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Uspekhi Mat. Nauk, 2016, Volume 71, Issue 5(431), Pages 3–112 (Mi umn9739)  

This article is cited in 48 scientific papers (total in 48 papers)

Boundary-value problems for elliptic functional-differential equations and their applications

A. L. Skubachevskii

RUDN University, Moscow, Russia

Abstract: Boundary-value problems are considered for strongly elliptic functional-differential equations in bounded domains. In contrast to the case of elliptic differential equations, smoothness of generalized solutions of such problems can be violated in the interior of the domain and may be preserved only on some subdomains, and the symbol of a self-adjoint semibounded functional-differential operator can change sign. Both necessary and sufficient conditions are obtained for the validity of a Gårding-type inequality in algebraic form. Spectral properties of strongly elliptic functional-differential operators are studied, and theorems are proved on smoothness of generalized solutions in certain subdomains and on preservation of smoothness on the boundaries of neighbouring subdomains. Applications of these results are found to the theory of non-local elliptic problems, to the Kato square-root problem for an operator, to elasticity theory, and to problems in non-linear optics.
Bibliography: 137 titles.

Keywords: elliptic functional-differential equations, spectral properties, smoothness of generalized solutions, non-local elliptic problems, Kato square-root problem, three-layer plates, non-linear optical systems with two-dimensional feedback.

Funding Agency Grant Number
Russian Foundation for Basic Research 14-01-00265
Ministry of Education and Science of the Russian Federation -4479.2014.1
This research was supported by the Russian Foundation for Basic Research (grant no. 14-01-00265) and the Programme "Leading Scientific Schools" (grant no. -4479.2014.1).


DOI: https://doi.org/10.4213/rm9739

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English version:
Russian Mathematical Surveys, 2016, 71:5, 801–906

Bibliographic databases:

UDC: 517.9
MSC: Primary 35J25; Secondary 35B65
Received: 30.11.2015
Revised: 03.06.2015

Citation: A. L. Skubachevskii, “Boundary-value problems for elliptic functional-differential equations and their applications”, Uspekhi Mat. Nauk, 71:5(431) (2016), 3–112; Russian Math. Surveys, 71:5 (2016), 801–906

