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Mat. Sb., 1994, Volume 185, Number 1, Pages 121–160 (Mi msb876)  

This article is cited in 53 scientific papers (total in 53 papers)

On solvability of nonlocal problems for a second-order elliptic equation

A. K. Gushchina, V. P. Mikhailov

a Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: This article is an investigation of the solvability of nonlocal problems for an elliptic equation, in which the values of the solution on the boundary of the domain $\mathcal{Q}$ under consideration are expressed in terms of its values at interior points and other points of the boundary.
A new concept of solution (in the space of $(n-1)$-dimensionally continuous functions) is introduced, broader than concepts considered previously, and sufficient conditions are established for the problem to be Fredholm with index zero. The connection between solvability of the problem in this formulation and in the classical formulation is studied. In particular, there is a class of nonlocal problems (including some problems studied previously) that are Fredholm with index zero in the formulation introduced but not in the classical formulation (sometimes not even Fredholm). For a certain class of problems a theorem on unique solvability is proved.

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English version:
Russian Academy of Sciences. Sbornik. Mathematics, 1995, 81:1, 101–136

Bibliographic databases:

UDC: 517.9
MSC: Primary 35J25; Secondary 47A53
Received: 24.06.1993

Citation: A. K. Gushchin, V. P. Mikhailov, “On solvability of nonlocal problems for a second-order elliptic equation”, Mat. Sb., 185:1 (1994), 121–160; Russian Acad. Sci. Sb. Math., 81:1 (1995), 101–136

