Skip to main content
SpringerLink
Log in

Nonclassical boundary-value problems. I

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

References

  1. S. Agmon, A. Douglis, and L. Nirenberg, “Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I,” Commun. Pure Appl. Math., 12, 623–727 (1959).

    MATH  MathSciNet  Google Scholar 

  2. M. S. Agranovich and M. I. Vishik, “Elliptic problems with a parameter and parabolic problems of general type,” Uspekhi Mat. Nauk, 19, No. 3, 53–161 (1964); English transl.: Russian Math. Surveys, 19 (1964).

    MATH  Google Scholar 

  3. W. G. Bade and R. S. Freeman, “Closed extensions of the Laplace operator determined by a general class of boundary conditions,” Pacific J. Math., 12, No. 2, 395–410 (1962).

    MATH  MathSciNet  Google Scholar 

  4. R. Beals, “Nonlocal elliptic boundary-value problems,” Bull. Amer. Math. Soc., 70, No. 5, 693–696 (1964).

    MATH  MathSciNet  Google Scholar 

  5. A. Bensoussan, “On the Hamilton-Jacobi approach for the optimal control of diffusion processes with jumps,” in: Stochastic Analysis (A. Freedman, M. Pinsky, Eds). Academic Press, New York, 25–55 (1978).

    Google Scholar 

  6. A. Bensoussan and J. L. Lions, Impulse Control and Quasi-Variational Inequalities, Gauthier-Villars, Bordas, Paris (1984).

    Google Scholar 

  7. O. V. Besov, V. P. Il'in, and S. M. Nikol'skii, Integral Representations of Functions and Embedding Theorems. I, Wiley, New York (1978).

    Google Scholar 

  8. A. V. Bitsadze, “On the theory of nonlocal boundary-value problems,” Dokl. Akad. Nauk SSSR, 277, No. 1, 17–19 (1984); English transl.: Soviet Math. Dokl. 30 (1984).

    MathSciNet  Google Scholar 

  9. A. V. Bitsadze, “On some class of conditionally solvable nonlocal boundary-value problems for harmonic functions,” Dokl. Akad. Nauk SSSR, 280, No. 3, 521–524 (1985); English transl.: Soviet Math. Dokl., 31 (1985).

    MathSciNet  Google Scholar 

  10. A. V. Bitsadze and A. A. Samarskii, “On some simple generalization of linear elliptic boundary-value problems,” Dokl. Akad. Nauk SSSR, 185, No. 4, 739–740 (1969); English transl.: Soviet Math. Dokl., 10 (1969).

    MathSciNet  Google Scholar 

  11. P. M. Blekher, “Operators depending meromorphically on a parameter,” Vestnik Moskov. Univ. Ser. I. Mat. Mekh., No. 5, 30–36 (1969); English transl.: Moscow Univ. Math. Bull., 24 (1969).

  12. J. M. Bony, P. Courrege, and P. Priouret, “Semi groups de Feller sur une variété à bord compacte et problèmes aux limites intégro-différentiels du second ordre donnant lieu au principe du maximum,” Ann. Inst. Fourier (Grenoble), 18, No. 2, 369–521 (1968).

    MATH  MathSciNet  Google Scholar 

  13. V. M. Borok and M. A. Perel'man, “On classes of uniqueness for solutions to multipoint boundary-value problem in infinite layer,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 8, 29–34 (1973); English transl.: Soviet Math. (Izv. VUZ), No. 8 (1973).

  14. F. Browder, “Non-local elliptic boundary value problems,” Amer. J. Math., 86, No. 4. 735–750 (1964).

    MATH  MathSciNet  Google Scholar 

  15. F. Browder, “Asymptotic distributions of eigenvalues and eigenfunctions for non-local elliptic boundary value problems. I,” Amer. J. Math., 87, No. 1, 175–195 (1965).

    MATH  MathSciNet  Google Scholar 

  16. J. W. Calkin, “General self-adjoint boundary conditions for certain partial differential operators,” Proc. Natl. Acad. Sci. USA, 25, No. 4, 201–206 (1939).

    MATH  Google Scholar 

  17. C. Cancelier, “Problèmes aux limites pseudo-différentiels donnant lieu au principe du maximum,” Comm. Partial Differential Equations, 11, 1677–1726 (1986).

