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

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

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
2. A. V. Aminova, “Pseudo–Riemannian manifolds with common geodesies”, Russian Math. Surveys, 48:2 (1993), 105–160
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
4. P. A. Zalesskii, “On virtually projective groups”, crll, 2004:572 (2004), 97
5. Amalendu Krishna, Marc Levine, “Additive higher Chow groups of schemes”, crll, 2008:619 (2008), 75
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
7. A. Yu. Savin, B. Yu. Sternin, “Noncommutative elliptic theory. Examples”, Proc. Steklov Inst. Math., 271 (2010), 193–211
8. Savin A.Yu., “O simvole nelokalnykh operatorov v prostranstvakh soboleva”, Differentsialnye uravneniya, 47:6 (2011), 890–893
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
10. N. R. Izvarina, “On the symbol of nonlocal operators associated with a parabolic diffeomorphism”, Eurasian Math. J., 9:2 (2018), 34–43
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