Geodesic
Boundary Value Problems with Non-Local Conditions
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 1 (2008) no. 2, pp. 158-187.

Voir la notice de l'article provenant de la source Math-Net.Ru

We describe a new algebra of boundary value problems which contains Lopatinskii elliptic as well as Toeplitz type conditions. These latter are necessary, if an analogue of the Atiyah–Bott obstruction does not vanish. Every elliptic operator is proved to admit up to a stabilisation elliptic conditions of such a kind. Corresponding boundary value problems are then Fredholm in adequate scales of spaces. The crucial novelty consists of the new type of weighted Sobolev spaces which fit well to the nature of pseudodifferential operators.
Keywords: pseudodifferential operators, boundary values problems, Toeplitz operators. 
@article{JSFU_2008_1_2_a5,
     author = {Nikolai N. Tarkhanov},
     title = {Boundary {Value} {Problems} with {Non-Local} {Conditions}},
     journal = {\v{Z}urnal Sibirskogo federalʹnogo universiteta. Matematika i fizika},
     pages = {158--187},
     publisher = {mathdoc},
     volume = {1},
     number = {2},
     year = {2008},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JSFU_2008_1_2_a5/}
}
TY  - JOUR
AU  - Nikolai N. Tarkhanov
TI  - Boundary Value Problems with Non-Local Conditions
JO  - Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika
PY  - 2008
SP  - 158
EP  - 187
VL  - 1
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/JSFU_2008_1_2_a5/
LA  - en
ID  - JSFU_2008_1_2_a5
ER  - 
%0 Journal Article
%A Nikolai N. Tarkhanov
%T Boundary Value Problems with Non-Local Conditions
%J Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika
%D 2008
%P 158-187
%V 1
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/JSFU_2008_1_2_a5/
%G en
%F JSFU_2008_1_2_a5
Nikolai N. Tarkhanov. Boundary Value Problems with Non-Local Conditions. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 1 (2008) no. 2, pp. 158-187. http://geodesic.mathdoc.fr/item/JSFU_2008_1_2_a5/

[1] M. F. Atiyah, R. Bott, “The index problem for manifolds with boundary”, Differential Analysis, (papers presented at the Bombay Colloquium 1964), Oxford University Press, 1964, 175–186 | MR

[2] M. Z. Solomyak, “On linear elliptic systems of first order”, Dokl. Akad. Nauk SSSR, 150:1 (1963), 48–51 ; 79 (1976), 315–330 (Russian) | MR | Zbl

[3] A. P. Calderón, “Boundary value problems for elliptic equations”, Outlines of the Joint Soviet-American Symp. on Part. Diff. Eq., Nauka, Novosibirsk, 1963, 303–304 | MR

[4] R. Seeley, “Topics in pseudo-differential operators”, Pseudo-Differential Operators, CIME, Cremonese, Roma, 1969, 167–306 | MR

[5] M. F. Atiyah, V. K. Patodi, I. M. Singer, “Spectral asymmetry and Riemannian geometry. I”, Math. Proc. Camb. Phil. Soc., 77 (1975), 43–69 ; “II”, 78 (1975), 405–432 ; “III”, 79 (1976), 71–99 | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl

[6] R. B. Melrose, The Atiyah–Patodi–Singer Index Theorem, A. K. Peters, Wellesley, MA, 1993 | MR | Zbl

[7] B. V. Fedosov, B. W. Schulze, N. N. Tarkhanov, “The index of elliptic operators on manifolds with conical points”, Sel. Math., New ser., 5 (1999), 1–40 | DOI | MR

[8] B. W. Schulze, “An algebra of boundary value problems not requiring Shapiro–Lopatinskij conditions”, J. Funct. Anal., 179 (2001), 374–408 | DOI | MR | Zbl

[9] L. Boutet de Monvel, “Boundary problems for pseudo-differential operators”, Acta Math., 126:1–2 (1971), 11–51 | MR | Zbl

[10] B. W. Schulze, Boundary Value Problems and Singular Pseudo-Differential Operators, J. Wiley, Chichester, 1998 | MR

[11] B. W. Schulze, J. Seiler, “Boundary value problems with global projection conditions”, J. Funct. Anal., 206 (2004), 449–498 | DOI | MR | Zbl

[12] B. W. Schulze, Pseudo-Differential Boundary Value Problems, Conical Singularities, and Asymptotics, Akademie–Verlag, Berlin, 1994 | MR

[13] B. W. Schulze, N. Tarkhanov, “Elliptic complexes of pseudodifferential operators on manifolds with edges”, Evolution equations, Feshbach resonances, singular Hodge theory, Math. Top., 16, Wiley-VCH, Berlin et al., 1999, 287–431 | MR | Zbl

[14] G. I. Eskin, Boundary Value Problems for Elliptic Pseudodifferential Operators, Math. Monogr., 52, Amer. Math. Soc., Providence, RI, 1980

[15] St. Rempel, B. W. Schulze, “Parametrices and boundary symbolic calculus for elliptic boundary problems without the transmission property”, Math. Nachr., 105 (1982), 45–149 | DOI | MR | Zbl