    MATH  MathSciNet  Google Scholar 

  18. T. Carleman, “Sur la theorie des equations integrales et ses applications,” Verhandlungen des Internat. Math. Kongr., Zürich., 1, 138–151 (1932).

    Google Scholar 

  19. K. Cooke, L. E. Rossovskii, and A. L. Skubachevskii, “A boundary-value problem for a functional-differential equation with a linearly transformed argument,” Differ. Uravn., 31, No. 8, 1348–1352 (1995); English transl.: Differ. Equ., 31 (1995).

    MathSciNet  Google Scholar 

  20. L. S. Darbinyan, “Solution of some functional equation in a class of analytic functions and their application” [in Russian], Erevansk. Gos. Univ. Uchen. Zap., 151, No. 3, 13–19 (1982).

    MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  22. N. Dunford and J. T. Schwartz, Linear Operators, Part. II: Spectral Theory, Interscience Publishers, New York-London (1963).

    MATH  Google Scholar 

  23. W. Feller, “The parabolic differential equations and associated semi-groups of transformations,” Annal. Math., 55, No. 3, 468–518 (1952).

    MathSciNet  Google Scholar 

  24. W. Feller, “Diffusion processes in one dimension,” Trans. Amer. Math. Soc., 77, No. 1, 1–31 (1954).

    MATH  MathSciNet  Google Scholar 

  25. R. S. Freeman, “Closed operators and their adjoints associated with elliptic differential operators,” Pacific J. Math., 22, No. 1, 71–97 (1967).

    MATH  MathSciNet  Google Scholar 

  26. E. I. Galakhov and A. L. Skubachevskii, “A nonlocal spectral problem,” Differ. Uravn., 33, No. 1, 25–32 (1997); English transl.: Differ. Equ., 33 (1997).

    MathSciNet  Google Scholar 

  27. E. I. Galakhov and A. L. Skubachevskii, “On non-negative contractive semigroups with nonlocal conditions,” Mat. Sb., 189, No. 1, 45–78 (1998); English transl.: Sb. Math., 189 (1998).

    MATH  MathSciNet  Google Scholar 

  28. E. I. Galakhov and A. L. Skubachevskii, “On Feller semigroups generated by elliptic operators with integro-differential boundary conditions,” J. Differential Equations, 176, 315–355 (2001).

    MATH  MathSciNet  Google Scholar 

  29. M. G. Garroni and J. L. Menaldi, “Green's function and invariant density for an integro-differential operator of second order,” Ann. Mat. Pura. Appl. (4), 154, 147–222 (1989).

    MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  31. F. Gimpert and P. L. Lions, “Existence and regularity results for solutions of second-order elliptic integro-differential operators,” Ricerche Mat., 33, No. 2, 315–358 (1984).

    MathSciNet  Google Scholar 

  32. I. C. Gohberg and E. I. Sigal, “An operator generalization of the logarithmic residue theorem and the theorem of Rouche,” Mat. Sb., 84(126), No. 4, 607–629 (1971); English transl.: Math. USSR Sb., 13 (1972).

    MathSciNet  Google Scholar 

  33. G. Grubb, “A characterization of the non-local boundary value problems associated with elliptic operator,” Ann. Sc. Norm. Super. Pisa Cl. Sci. (3), 22, No. 3, 425–513 (1968).

    MATH  MathSciNet  Google Scholar 

  34. P. L. Gurevich, “Solvability of the boundary-value problem for some differential-difference equations,” Funct. Differ. Equ., 5, No. 1–2, 139–157 (1998).

    MATH  MathSciNet  Google Scholar 

  35. P. L. Gurevich, “Nonlocal elliptic problems in dihedral angles and the Green formula,” Dokl. Akad. Nauk, 379, No. 6, 735–738 (2001); English transl.: Russian Acad. Sci. Dokl. Math., 64, No. 1 (2001).

    MathSciNet  Google Scholar 

  36. P. L. Gurevich, “Nonlocal problems for elliptic equations in dihedral angles and the Green formula,” Mitt. Math. Sem. Giessen, 247, 1–74 (2001).

