. Equ., 186, 485–508 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  16. V. I. Burenkov and P.D. Lamberti, “Spectral stability of general nonnegative self-adjoint operators with applications to Neumann-type operators,” J. Differ. Equ., 233, 345–379 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  17. T. Carleman, “Sur la théorie des équations intégrales et ses applications,” Verhandlungen des Int. Math. Kongr. Z¨urich, 1, 138–151 (1932).

  18. K. Cooke, L. E. Rossovskii, and A. L. Skubačevskiĭ, “A boundary value problem for a functional differential equation with a linearly transformed argument,” Differ. Equ., 31, No. 8, 1294–1299 (1995).

    MathSciNet  Google Scholar 

  19. R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. I, Interscience, New York (1953).

  20. E.B. Davies, “Eigenvalue stability bounds via weighted Sobolev spaces,” Math. Z., 214, 357–371 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  21. E. B. Davies, Spectral Theory and Differential Operators, Cambridge University Press, Cambridge (1995).

    Book  MATH  Google Scholar 

  22. G. Derfel and A. Iserles, “The pantograph equation in the complex plane,” J. Math. Anal. Appl., 213, 117–132 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  23. N. Dunford and J. Schwartz, Linear Operators. Part II: Spectral Theory. Self-Adjoint Operators in Hilbert Space, Interscience, New York (1963).

  24. L. Gårding, “Dirichlet’s problem for linear elliptic partial differential equations,” Math. Scand., 1, 55–72 (1953).

    Article  MathSciNet  Google Scholar 

  25. J.K. Hale, “Eigenvalues and perturbed domains,” Ten Mathematical Essays on Approximation in Analysis and Topology, Elsevier B.V., Amsterdam, 95–123 (2005).

  26. A. J. Hall and G.C. Wake, “A functional differential equation arising in the modelling of cell growth,” J. Aust. Math. Soc. Ser. B, 30, 424–435 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  27. L. Hӧrmander, The Analysis of Linear Partial Differential Operators. Vol. 3. Pseudodifferential Operators, Springer, Berlin (1985).

    Google Scholar 

  28. A. Iserles, “On the generalized pantograph functional differential equation,” Eur. J. Appl. Math., 4, 1–38 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  29. A. Iserles and Y. Liu, “On neutral functional differential equations with proportional delays,” J. Math. Anal. Appl., 207, No. 1, 73–95 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  30. G. A. Kamenskiĭ, “Variational and boundary value problems with deviating argument,” Differ. Uravn., 6, 1349–1358 (1970).

    MathSciNet  Google Scholar 

  31. G. A. Kamenskiĭ, A. D. Myškis, and A. L. Skubačevskiĭ, “On the minimum of a quadratic functional and on linear boundary value problems of elliptic type with deviating arguments,” Differ. Uravn., 16, No. 8, 1469–1473 (1980).

    MathSciNet  Google Scholar 

  32. T. Kato, “Fractional powers of dissipative operators,” J. Math. Soc. Jpn., 13, No. 3, 246–274 (1961).

    Article  MathSciNet  MATH  Google Scholar 

  33. T. Kato, Perturbation Theory for Linear Operators [Russian translation], Mir, Moscow (1972).

    Google Scholar 

  34. T. Kato and J.B. McLeod, “Functional differential equation = ay(λt)+by(t),Bull. Am. Math. Soc., 77, No. 6, 891–937 (1971).

    Article  MathSciNet  MATH  Google Scholar 

  35. V. A. Kondratiev, “Boundary value problems for elliptic equaitons in domains with conical or angular points,” Tr. Mosk. Mat. Obshch., 16, 209–292 (1967).

    Google Scholar 

  36. N. N. Krasovskii, Theory of Control of Motion [in Russian], Nauka, Moscow (1968).

    Google Scholar 

  37. P. D. Lamberti and M. Lanza de Cristoforis, “A global Lipschitz continuity result for a domain dependent Dirichlet eigenvalue problem for the Laplace operator,” Z. Anal. Anwend., 24, 277–304 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  38. J. L. Lions, “Espaces d’interpolation et domaines de puissances fractionnaires d’opérateurs,” J. Math. Soc. Jpn., 14, No. 2, 233–241 (1962).

    Article  MATH  Google Scholar 

  39. J.-L. Lions and E. Magenes, Problèmes aux Limites Non Homogènes et Applications. Vol. 1, Dunod, Paris (1968).

  40. A. McIntosh, “On the comparability of A 1/2 and A 1/2 ,Proc. Am.Math. Soc., 32, No. 2, 430–434 (1972).

    MATH  Google Scholar 

  41. A.D. Myshkis, Linear Differential Equations with Retarded Argument [in Russian], Gostekhizdat, Moscow–Leningrad (1951).

    MATH  Google Scholar 

  42. V. E. Nazaikinskii, A.Yu. Savin, and B.Yu. Sternin, Elliptic Theory and Noncommutative Geometry, Birkhäuser, Basel (2008).