Citation in format AMSBIB
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\by A.~K.~Gushchin, V.~P.~Mikhailov
\paper On solvability of nonlocal problems for a~second-order elliptic equation
\jour Mat. Sb.
\yr 1994
\vol 185
\issue 1
\pages 121--160
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\transl
\jour Russian Acad. Sci. Sb. Math.
\yr 1995
\vol 81
\issue 1
\pages 101--136
\crossref{https://doi.org/10.1070/SM1995v081n01ABEH003617}
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. K. Gushchin, V. P. Mikhailov, “On the continuity of the solutions of a class of non-local problems for an elliptic equation”, Sb. Math., 186:2 (1995), 197–219  mathnet  crossref  mathscinet  zmath  isi
    2. Gushchin A. Mikhailov V., “On Unique Solvability of Non-Local Problems for Elliptic Equation”, Dokl. Akad. Nauk, 351:1 (1996), 7–8  mathnet  mathscinet  zmath  isi
    3. A. K. Gushchin, “Some properties of the solutions of the Dirichlet problem for a second-order elliptic equation”, Sb. Math., 189:7 (1998), 1009–1045  mathnet  crossref  crossref  mathscinet  zmath  isi
    4. E. I. Galakhov, A. L. Skubachevskii, “On non-negative contractive semigroups with non-local conditions”, Sb. Math., 189:1 (1998), 43–74  mathnet  crossref  crossref  mathscinet  zmath  isi
    5. Gushchin A., “Inner Estimates of the Weak Solution of the Dirichlet Problem for a Second-Order Elliptic Equation”, Dokl. Akad. Nauk, 358:6 (1998), 731–733  mathnet  mathscinet  zmath  isi
    6. I. M. Petrushko, “Existence of boundary values for solutions of degenerate elliptic equations”, Sb. Math., 190:7 (1999), 973–1004  mathnet  crossref  crossref  mathscinet  zmath  isi
    7. O. A. Kovaleva, “On a property of the index of non-local elliptic problems”, Russian Math. Surveys, 54:4 (1999), 843–844  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    8. Kovaleva O., Skubachevskii A., “Nonlocal Elliptic Problems in Sobolev Spaces W-P(1)(Q)”, Dokl. Akad. Nauk, 366:4 (1999), 449–451  mathnet  mathscinet  zmath  isi
    9. O. A. Kovaleva, A. L. Skubachevskii, “Solvability of nonlocal elliptic problems in weight spaces”, Math. Notes, 67:6 (2000), 743–757  mathnet  crossref  crossref  mathscinet  zmath  isi
    10. Gushchin, AK, “A condition for complete continuity of the operators arising in nonlocal problems for elliptic equations”, Doklady Mathematics, 62:1 (2000), 32  mathscinet  zmath  isi  elib
    11. L. S. Pulkina, “A Nonlocal Problem with Integral Conditions for the Quasilinear Hyperbolic Equation”, Math. Notes, 70:1 (2001), 79–85  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    12. L. S. Pulkina, “A Nonlocal Problem for a Loaded Hyperbolic Equation”, Proc. Steklov Inst. Math., 236 (2002), 285–290  mathnet  mathscinet  zmath
    13. A. K. Gushchin, “A condition for the compactness of operators in a certain class and its application to the analysis of the solubility of non-local problems for elliptic equations”, Sb. Math., 193:5 (2002), 649–668  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    14. M. A. Skryabin, “Nonlocal Elliptic Problems in Dihedral Angles and Functional Differential Equations”, Journal of Mathematical Sciences, 129:5 (2005), 4227–4249  mathnet  crossref  mathscinet  zmath  elib
    15. P. L. Gurevich, “Non-local elliptic problems with non-linear argument transformations near the points of conjugation”, Izv. Math., 67:6 (2003), 1149–1186  mathnet  crossref  crossref  mathscinet  zmath  isi
    16. Gurevich P., “Solvability of Nonlocal Elliptic Problems in Sobolev Spaces”, Russ. J. Math. Phys., 10:4 (2003), 436–466  mathscinet  zmath  isi
    17. Gushchin, AK, “Carleson-type estimates for solutions to second-order elliptic equations”, Doklady Mathematics, 69:3 (2004), 329  mathscinet  zmath  isi  elib
    18. Gurevich P., “Smoothness of Generalized Solutions to Nonlocal Elliptic Problems on the Plane”, Dokl. Math., 70:2 (2004), 714–718  mathscinet  isi
    19. Gurevich P., “Solvability of Nonlocal Elliptic Problems in Sobolev Spaces, II”, Russ. J. Math. Phys., 11:1 (2004), 1–44  mathscinet  zmath  isi
    20. Dehghan M., “The Solution of a Nonclassic Problem for One-Dimensional Hyperbolic Equation Using the Decomposition Procedure”, Int. J. Comput. Math., 81:8 (2004), 979–989  crossref  mathscinet  zmath  isi
    21. A. K. Gushchin, “On the interior smoothness of solutions to second-order elliptic equations”, Siberian Math. J., 46:5 (2005), 826–840  mathnet  crossref  mathscinet  zmath  isi  elib
    22. P. L. Gurevich, “Generalized Solutions of Nonlocal Elliptic Problems”, Math. Notes, 77:5 (2005), 614–629  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    23. Gushchin, AK, “On the interior smoothness of solutions to second-order elliptic equations”, Doklady Mathematics, 72:2 (2005), 665  mathscinet  zmath  isi  elib
    24. Dehghan M., “On the Solution of an Initial-Boundary Value Problem That Combines Neumann and Integral Condition for the Wave Equation”, Numer. Meth. Part Differ. Equ., 21:1 (2005), 24–40  crossref  mathscinet  zmath  isi
    25. P. L. Gurevich, “On the Stability of the Index of Unbounded Nonlocal Operators in Sobolev Spaces”, Proc. Steklov Inst. Math., 255 (2006), 108–126  mathnet  crossref  mathscinet  elib
    26. Berikelashvili G., “To a Nonlocal Generalization of the Dirichlet Problem”, J. Inequal. Appl., 2006, 93858  crossref  mathscinet  zmath  isi
    27. Gushchin, AK, “Smoothness of solutions to the Dirichlet problem for a second-order elliptic equation with a square integrable boundary function”, Doklady Mathematics, 76:1 (2007), 486  crossref  mathscinet  zmath  isi  elib