    Google Scholar 

  37. P. L. Gurevich, “Asymptotics of solutions for nonlocal elliptic problems in plane angles,” Tr. Semin. im. I. G. Petrovskogo, 23, 93–126 (2003); English transl.: J. Math. Sci. (N.Y.), 120, No. 3 (2004).

    Google Scholar 

  38. P. L. Gurevich, “Nonlocal elliptic problems with nonlinear transformations of variables near conjugation points,” Izv. Ross. Akad. Nauk Ser. Mat., 67, No. 6, 71–110 (2003); English transl.: Izv. Math., 67, No. 6 (2003).

    MathSciNet  Google Scholar 

  39. P. L. Gurevich, “Asymptotics of solutions for nonlocal elliptic problems in plane bounded domains,” Funct. Differ. Equ., 10, No. 1–2, 175–214 (2003).

    MATH  MathSciNet  Google Scholar 

  40. P. L. Gurevich, “Solvability of nonlocal elliptic problems in Sobolev spaces, I,” Russ. J. Math. Phys., 10, No. 4, 436–466 (2003).

    MATH  MathSciNet  Google Scholar 

  41. P. L. Gurevich, “Solvability of nonlocal elliptic problems in Sobolev spaces, II,” Russ. J. Math. Phys., 11, No. 1, 1–44 (2004).

    MATH  MathSciNet  Google Scholar 

  42. P. L. Gurevich, “Generalized solutions of nonlocal elliptic problems,” Mat. Zametki, 77, No. 5, 665–682 (2005); English transl.: Math. Notes, 77, No. 5–6 (2005).

    MathSciNet  Google Scholar 

  43. P. L. Gurevich, “On stability of index for unbounded nonlocal operators in Sobolev spaces,” Tr. Mat. Inst. Steklova, 255, 116–135 (2006); English transl.: Proc. Steklov Inst. Math., 255 (2006)

    MathSciNet  Google Scholar 

  44. P. L. Gurevich, “The consistency conditions and the smoothness of generalized solutions of nonlocal elliptic problems,” Adv. Differential Equations, 11, No. 3, 305–360 (2006).

    MATH  MathSciNet  Google Scholar 

  45. P. L. Gurevich and A. L. Skubachevskii, “On the Fredholm and unique solvability of nonlocal elliptic problems in multidimensional domains,” Tr. Moskov. Mat. Obs., 68, 288–373 (2007); English transl.: Trans. Moscow Math. Soc., 68 (2007).

    MathSciNet  Google Scholar 

  46. A. K. Gushchin, “The compactness condition for a class of operators and its application to the investigation of the solvability of nonlocal problems for elliptic equations,” Mat. Sb., 193, No. 5, 17–36 (2002); English transl.: Sb. Math., 193, No. 5–6 (2003).

    MathSciNet  Google Scholar 

  47. A. K. Gushchin and V. P. Mikhailov, “The Fredholmness of a class of nonlocal problems for a second-order elliptic equation,” Dokl. Akad. Nauk, 333, No. 3, 290–292 (1993); English transl.: Russian Acad. Sci. Dokl. Math., 48, No. 3 (1993).

    MathSciNet  Google Scholar 

  48. A. K. Gushchin and V. P. Mikhailov, “On the solvability of nonlocal problems for a second-order elliptic equation,” Math. Sb., 185, No. 4, 121–160 (1994); English transl.: Sb. Math., 81 (1995).

    Google Scholar 

  49. V. A. Il'in, “The necessary and sufficient conditions for property of being a Riesz basis for root functions of discontinuous second order operator,” Differ. Uravn., 22, No. 12, 2059–2071 (1986); English transl.: Differ. Equ., 22 (1986).

    Google Scholar 

  50. V. A. Il'in and E. I. Moiseev, “An apriori bound for a solution of the problem conjugate to a non-local boundary-value problem of the first kind,” Differ. Uravn., 24, No. 5, 795–804 (1988); English transl.: Differ. Equ., 24 (1988).

    MATH  Google Scholar 

  51. Y. Ishikawa, “A remark on the existence of a diffusion process with non-local boundary conditions,” J. Math. Soc. Japan, 42, No. 1. 171–184 (1990).

    MATH  MathSciNet  Google Scholar 

  52. A. G. Kamenskii, “Boundary-value problems for equations with formally symmetric differential-difference operators,” Differ. Uravn., 12, No. 5, 815–824 (1976); English transl.: Differ. Equ., 12 (1976).