    MATH  Google Scholar 

  43. J. R. Ockendon and A. B. Tayler, “The dynamics of a current collection system for an electric locomotive,” Proc. R. Soc. London Ser. A Math. Phys. Eng. Sci., 322, 447–468 (1971).

    Article  Google Scholar 

  44. G. G. Onanov and A. L. Skubachevskiĭ, “Differential equations with displaced arguments in stationary problems in the mechanics of a deformable body,” Sov. Appl. Mech., 15, 391–397 (1979).

    Article  MATH  Google Scholar 

  45. G. G. Onanov and E. L. Tsvetkov, “On the minimum of the energy functional with respect to functions with deviating argument in a stationary problem of elasticity theory,” Russ. J. Math. Phys., 3, 491–500 (1996).

    MathSciNet  MATH  Google Scholar 

  46. B. A. Plamenevskiĭ, Algebras of Pseudodifferential Operators [in Russian], Nauka, Moscow (1986).

    MATH  Google Scholar 

  47. V. V. Pod”yapol’skiĭ and A. L. Skubachevskiĭ, “On the completeness and basis property of a system of root functions of strongly elliptic functional differential operators,” Russ. Math. Surv., 51, No. 1, 169–170 (1996).

  48. V. V. Pod”yapol’skiĭ and A. L. Skubachevskiĭ, “Spectral asymptotics of strongly elliptic differential-difference operators,” Differ. Equ., 35, No. 6, 794–802 (1999).

  49. V.A. Popov and A. L. Skubachevskiĭ, “A priori estimates for elliptic differential-difference operators with degeneration,” J. Math. Sci. (N.Y.), 171, No. 1, 130–148 (2010).

  50. V.A. Popov and A. L. Skubachevskiĭ, “Smoothness of generalized solutions of elliptic differentialdifference equations with degeneration,” J. Math. Sci. (N.Y.), 190, No. 1, 135–146 (2013).

  51. D. Przeworska-Rolewicz, Equations with Transformed Argument: an Algebraic Approach, PWN, Warszawa (1973).

    MATH  Google Scholar 

  52. V. S. Rabinovič, “The solvability of differential-difference equations on R n and in a half-space,” Dokl. Akad. Nauk SSSR, 243, No. 5, 1134–1137 (1978).

    MathSciNet  Google Scholar 

  53. L. E. Rossovskii, “Dum** problem for systems with delay depending on time,” In: Problems of Contemporary Mathematics and Applications to Problems of Physics and Mechanics, 172–182, MIPT, Moscow (1995).

  54. L. E. Rossovskii, “Coercivity of functional differential equations,” Math. Notes, 59, No. 1-2, 75–82 (1996).

    Article  MathSciNet  Google Scholar 

  55. L.E. Rossovskii, “Coercivity of a class of functional differential equations,” Funct. Anal. Appl., 30, No. 1, 62–64 (1996).

    Article  MathSciNet  Google Scholar 

  56. L. E. Rossovskii, “Boundary value problems for elliptic functional differential equations with dilatation and contraction of the arguments,” Trans. Moscow Math. Soc., 2001, 185–212 (2001).

    MathSciNet  MATH  Google Scholar 

  57. L. E. Rossovskii, “Strongly elliptic differential-difference operators in a semibounded cylinder,” Fundam. Prikl. Mat., 7, No. 1, 289–293 (2001).

    MathSciNet  MATH  Google Scholar 

  58. L. E. Rossovskii, “On the boundary value problem for the elliptic functional differential equation with contractions,” Funct. Differ. Equ., 8, No. 3-4, 395–406 (2001).

    MathSciNet  MATH  Google Scholar 

  59. L. E. Rossovskii, “On boundary value problems for a class of functional differential equations,” Nonlinear Anal., 49, No. 6, 799–816 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  60. L. E. Rossovskii, “An elliptic functional differential equation with contractions of the arguments,” Dokl. Akad. Nauk, 411, No. 2, 161–163 (2006).

    MathSciNet  MATH  Google Scholar 

  61. L. E. Rossovskii, “Spectral properties of some functional differential operators, and a Gårding-type inequality,” Dokl. Math., 82, No. 2, 765–768 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  62. L. E. Rossovskii, “On the spectral stability of functional differential equations,” Math. Notes, 90, No. 5-6, 867–881 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  63. L.E. Rossovskii, “On a class of sectorial functional differential operators,” Differ. Equ., 48, No. 2, 234–245 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  64. L.E. Rossovskii, “On the coercivity of functional differential equations,” J. Math. Sci. (N.Y.), 201, No. 5, 663–672 (2014).

  65. L. E. Rossovskii and A. L. Skubachevskii, “Boundary value problems for functional differential equations with linearly transformed argument,” Spectr. Evol. Probl., 4, 77–82 (1995).