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    This publication is cited in the following articles:
    1. Liiko V.V. Skubachevskii A.L., “Smoothness of Solutions to the Mixed Problem For Elliptic Differential-Difference Equation in Cylinder”, Complex Var. Elliptic Equ.  crossref  isi
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    5. V. A. Popov, “Sledy obobschennykh reshenii ellipticheskikh differentsialno-raznostnykh uravnenii s vyrozhdeniem”, Trudy seminara po differentsialnym i funktsionalno-differentsialnym uravneniyam v RUDN pod rukovodstvom A. L. Skubachevskogo, SMFN, 62, RUDN, M., 2016, 124–139  mathnet
    6. A. Ashyralyev, Kh. Belakroum, A. Guezane-Lakoud, “Stability of boundary-value problems for third-order partial differential equations”, Electron. J. Differential Equations, 2017, 53, 11 pp.  mathscinet  zmath  isi
    7. A. Ashyralyev, F. Emharab, “Source identification problems for hyperbolic differential and difference equations”, International Conference Functional Analysis In Interdisciplinary Applications (FAIA 2017), AIP Conf. Proc., 1880, Amer. Inst. Phys., 2017, 040001  crossref  isi  scopus
    8. A. Ashyralyev, Kh. Belakroum, A. Guezane-Lakoud, “Numerical algorithm for the third-order partial differential equation with local boundary conditions”, International Conference Functional Analysis In Interdisciplinary Applications (FAIA 2017), AIP Conf. Proc., 1880, Amer. Inst. Phys., 2017, 040008  crossref  isi  scopus
    9. A. Ashyralyev, Kh. Belakroum, A. Guezane-Lakoud, “Numerical algorithm for the third-order partial differential equation with nonlocal boundary conditions”, International Conference Functional Analysis In Interdisciplinary Applications (FAIA 2017), AIP Conf. Proc., 1880, Amer. Inst. Phys., 2017, 040012  crossref  isi  scopus
    10. A. Muravnik, “On the half-plane Dirichlet problem for differential-difference elliptic equations with several nonlocal terms”, Math. Model. Nat. Phenom., 12:6 (2017), 130–143  crossref  zmath  isi  scopus
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    15. A. L. Skubachevskii, “The Kato conjecture for elliptic differential-difference operators with degeneration in a cylinder”, Dokl. Math., 97:1 (2018), 32–34  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  scopus
    16. Kh. Belakroum, A. Ashyralyev, A. Guezane-Lakoud, “A note on the nonlocal boundary value problem for a third order partial differential equation”, Filomat, 32:3 (2018), 801–808  crossref  isi  scopus
    17. A. L. Skubachevskiǐ, “On a class of functional-differential operators satisfying the Kato conjecture”, St. Petersburg Math. J., 30:2 (2019), 329–346  mathnet  crossref  mathscinet  isi  elib
    18. O. V. Solonukha, “On an Elliptic Differential-Difference Equation with Nonsymmetric Shift Operator”, Math. Notes, 104:4 (2018), 572–586  mathnet  crossref  crossref  mathscinet  isi  elib
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    20. A. Ashyraliev, Kh. Belakrum, “Ustoichivaya raznostnaya skhema dlya uravneniya v chastnykh proizvodnykh tretego poryadka”, Differentsialnye i funktsionalno-differentsialnye uravneniya, SMFN, 64, no. 1, Rossiiskii universitet druzhby narodov, M., 2018, 1–19  mathnet  crossref
    21. V. A. Popov, “Otsenki reshenii ellipticheskikh differentsialno-raznostnykh uravnenii s vyrozhdeniem”, Differentsialnye i funktsionalno-differentsialnye uravneniya, SMFN, 64, no. 1, Rossiiskii universitet druzhby narodov, M., 2018, 131–147  mathnet  crossref
    22. A. Aibeche, N. Amroune, S. Maingot, “General non local boundary value problem for second order elliptic equation”, Math. Nachr., 291:10 (2018), 1470–1485  crossref  mathscinet  zmath  isi  scopus
    23. Ch. Ashyralyyev, A. Çay, “Well-posedness of Neumann-type elliptic overdetermined problem with integral condition”, International Conference on Analysis and Applied Mathematics (ICAAM 2018), AIP Conf. Proc., 1997, Amer. Inst. Phys., 2018, 020026-1  crossref  isi  scopus
    24. Ch. Ashyralyyev, G. Akyuz, “A third order of accuracy difference scheme for Bitsadze-Samarskii type elliptic overdetermined multi-point problem”, International Conference on Analysis and Applied Mathematics (ICAAM 2018), AIP Conf. Proc., 1997, Amer. Inst. Phys., 2018, 020009-1  crossref  isi  scopus
    25. A. L. Skubachevskii, “Elliptic differential-difference operators with degeneration and the Kato square root problem”, Math. Nachr., 291:17-18 (2018), 2660–2692  crossref  mathscinet  zmath  isi
    26. A. Ashyralyev, O. Gercek, E. Zusi, “A note on the second order of accuracy difference scheme for elliptic-parabolic equations in Holder spaces”, Bull. Karaganda Univ. Math., 91:3 (2018), 108–116  crossref  isi