    MathSciNet  Google Scholar 

  53. G. A. Kamenskii and A. D. Myshkis, “On the formulation of boundary value problems for differential equations with deviating arguments and several highest terms,” Differ. Uravn., 10, No. 3, 409–418 (1974); English transl.: Differ. Equ., 10 (1974).

    MATH  Google Scholar 

  54. G. A. Kamenskii, A. D. Myshkis, and A. L. Skubachevskii, “Smooth solutions of a boundary-value problem for a differential-difference equation of neutral type,” Ukrain. Mat. Zh., 37, No. 5, 581–589 (1985); English transl.: Ukrainian Math. J., 37 (1986).

    MathSciNet  Google Scholar 

  55. L. I. Kamynin, “Uniqueness of solutions to boundary-value problems for second order elliptic differential equation with degeneration,” Differ. Uravn., 14, No. 1, 39–49 (1978); English transl.: Differ. Equ., 14 (1978).

    MATH  MathSciNet  Google Scholar 

  56. Yu. I. Karlovich, “On integral operators with shift of integration contour into domain,” Dokl. Akad. Nauk SSSR, 216, No. 1, 32–35 (1974).

    MathSciNet  Google Scholar 

  57. Yu. I. Karlovich, “On singular integral operators with shift of the contour-support into domain” [in Russian], Tr. Tbilisskogo Matematicheskogo Instituta, 44, 113–124 (1974).

    MATH  Google Scholar 

  58. Yu. I. Karlovich, V. G. Kravchenko, and G. S. Litvinchuk, “Noether's theory of singular integral operators with shift,” Izv. Vyssh. Uchebn. Zaved. Mat., 27, 3–27 (1983); English transl.: Soviet Math. (Izv. VUZ), 27 (1983).

    MathSciNet  Google Scholar 

  59. T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin-Heideberg-New York (1966).

    MATH  Google Scholar 

  60. K. Yu. Kishkis, “On some nonlocal problem for harmonic functions in multiply connected domain,” Differ. Uravn., 23, No. 1, 174–177 (1987); English transl.: Differ. Equ., 23 (1987).

    MATH  MathSciNet  Google Scholar 

  61. K. Yu. Kishkis, “The index of the Bitsadze-Samarskii problem for harmonic functions,” Differ. Uravn., 24, No. 1, 105–110 (1988); English transl.: Differ. Equ., 24 (1988).

    MathSciNet  Google Scholar 

  62. K. Yu. Kishkis, “On the theory of nonlocal problems for the Laplace equation,” Differ. Uravn., 25, No. 1, 59–64 (1989); English transl.: Differ. Equ., 25 (1989).

    MathSciNet  Google Scholar 

  63. V. A. Kondrat'ev, “Boundary-value problems for parabolic equations in closed domains,” Tr. Mosk. Mat. Obs., 15, 400–451 (1966); English transl.: Trans. Moscow Math. Soc., 15 (1966).

    MATH  Google Scholar 

  64. V. A. Kondrat'ev, “Boundary-value problems for elliptic equations in domains with conical or angular points,” Tr. Mosk. Mat. Obs., 16, 209–292 (1967); English transl.: Trans. Moscow Math. Soc., 16 (1967).

    MATH  Google Scholar 

  65. V. A. Kondrat'ev and O. A. Oleynik, “Boundary problems for partial differential equations in non-smooth domains,” Uspekhi Mat. Nauk, 38, No. 2, 3–76 (1983); English transl.: Russian Math. Surveys, 38 (1983).

    MATH  MathSciNet  Google Scholar 

  66. O. A. Kovaleva and A. L. Skubachevskii, “Nonlocal elliptic problems in the Sobolev spaces W p l (Q),” Dokl. Akad. Nauk, 366, No. 4, 449–451 (1999).

    MATH  MathSciNet  Google Scholar 

  67. O. A. Kovaleva and A. L. Skubachevskii, “Solvability of nonlocal elliptic problems in weight spaces,” Mat. Zametki, 67, No. 6, 882–898 (2000); English transl.: Math. Notes., 67 (2000).