    MathSciNet  MATH  Google Scholar 

  66. L. E. Rossovskii and A. L. Skubachevskiĭ, “Solvability and regularity of solutions of some classes of elliptic functional differential equations,” In: Differ. Equ., 114–192, VINITI, Moscow (1999).

  67. W. Rudin, Functional Analysis [Russian translation], Mir, Moscow (1971).

    MATH  Google Scholar 

  68. A.Yu. Savin, “On the index of nonlocal elliptic operators corresponding to a nonisometric diffeomorphism,” Math. Notes, 90, No. 5-6, 701–714 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  69. A.Yu. Savin and B.Yu. Sternin, “On the index of elliptic operators for a group of dilations,” Sb. Math., 202, No. 9-10, 1505–1536 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  70. R. V. Shamin, “Spaces of initial data for differential equations in a Hilbert space,” Sb. Math., 194, No. 9-10, 1411–1426 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  71. M. A. Skryabin, “Partition of unity and the strong ellipticity problem for functional differential operators,” J. Math. Sci. (N.Y.), 170, No. 2, 270–282 (2010).

  72. A. L. Skubachevskii, “Some nonlocal elliptic boundary value problems,” Differ. Uravn., 18, No. 9, 1590–1599 (1982).

    MathSciNet  Google Scholar 

  73. A. L. Skubachevskii, “On the spectrum of some nonlocal elliptic boundary value problems,” Mat. Sb., 117, No. 4, 548–558 (1982).

    MathSciNet  Google Scholar 

  74. A. L. Skubachevskii, “Nonlocal elliptic boundary value problems with degeneration,” Differ. Uravn., 19, No. 3, 457–470 (1983).

    MathSciNet  Google Scholar 

  75. A. L. Skubachevskii, “Smoothness of solutions of the first boundary value problem for an elliptic differential-difference equation,” Mat. Zametki, 34, No. 1, 105–112 (1983).

    MathSciNet  Google Scholar 

  76. A. L. Skubachevskii, “Nonlocal boundary value problems with a shift,” Mat. Zametki, 38, No. 4, 587–598 (1985).

    MathSciNet  Google Scholar 

  77. A. L. Skubachevskii, “Elliptic problems with nonlocal conditions near the boundary,” Mat. Sb., 129, No. 2, 279–302 (1986).

    MathSciNet  Google Scholar 

  78. A. L. Skubachevskii, “The first boundary value problem for strongly elliptic differential-difference equations,” J. Differ. Equ., 63, 332–361 (1986).

    Article  MathSciNet  Google Scholar 

  79. A. L. Skubachevskii, “Some problems for multidimensional diffusion processes,” Sov. Math. Dokl., 40, No. 1, 75–79 (1990).

    MathSciNet  Google Scholar 

  80. A. L. Skubachevskii, “On the problem of dam** a control system with aftereffect,” Dokl. Math., 49, No. 2, 282–286 (1994).

    MathSciNet  MATH  Google Scholar 

  81. A. L. Skubachevskii, Elliptic Functional Differential Equations and Applications, Birkhäuser Verlag, Basel (1997).

    MATH  Google Scholar 

  82. A. L. Skubachevskii and R.V. Shamin, “The mixed boundary value problem for parabolic differential-difference equation,” Funct. Differ. Equ., 8, No. 3-4, 407–424 (2001).

    MathSciNet  MATH  Google Scholar 

  83. A. L. Skubachevskii and E. L. Tsvetkov, “The second boundary value problem for elliptic differential-difference equations,” Differ. Equ., 25, No. 10, 1245–1254 (1990).

    MathSciNet  Google Scholar 

  84. A. L. Skubachevskii and E. L. Tsvetkov, “General boundary-value problems for elliptic differentialdifference equations,” Tr. St. Petersburg Mat. Obshch., 5, 153–199 (1998).

    Google Scholar 

  85. M. Taylor, Pseudodifferential Operators, Princeton Univ. Press, Princeton (1981).

    MATH  Google Scholar 

  86. E. L. Tsvetkov, “Solvability and the spectrum of the third boundary value problem for an elliptic differential-difference equation,” Math. Notes, 51, No. 5-6, 599–603 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  87. E. L. Tsvetkov, “On the smoothness of generalized solutions of the third boundary value problem for an elliptic differential-difference equation,” Ukr. Math. J., 45, No. 8, 1272–1284 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  88. M. I. Vishik, “On strongly elliptic systems of differential equations,” Mat. Sb., 29, No. 3, 615–676 (1951).

    MathSciNet  Google Scholar 

  89. L. R. Volevich, “Solvability of boundary-value problems for general elliptic systems,” Mat. Sb., 68, 373–416 (1965).

    MathSciNet  Google Scholar 

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Correspondence to L. E. Rossovskii.

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Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 54, Functional Differential Equations, 2014.

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Rossovskii, L.E. Elliptic Functional Differential Equations with Contractions and Extensions of Independent Variables of the Unknown Function. J Math Sci 223, 351–493 (2017). https://doi.org/10.1007/s10958-017-3360-1

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