Moscow Mathematical Society

On the Fredholm and unique solvability of nonlocal elliptic problems in multidimensional domains

Author(s): P. L. Gurevich; A. L. Skubachevskii
Translated by: the authors
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 68 (2007).
Journal: Trans. Moscow Math. Soc. 2007, 261-336.
MSC (2000): Primary 35J40
Posted: November 21, 2007
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1.: S. Agmon, A. Douglis, L. Nirenberg, ``Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I,'' Comm. Pure Appl. Math., 12, 623-727 (1959). MR 0125307 (23:A2610)
2.: M. S. Agranovich and M. I. Vishik, ``Elliptic problems with a parameter and parabolic problems of general type,'' Uspekhi Mat. Nauk, 19, 53-161 (1964); English transl.: Russian Math. Surveys, 19 (1964). MR 0192188 (33:415)
3.: A. V. Bitsadze, A. A. Samarski ${\u{\i\/}}\kern.15em$ , ``On some simple generalizations of linear elliptic boundary value problems,'' Dokl. Akad. Nauk SSSR., 185, 739-740 (1969); English transl.: Soviet Math. Dokl., 10 (1969). MR 0247271 (40:540)
4.: I. C. Gohberg and E. I. Sigal, ``An operator generalization of the logarithmic residue theorem and the theorem of Rouché,'' Mat. Sb., 84 (126), 607-629 (1971); English transl.: Math. USSR Sb., 13, 603-625 (1971). MR 0313856 (47:2409)
5.: P. L. Gurevich, ``Nonlocal problems for elliptic equations in dihedral angles and the Green formula,'' Mitteilungen aus dem Mathem. Seminar Giessen, Math. Inst. Univ. Giessen, Germany, 247, 1-74 (2001). MR 1867955 (2003a:35047)
6.: P. L. Gurevich, ``Asymptotics of solutions for nonlocal elliptic problems in plane angles,'' Trudy Seminara imeni I. G. Petrovskogo, 23, 93-126 (2003); English transl.: J. Math. Sci., 120, No. 3, 1295-1312 (2004). MR 2085182 (2005e:35049)
7.: P. L. Gurevich, ``Non-local elliptic problems with non-linear argument transformations near the points of conjugation,'' Izv. Ross. Akad. Nauk. Ser. Mat., 67, No. 6, 71-110 (2003); English transl.: Izv. Math., 67, No. 6, 1149-1186 (2003). MR 2032091 (2005j:35037)
8.: P. L. Gurevich, ``Solvability of nonlocal elliptic problems in Sobolev spaces, I'' Russ. J. Math. Phys., 10, No. 4, 436-466 (2003). MR 2096605 (2005g:35062)
9.: P. L. Gurevich, ``Solvability of nonlocal elliptic problems in Sobolev spaces, II'' Russ. J. Math. Phys., 11, No. 1, 1-44 (2004). MR 2134647 (2005k:35086)
10.: P. L. Gurevich, ``Smoothness of Generalized Solutions to Nonlocal Elliptic Problems on the Plane'' Dokl. Akad. Nauk, 398, No. 3, 295-299 (2004); English transl.: Russian Acad. Sci. Dokl. Math., 70, 714-718 (2004). MR 2116507 (2005k:35080)
11.: 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,'' Mat. Sb., 193, No. 5, 17-36 (2002); English transl.: Sb. Math., 193, No. 5, 649-668 (2002). MR 1918245 (2004c:35147)
12.: A. K. Gushchin, V. P. Mikhailov, ``On solvability of nonlocal problems for elliptic equations of second order,'' Mat. Sb., 185, 121-160 (1994); English transl.: Russian Acad. Sci. Sb. Math., 81, 101-136 (1995). MR 1264079 (94m:35088)
13.: T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin-Heidelberg-New York, 1966. MR 0203473 (34:3324)
14.: K. Yu. Kishkis, ``The index of a Bitsadze-Samarskii Problem for harmonic functions,'' Differentsial'nye Uravneniya., 24, No. 1, 105-110 (1988); English transl.: Diff. Equ., 24, 83-87 (1988). MR 930160 (89h:35092)
15.: V. A. Kondrat'ev, ``Boundary value problems for elliptic equations in domains with conical or angular points,'' Trudy Moskov. Mat. Obshch., 16, 209-292 (1967); English transl.: Trans. Moscow Math. Soc., 16 (1967). MR 0226187 (37:1777)
16.: V. A. Kondrat'ev and O. A. Oleinik, ``Boundary value problems for partial differential equations in non-smooth domains,'' Uspekhi Mat. Nauk., 38, 3-75 (1983); English transl.: Russian Math. Surveys, 38, 1-86 (1983). MR 695471 (85j:35002)
17.: O. A. Kovaleva and A. L. Skubachevskii, ``Solvability of nonlocal elliptic problems in weighted spaces,'' Mat. Zametki., 67, 882-898 (2000); English transl.: Math. Notes., 67 (2000). MR 1820643 (2002j:35089)
18.: S. G. Krein, Linear Equations in a Banach Space, Nauka, Moscow, 1971; English transl.: Birkhäuser, Boston, 1982. MR 0374949 (51:11145)
19.: J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. 1, Springer-Verlag, New York-Heidelberg-Berlin, 1972. MR 0350177 (50:2670)
20.: V. G. Maz'ya and B. A. Plamenevskii, ``-estimates of solutions of elliptic boundary value problems in domains with edges,'' Trudy Moskov. Mat. Obshch., 37, 49-93 (1978); English transl.: Trans. Moscow Math. Soc., 37 (1980). MR 514327 (81b:35027)
21.: C. Miranda, Equazioni Alle Derivate Parziali di Tipo Ellittico, Springer-Verlag, Berlin-Göttingen-Hiedelberg, 1955. MR 0087853 (19:421d)
22.: S. A. Nazarov and B. A. Plamenevskii, Elliptic Problems in Domains with Piecewise Smooth Boundaries, Moscow: Nauka, 1991; English transl. in De Gruyter Expositions in Mathematics, 13. Walter de Gruyter Publishers, Berlin-New York, 1994. MR 1283387 (95h:35001)
23.: M. A. Skryabin, ``Nonlocal Elliptic Problems in Dihedral Angles and Functional Differential Equations,'' Sovremennaya Matematika. Fundamental'nye Napravleniya, 4, 121-143 (2003); English transl.: J. Math. Sci., 129, No. 5, 4227-4249 (2005). MR 2122347 (2005j:35075)
24.: A. L. Skubachevskii, ``Elliptic problems with nonlocal conditions near the boundary,'' Mat. Sb., 129 (171), 279-302 (1986); English transl.: Math. USSR Sb., 57, 279-302 (1987). MR 832122 (87h:35089)
25.: A. L. Skubachevskii, ``Model nonlocal problems for elliptic equations in dihedral angles,'' Differentsial'nye Uravneniya, 26, 119-131 (1990); English transl.: Differ. Equ., 26, 106-115 (1990). MR 1050367 (91h:35119)
26.: A. L. Skubachevskii, ``Truncation-function method in the theory of nonlocal problems,'' Differentsial'nye Uravneniya, 27, 128-139 (1991); English transl.: Diff. Equ., 27, 103-112 (1991). MR 1133518 (92k:35055)
27.: A. L. Skubachevskii, ``On the stability of index of nonlocal elliptic problems,'' J. of Mathematical Analysis and Applications, 160, 323-341 (1991). MR 1126121 (92m:58128)
28.: A. L. Skubachevskii, Elliptic Functional Differential Equations and Applications, Basel-Boston-Berlin, Birkhäuser, 1997. MR 1437607 (98c:35164)
29.: A. L. Skubachevskii, ``Regularity of solutions of a nonlocal elliptic problem,'' Russ. J. Math. Phys., 8, 365-374 (2001). MR 1930382 (2003g:35044)
30.: L. N. Slobodetskii, ``Generalized Sobolev spaces and their application to boundary problems for partial differential equations,'' Leningrad. Gos. Ped. Inst. Uchen.Zap., 197, 54-112 (1958); English transl.: Amer. Math. Soc. Transl. Ser. 2, 57, 207-275 (1966). MR 0203222 (34:3075)
31.: L. R. Volevich ``Solvability of boundary-value problems for general elliptic systems,'' Mat. Sb., 68, No. 3, 373-416 (1965). MR 0192191 (33:418)