    MathSciNet  Google Scholar 

  68. A. M. Krall, “The development of general differential and general boundary systems,” Rocky Mountain J. Math., 5, No. 4, 493–542 (1975).

    Article  MATH  MathSciNet  Google Scholar 

  69. S. G. Krein, Linear Equations in Banach Spaces, Birkhäuser, Boston (1982).

    MATH  Google Scholar 

  70. A. A. Lacey, “Thermal runaway in nonlocal problem modeling Ohmic heating, I: Model derivation and some special cases,” Eur. J. Appl. Math., 6, 127–144 (1995).

    MATH  MathSciNet  Google Scholar 

  71. A. A. Lacey, “Thermal runaway in nonlocal problem modeling Ohmic heating, II: General proof of blow-up and asymptotics of runaway,” Eur. J. Appl. Math., 6, 201–224 (1995).

    MATH  MathSciNet  Google Scholar 

  72. O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Acad. Press., New York-London (1968).

    MATH  Google Scholar 

  73. S. Lenhart, “Integro-differential operators associated with diffusion process with jumps,” Appl. Math. Optim., 9, 177–191 (1982).

    MATH  MathSciNet  Google Scholar 

  74. J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol. I, Springer, New York-Heidelberg-Berlin (1972).

    Google Scholar 

  75. V. G. Maz'ya and B. A. Plamenevskii, “L p -estimates of solutions of elliptic boundary-value problems in domains with edges,” Tr. Mosk. Mat. Obs., 37, 49–93 (1978); English transl.: Trans. Moscow Math. Soc., 37 (1980).

    MATH  MathSciNet  Google Scholar 

  76. C. Miranda, Equazioni alle Derivate Parziali di Tipo Ellittico, Springer, Berlin-Göttingen-Heidelberg (1955).

    MATH  Google Scholar 

  77. E. Mitidieri and S. I. Pohozaev, “Liouville-type theorems for certain nonlinear nonlocal problems,” Dokl. Akad. Nauk, 399, No. 6, 732–736 (2004); English transl.: Russian Acad. Sci. Dokl. Math., 70, No. 3 (2004)

    MathSciNet  Google Scholar 

  78. E. I. Moiseev, “On the basis property of eigenfunctions for some nonlocal boundary-value problem,” Dokl. Akad. Nauk SSSR, 313, No. 3, 556–559 (1990); English transl.: Soviet Math. Dokl (1990).

    Google Scholar 

  79. E. I. Moiseev, “Spectral problems of nonlocal boundary-value problems,” Differ. Uravn., 30, No. 4, 797–805 (1994); English transl.: Differ. Equ., 30 (1994).

    MATH  MathSciNet  Google Scholar 

  80. N. I. Muskhelishvili, Singular Integral Equations [in Russian], Fizmatgiz, Moscow (1962).

    MATH  Google Scholar 

  81. M. A. Mustafin, “On the Abel method of summability for series with respect to root vectors of nonlocal elliptic problems,” Differ. Uravn., 26, No. 1, 167–168 (1990); English transl.: Differ. Equ., 26 (1990).

    MATH  MathSciNet  Google Scholar 

  82. S. A. Nazarov and B. A. Plamenevskii, Elliptic Problems in Domains with Piecewise Smooth Boundaries [in Russian], Nauka, Moscow (1991).

    Google Scholar 

  83. G. G. Onanov and A. L. Skubachevskii, “Differential equations with deviating arguments in stationary problems in the mechanics of a deformed body,” Prikl. Mekh., 15, No. 5, 39–47 (1979); English transl.: Soviet Applied Mech., 15 (1979).

    MathSciNet  Google Scholar 

  84. 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 theorem,” Russ. J. Math. Phys., 3, No. 4, 491–500 (1996).

    MathSciNet  Google Scholar 

  85. B. P. Paneyakh, “Some nonlocal boundary value problems for linear differential operators,” Mat. Zametki, 35, No. 3, 425–434 (1984); English transl.: Math. Notes, 35 (1984).

    MathSciNet  Google Scholar 

  86. M. Picone, “I teoremi d'esistenza per gl'integrale di una equazione differenziale lineare ordinaria soddisfacenti ad uns nuova classe di condizioni,” Rend. Acc. Lincei., 17, 340–347 (1908).