    28. Denche M., Berkane A., “Boundary Value Problem for Abstract First Order Differential Equation with Integral Condition”, J. Math. Anal. Appl., 333:2 (2007), 657–666  crossref  mathscinet  zmath  isi
    29. A. L. Skubachevskii, “On Necessary Conditions for the Fredholm Solvability of Nonlocal Elliptic Problems”, Proc. Steklov Inst. Math., 260 (2008), 238–253  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    30. A. K. Gushchin, “A strengthening of the interior Hölder continuity property for solutions of the Dirichlet problem for a second-order elliptic equation”, Theoret. and Math. Phys., 157:3 (2008), 1655–1670  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    31. Gurevich P., “Smoothness of Generalized Solutions for Higher-Order Elliptic Equations with Nonlocal Boundary Conditions”, J. Differ. Equ., 245:5 (2008), 1323–1355  crossref  mathscinet  zmath  isi  elib
    32. Mesloub S., “On a Singular Two Dimensional Nonlinear Evolution Equation with Nonlocal Conditions”, Nonlinear Anal.-Theory Methods Appl., 68:9 (2008), 2594–2607  crossref  mathscinet  zmath  isi
    33. A. L. Skubachevskii, “Nonclassical boundary-value problems. II”, Journal of Mathematical Sciences, 166:4 (2010), 377–561  mathnet  crossref  mathscinet  elib
    34. Stikonas A., Stikoniene O., “Characteristic Functions for Sturm-Liouville Problems with Nonlocal Boundary Conditions”, Math. Model. Anal., 14:2 (2009), 229–246  crossref  mathscinet  zmath  isi  elib
    35. N. V. Beilina, “O suschestvovanii resheniya odnoi nelokalnoi zadachi dlya uravneniya Laplasa”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 1(20) (2010), 205–208  mathnet  crossref
    36. P. L. Gurevich, “Elliptic problems with nonlocal boundary conditions and Feller semigroups”, Journal of Mathematical Sciences, 182:3 (2012), 255–440  mathnet  crossref  mathscinet  zmath
    37. Berikelashvili G.K., Gordeziani D.G., “On a Nonlocal Generalization of the Biharmonic Dirichlet Problem”, Differ. Equ., 46:3 (2010), 321–328  crossref  mathscinet  zmath  isi
    38. A. K. Guschin, “Otsenki resheniya zadachi Dirikhle s granichnoi funktsiei iz $L_p$”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 1(22) (2011), 53–67  mathnet  crossref  elib
    39. Guschin A.K., “O razreshimosti zadachi dirikhle s granichnoi funktsiei iz l_{p} dlya ellipticheskogo uravneniya vtorogo poryadka”, Doklady Akademii nauk, 437:5 (2011), 583–586  elib
    40. Gushchin A.K., “Solvability of the Dirichlet Problem for a Second-Order Elliptic Equation with a Boundary Function From l-P”, Dokl. Math., 83:2 (2011), 219–221  crossref  mathscinet  zmath  isi  elib
    41. A. K. Gushchin, “The Dirichlet problem for a second-order elliptic equation with an $L_p$ boundary function”, Sb. Math., 203:1 (2012), 1–27  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    42. Gushchin A.K., “Estimates of the Nontangential Maximal Function for Solutions of a Second-Order Elliptic Equation”, Dokl. Math., 86:2 (2012), 667–669  mathnet  crossref  mathscinet  zmath  isi  elib
    43. A. K. Gushchin, “$L_p$-estimates for solutions of second-order elliptic equation Dirichlet problem”, Theoret. and Math. Phys., 174:2 (2013), 209–219  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    44. A. K. Guschin, “$L_p$-otsenki nekasatelnoi maksimalnoi funktsii dlya reshenii ellipticheskogo uravneniya vtorogo poryadka”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 1(30) (2013), 53–69  mathnet  crossref
    45. S. V. Kirichenko, “Ob odnoi kraevoi zadache dlya uravneniya smeshannogo tipa s nelokalnymi nachalnymi usloviyami v pryamougolnike”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 3(32) (2013), 185–189  mathnet  crossref  zmath
    46. A. K. Gushchin, “$L_p$-estimates for the nontangential maximal function of the solution to a second-order elliptic equation”, Sb. Math., 207:10 (2016), 1384–1409  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    47. A. I. Kozhanov, G. A. Lukina, “Nelokalnye zadachi s integralnym usloviem dlya differentsialnykh uravnenii nechetnogo poryadka”, Sib. elektron. matem. izv., 13 (2016), 452–466  mathnet  crossref  mathscinet
    48. A. K. Gushchin, “The Luzin area integral and the nontangential maximal function for solutions to a second-order elliptic equation”, Sb. Math., 209:6 (2018), 823–839  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    49. A. K. Gushchin, “A criterion for the existence of $L_p$ boundary values of solutions to an elliptic equation”, Proc. Steklov Inst. Math., 301 (2018), 44–64  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    50. S. V. Kirichenko, “Ob odnoi zadache s nelokalnym po peremennoi vremeni usloviem dlya giperbolicheskogo uravneniya”, Vestn. SamU. Estestvennonauchn. ser., 24:4 (2018), 24–28  mathnet  crossref  elib
    51. Kozhanov A.I., “Nonlocal Problems With Integral Conditions For Elliptic Equations”, Complex Var. Elliptic Equ., 64:5 (2019), 741–752  crossref  mathscinet  zmath  isi  scopus
    52. A. K. Gushchin, “The boundary values of solutions of an elliptic equation”, Sb. Math., 210:12 (2019), 1724–1752  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
    53. A. K. Gushchin, “On the Existence of $L_2$ Boundary Values of Solutions to an Elliptic Equation”, Proc. Steklov Inst. Math., 306 (2019), 47–65  mathnet  crossref  crossref  mathscinet  isi  elib
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