Retrieve articles in Transactions of the Moscow Mathematical Society with MSC (2000): 35J40

P. L. Gurevich
Affiliation: Department of Differential Equations and Mathematical Physics, Peoples' Friendship University of Russia, Miklukho-Maklaya St. 6, 117198 Moscow, Russia
Address at time of publication: Interdisciplinary Center for Scientific Computing of the University of Heidelberg Im Neuenheimer Feld 368, D-69120 Heidelberg, Germany
Email: pavel.gurevich@iwr.uni-heidelberg.de

A. L. Skubachevskii
Affiliation: Department of Differential Equations and Mathematical Physics, Peoples' Friendship University of Russia, Miklukho-Maklaya St. 6, 117198 Moscow, Russia
Email: skub@lector.ru

DOI: 10.1090/S0077-1554-07-00164-1
PII: S 0077-1554(07)00164-1
Keywords: Elliptic problems, nonlocal boundary conditions, solvability, nonsmooth domains
Posted: November 21, 2007
Additional Notes: Supported by the Russian Foundation for Basic Research (project No. 04-01-00256) and the Russian President Grant (project No. MK-980.2005.1). The first author was also supported by the Alexander von Humboldt Foundation.
Copyright of article: Copyright 2007, American Mathematical Society

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ISSN: 1547-738X(e) ISSN: 0077-1554(p)

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