    Google Scholar 

  87. M. Picone, “Equazione integrale traducente il piu generale problema lineare per le equation differentiali lineari ordinarie di qualsivoglia ordine,” Accad. Naz. dei Lincei. Atti dei Convegni. Roma, 15, No. 6, 942–948 (1932).

    MATH  Google Scholar 

  88. V. V. Pod'yapol'skii, “The Abel summability of system of root functions for some nonlocal problem with integral conditions,” Mat. Zametki, 65, No. 5, 797–800 (1999); English transl.: Math. Notes, 65 (1999).

    MathSciNet  Google Scholar 

  89. V. V. Pod'yapol'skii and A. L. Skubachevskii, “The spectral asymptotics of strongly elliptic differential-difference operators,” Differ. Uravn., 35, No. 6, 793–800 (1999); English transl.: Differ. Equ., 35 (1999).

    MathSciNet  Google Scholar 

  90. Ya. A. Roitberg and Z. G. Sheftel', “On a class of general nonlocal elliptic problems,” Dokl. Akad. Nauk SSSR, 192, No. 3, 511–513 (1970); English transl.: Soviet Math. Dokl., 11 (1970).

    MathSciNet  Google Scholar 

  91. Ya. A. Roitberg and Z. G. Sheftel, “Nonlocal problems for elliptic equations and systems,” Sibirsk. Mat. Zh., 13, No. 1, 165–181 (1972). English transl.: Siberian Math. J., 13 (1972).

    Google Scholar 

  92. L. E. Rossovskii, “On boundary-value problems for a class of functional differential equations,” Nonlinear Anal., 49, 799–816 (2002).

    MATH  MathSciNet  Google Scholar 

  93. A. A. Samarskii, “Some problems of the theory of differential equations,” Differ. Uravn., 16, No. 11, 1925–1935 (1980); English transl.: Differ. Equ., 16 (1980).

    MathSciNet  Google Scholar 

  94. K. Sato and T. Ueno, “Multidimensional diffusion and the Markov process on the boundary,” J. Math. Kyoto Univ., 4, No. 3, 529–605 (1965).

    MATH  MathSciNet  Google Scholar 

  95. M. Schechter, “Nonlocal elliptic boundary value problems,” Ann. Sc. Norm. Super. Pisa Cl. Sci. (3), 20, No. 2, 421–441 (1966).

    MATH  MathSciNet  Google Scholar 

  96. Yu. G. Sentsov, “On the Riesz basis property of a system of eigen-and associated functions for a differential operator with integral conditions,” Mat. Zametki, 65, No. 6, 948–952 (1999); English transl.: Math. Notes, 65 (1999).

    MathSciNet  Google Scholar 

  97. A. A. Shkalikov, “Basis property of eigenfunctions of ordinary differential operators with integral boundary conditions,” Vestnik Moskov. Univ. Ser. I Mat. Mekh., No. 6, 12–21 (1982); English transl.: Moscow Univ. Math. Bull., 37, No. 6 (1982).

  98. Yu. T. Sil'chenko, “Eigenvalues and eigenfunctions of differential operator with nonlocal boundary conditions,” Differ. Uravn., 42, No. 6, 764–768 (2006); English transl.: Differ. Equ., 42 (2006).

    MathSciNet  Google Scholar 

  99. A. L. Skubachevskii, “Some nonlocal elliptic boundary-value problems,” Differ. Uravn., 18, No. 9, 1590–1599 (1982); English transl.: Differ. Equ., 18 (1982).

    MathSciNet  Google Scholar 

  100. A. L. Skubachevskii, “Nonlocal elliptic boundary-value problems with degeneration,” Differ. Uravn., 19, No. 3, 457–470 (1983); English transl.: Differ. Equ., 19 (1983).

    MathSciNet  Google Scholar 

  101. A. L. Skubachevskii, “Nonlocal elliptic problems with a parameter,” Mat. Sb., 121(163), No. 6, 201–210 (1983); English transl.: Math. USSR Sb., 49 (1984).

    MathSciNet  Google Scholar 

  102. A. L. Skubachevskii, “The elliptic problems of A. V. Bitsadze and A. A. Samarskii,” Dokl. Akad. Nauk SSSR, 278, No. 4, 813–816 (1984): English transl.: Soviet Math. Dokl., 30 (1984).

