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Izv. Akad. Nauk SSSR Ser. Mat., 1989, Volume 53, Issue 1, Pages 3–24 (Mi izv1159)  

This article is cited in 10 scientific papers (total in 10 papers)

Boundary value problems with strong nonlocalness for elliptic equations

A. B. Antonevich


Abstract: Nonlocal boundary value problems are considered for elliptic equations of the following form. A nonperiodic mapping $g$ of the boundary to itself is given, and the boundary condition connects the values of the unknown function and its derivatives at the points $x,g(x),g(g(x)),…$. The author obtains necessary and sufficient conditions for the problem to be Noetherian (i.e. for the operator to be Fredholm) in terms of the invertibility of an auxiliary functional operator (the symbol of the problem), acting in a function space on the bundle of unit cotangent vectors to the boundary. Explicit necessary and sufficient conditions for the Noether property are presented for a number of examples. The main constructions and proofs are based on the theory of $C^*$-algebras generated by dynamical systems.
Bibliography: 37 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1990, 34:1, 1–21

Bibliographic databases:

UDC: 517.9
MSC: Primary 35J40; Secondary 46L05
Received: 18.03.1987

Citation: A. B. Antonevich, “Boundary value problems with strong nonlocalness for elliptic equations”, Izv. Akad. Nauk SSSR Ser. Mat., 53:1 (1989), 3–24; Math. USSR-Izv., 34:1 (1990), 1–21

Citation in format AMSBIB
\Bibitem{Ant89}
\by A.~B.~Antonevich
\paper Boundary value problems with strong nonlocalness for elliptic equations
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1989
\vol 53
\issue 1
\pages 3--24
\mathnet{http://mi.mathnet.ru/izv1159}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=992976}
\zmath{https://zbmath.org/?q=an:0798.35045}
\transl
\jour Math. USSR-Izv.
\yr 1990
\vol 34
\issue 1
\pages 1--21
\crossref{https://doi.org/10.1070/IM1990v034n01ABEH000575}


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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. Yu. D. Latushkin, A. M. Stepin, “Weighted translation operators and linear extensions of dynamical systems”, Russian Math. Surveys, 46:2 (1991), 95–165  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    2. A. V. Aminova, “Pseudo–Riemannian manifolds with common geodesies”, Russian Math. Surveys, 48:2 (1993), 105–160  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    3. A. K. Ratyni, “An elliptic boundary value problem with a superposition operator in the boundary condition. I”, Russian Math. (Iz. VUZ), 44:4 (2000), 34–38  mathnet  mathscinet  zmath  elib
    4. P. A. Zalesskii, “On virtually projective groups”, crll, 2004:572 (2004), 97  crossref  mathscinet  zmath
    5. Amalendu Krishna, Marc Levine, “Additive higher Chow groups of schemes”, crll, 2008:619 (2008), 75  crossref  mathscinet  zmath  isi
    6. Ratyni A.K., “On the solvability of the first nonlocal boundary value problem for an elliptic equation”, Differ. Equ., 45:6 (2009), 862–872  crossref  mathscinet  zmath  isi  elib  elib
    7. A. Yu. Savin, B. Yu. Sternin, “Noncommutative elliptic theory. Examples”, Proc. Steklov Inst. Math., 271 (2010), 193–211  mathnet  crossref  mathscinet  isi  elib
    8. Savin A.Yu., “O simvole nelokalnykh operatorov v prostranstvakh soboleva”, Differentsialnye uravneniya, 47:6 (2011), 890–893  elib
    9. L. E. Rossovskii, “Elliptic functional differential equations with contractions and extensions of independent variables of the unknown function”, Journal of Mathematical Sciences, 223:4 (2017), 351–493  mathnet  crossref
    10. N. R. Izvarina, “On the symbol of nonlocal operators associated with a parabolic diffeomorphism”, Eurasian Math. J., 9:2 (2018), 34–43  mathnet
  • Известия Академии наук СССР. Серия математическая Izvestiya: Mathematics
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