    27. E. P. Ivanova, “On Smooth Solutions of Differential-Difference Equations with Incommensurable Shifts of Arguments”, Math. Notes, 105:1 (2019), 140–144  mathnet  crossref  crossref  mathscinet  isi  elib
    28. A. L. Skubachevskii, V. V. Liiko, “On a certain property of a regular difference operator with variable coefficients”, Complex Var. Elliptic Equ., 64:5 (2019), 852–865  crossref  mathscinet  zmath  isi  scopus
    29. V A. Razgulin , V S. Sazonova, “Hopf bifurcation in diffusive model of nonlinear optical system with matrix Fourier filtering”, Commun. Nonlinear Sci. Numer. Simul., 77 (2019), 288–304  crossref  isi
    30. V. V. Liiko, A. L. Skubachevskii, “Silno ellipticheskie differentsialno-raznostnye uravneniya so smeshannymi kraevymi usloviyami v tsilindricheskoi oblasti”, Trudy Matematicheskogo instituta im. S.M. Nikolskogo RUDN, SMFN, 65, no. 4, Rossiiskii universitet druzhby narodov, M., 2019, 635–654  mathnet  crossref
    31. D. A. Neverova, “Gladkost obobschennykh reshenii vtoroi i tretei kraevykh zadach dlya silno ellipticheskikh differentsialno-raznostnykh uravnenii”, Trudy Matematicheskogo instituta im. S.M. Nikolskogo RUDN, SMFN, 65, no. 4, Rossiiskii universitet druzhby narodov, M., 2019, 655–671  mathnet  crossref
    32. L. E. Rossovskii, A. A. Tovsultanov, “Elliptic functional differential equation with affine transformations”, J. Math. Anal. Appl., 480:2 (2019), 123403  crossref  isi
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    34. V. V. Liiko, A. L. Skubachevskii, “Mixed Problems for Strongly Elliptic Differential-Difference Equations in a Cylinder”, Math. Notes, 107:5 (2020), 770–790  mathnet  crossref  crossref  mathscinet  isi  elib
    35. A. M. Selitskii, “On solvability of parabolic functional differential equations in Banach spaces (II)”, Eurasian Math. J., 11:2 (2020), 86–92  mathnet  crossref
    36. N. B. Zhuravlev, L. E. Rossovskii, “The spectral radius of a certain parametric family of functional operators”, Russian Math. Surveys, 75:5 (2020), 971–973  mathnet  crossref  crossref  mathscinet  isi  elib
    37. A. B. Muravnik, “Elliptic Differential-Difference Equations in the Half-Space”, Math. Notes, 108:5 (2020), 727–732  mathnet  crossref  crossref  mathscinet  isi  elib
    38. D. A. Neverova, “Gladkost obobschennykh reshenii zadachi Neimana dlya silno ellipticheskogo differentsialno-raznostnogo uravneniya na granitse sosednikh podoblastei”, Trudy Krymskoi osennei matematicheskoi shkoly-simpoziuma, SMFN, 66, no. 2, Rossiiskii universitet druzhby narodov, M., 2020, 272–291  mathnet  crossref
    39. A. Ashyralyev, C. Ashyralyyev, V. G. Zvyagin, “A note on well-posedness of source identification elliptic problem in a Banach space”, Bull. Karaganda Univ-Math., 99:3 (2020), 96–104  crossref  isi
    40. V N. Zaitseva, “Global classical solutions of some two-dimensional hyperbolic differential-difference equations”, Differ. Equ., 56:6 (2020), 734–739  crossref  mathscinet  zmath  isi
    41. V. A. Popov, “Elliptic functional differential equations with degenerations”, Lobachevskii J. Math., 41:5, SI (2020), 869–894  crossref  mathscinet  zmath  isi
    42. V N. Zaitseva, “On global classical solutions of hyperbolic differential-difference equations”, Dokl. Math., 101:2 (2020), 115–116  crossref  isi
    43. Zaitseva N.V., “Classical Solutions of Hyperbolic Differential-Difference Equations With Several Nonlocal Terms”, Lobachevskii J. Math., 42:1 (2021), 231–236  crossref  mathscinet  zmath  isi  scopus
    44. A. B. Muravnik, “Elliptic Differential-Difference Equations of General Form in the Half-Space”, Math. Notes, 110:1 (2021), 92–99  mathnet  crossref  crossref  isi
    45. E. P. Ivanova, “Differentsialno-raznostnye uravneniya s nesoizmerimymi sdvigami argumentov”, Materialy Voronezhskoi vesennei matematicheskoi shkoly Sovremennye metody teorii kraevykh zadach. Pontryaginskie chteniyaXXX. Voronezh, 39 maya 2019 g. Chast 2, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 191, VINITI RAN, M., 2021, 92–100  mathnet  crossref
    46. A. A. Tovsultanov, “Funktsionalno-differentsialnoe uravnenie s rastyazheniem i povorotom”, Vladikavk. matem. zhurn., 23:1 (2021), 77–87  mathnet  crossref
    47. Nikolai B. Zhuravlev, Leonid E. Rossovskii, “Spectral Radius Formula for a Parametric Family of Functional Operators”, Regul. Chaotic Dyn., 26:4 (2021), 392–401  mathnet  crossref  mathscinet
    48. O. V. Solonukha, “O razreshimosti lineinoi parabolicheskoi zadachi s nelokalnymi kraevymi usloviyami”, Posvyaschaetsya pamyati professora N.D. Kopachevskogo, SMFN, 67, no. 2, Rossiiskii universitet druzhby narodov, M., 2021, 349–362  mathnet  crossref
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