    MathSciNet  Google Scholar 

  103. A. L. Skubachevskii, “Solvability of elliptic problems with boundary conditions of Bitsadze-Samarskii type,” Differ. Uravn., 21, No. 4, 701–706 (1985); English transl.: Differ. Equ., 21 (1985).

    MathSciNet  Google Scholar 

  104. A. L. Skubachevskii, “Elliptic problems with nonlocal conditions near the boundary,” Mat. Sb., 129(171), No. 2, 279–302 (1986); English transl.: Math. USSR Sb., 57 (1987).

    MathSciNet  Google Scholar 

  105. A. L. Skubachevskii, “Solvability of elliptic problems wilh nonlocal boundary conditions,” Dokl. Akad. Nauk SSSR, 291, No. 3, 551–555 (1986): English transl.: Soviet Math. Dokl., 34 (1987).

    MathSciNet  Google Scholar 

  106. A. L. Skubachevskii, “Eigenvalues and eigenfunctions of some nonlocal boundary-value problems,” Differ. Uravn., 25, No. 1, 127–136 (1989); English transl.: Differ. Equ., 25 (1989).

    MathSciNet  Google Scholar 

  107. A. L. Skubachevskii, “On some problems for multidimensional diffusion processes,” Dokl. Akad. Nauk SSSR, 307, No. 2, 287–291 (1989); English transl.: Soviet Math. Dokl., 40 (1990).

    Google Scholar 

  108. A. L. Skubachevskii, “Model nonlocal problems for elliptic equations in dihedral angles,” Differ. Uravn., 26, No. 1, 120–131 (1990); English transl.: Differ. Equ., 26 (1990).

    MathSciNet  Google Scholar 

  109. A. L. Skubachevskii, “Truncation-function method in the theory of nonlocal problems,” Differ. Uravn., 27, No. 1, 128–139 (1991); English transl.: Differ. Equ., 27 (1991).

    MathSciNet  Google Scholar 

  110. A. L. Skubachevskii, “On the stability of index of nonlocal elliptic problems,” J. Math. Anal. Appl., 160, No. 2, 323–341 (1991).

    MATH  MathSciNet  Google Scholar 

  111. A. L. Skubachevskii, “Boundary-value problems for differential-difference equations with incommensurable shifts,” Dokl. Akad. Nauk, 324, No. 6, 1155–1158 (1992): English transl.: Russian Acad. Sci. Dokl. Math., 45 (1992).

    MathSciNet  Google Scholar 

  112. A. L. Skubachevskii, “On the Feller semigroups for multidimensional diffusion processes,” Dokl. Akad. Nauk, 341, No. 2, 173–176 (1995): English transl.: Russian Acad. Sci. Dokl. Math., 51 (1995).

    MathSciNet  Google Scholar 

  113. A. L. Skubachevskii, “Nonlocal elliptic problems and multidimensional diffusion processes,” Russ. J. Math. Phys., 3, No. 3, 327–360 (1995).

    MathSciNet  Google Scholar 

  114. A. L. Skubachevskii, Elliptic Functional Differential Equations and Applications, Birkhäuser, Basel-Boston-Berlin (1997).

    MATH  Google Scholar 

  115. A. L. Skubachevskii, “Regularity of solutions for some nonlocal elliptic problem,” Russ. J. Math. Phys., 8, No. 3, 365–374 (2001).

    MATH  MathSciNet  Google Scholar 

  116. A. L. Skubachevskii, “Nonlocal boundary-value problems for elliptic systems in angles,” Tr. Semin. im. I. G. Petrovskogo, 22, 283–298 (2002); English transl.: J. Math. Sci. (N. Y.), 114, No. 4 (2003).

    Google Scholar 

  117. A. L. Skubachevskii, “On the solvability of nonlocal problems for elliptic systems in infinite angles,” Dokl. Akad. Nauk, 412, No. 3, 317–320 (2007).

    MathSciNet  Google Scholar 

  118. A. L. Skubachevskii and G. M. Steblov, “On the spectrum of differential operators with domain nondense in L 2(0, 1),” Dokl. Akad. Nauk SSSR, 321, No. 6, 1158–1163 (1991); English transl.: Soviet Math. Dokl., 44 (1992).

    Google Scholar 

  119. L. N. Slobodetskii, “Generalized Sobolev spaces and their application to boundary problems for partial differential equations,” Leningrad. Gos. Ped. Inst. Uchen. Zap., 197, 54–112 (1958); English transl.: Amer. Math. Soc. Transl. Ser. 2, 57 (1966).

    Google Scholar 

  120. S. L. Sobolev, Application of Functional Analysis in Mathematical Physics, AMS, Providence, (1963).

    Google Scholar 

  121. A. P. Soldatov, “The Bitsadze-Samarskii problem for functions analytic in the sense of Douglis,” Differ. Uravn., 41, No. 3, 396–407 (2005); English transl.: Differ. Equ., 41 (2005).

    MathSciNet  Google Scholar 

  122. V. A. Solonnikov, “General boundary value problems for systems elliptic in the sense of A. Douglis and L. Nirenberg,” Trudy Mat. Inst. Steklova, 92, 233–297 (1966).

    MathSciNet  Google Scholar 

  123. A. Sommerfeld, “Ein Beitrag zur hydrodynamischen Erklärung der turbulenten Flussigkeitsbewegungen,” Atti del IV Congr. Intern dei Matem. Rome, 3, 116–124 (1909).

    Google Scholar 

  124. B. Yu. Sternin and B. E. Shatalov, “Extension of algebra of pseudodifferential operators and some nonlocal elliptic problems,” Mat. Sb., 185, No. 3, 117–160 (1994); English transl.: Sb. Math., 81, No. 2 (1995).

    Google Scholar 

  125. K. Taira, Diffusion Processes and Partial Differential Equations, London, Academic Press (1988).

    MATH  Google Scholar 

  126. K. Taira, “On the existence of Feller semigroups with boundary conditions,” Mem. Amer. Math. Soc., 99, No. 475, 1–65 (1992).

    MathSciNet  Google Scholar 

  127. Ya. D. Tamarkin, Some General Problems of the Theory of Ordinary Linear Differential Equations and Expansion of an Arbitrary Function in Series of Fundamental Functions, Petrograd (1917); abridged English transl.: Math. Z., 27 (1927/1928).

  128. H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, VEB Deutscher Verlag der Wissenschaften, Berlin (1978).

    Google Scholar 

  129. A. D. Ventsel', “On boundary conditions for multidimensional diffusion processes,” Teor. Veroyatn. Primen., 14, No. 2, 172–185 (1959).

    Google Scholar 

  130. M. I. Vishik, “On strongly elliptic systems of differential equations” [in Russian], Mat. Sb., 29(71), No. 3, 615–676 (1951).

    Google Scholar 

  131. M. I. Vishik, “On general boundary value problems for elliptic differential equations” [in Russian], Tr. Mosk. Mat. Obs., 1, 187–246 (1952).

    MATH  Google Scholar 

  132. L. R. Volevich, “Solvability of boundary-value problems for general elliptic systems” [in Russian], Mat. Sb., 68, No. 3, 373–416 (1965).

    MathSciNet  Google Scholar 

  133. K. Yosida, Functional Analysis, Springer, Berlin-Heidelberg-New York (1971).

    MATH  Google Scholar 

  134. N. V. Zhitarashu and S. D. Eidel'man, “On nonlocal boundary-value problems for elliptic equations” [in Russian], Mat. Issled. Kishinev, 6, No. 2, 63–73 (1971).

    MATH  Google Scholar 

  135. N. A. Zhura, “Boundary value problems of Bitsadze-Samarskii type for systems that are elliptic in the Douglis-Nirenberg sense,” Differ. Uravn., 28, No. 1, 81–91 (1992); English transl.: Differ. Equ., 28 (1992).

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. People’s Friendship University of Russia, Moscow, Russia

    A. L. Skubachevskii

Authors
  1. A. L. Skubachevskii
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. L. Skubachevskii.

Additional information

__________

Translated from Sovremennaya Matematika. Fundamental'nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 26, Nonclassical boundary-value problems. I, 2008.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Skubachevskii, A.L. Nonclassical boundary-value problems. I. J Math Sci 155, 199–334 (2008). https://doi.org/10.1007/s10958-008-9218-9

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-008-9218-9

Keywords

Search

Navigation