Abstract
In the paper we investigate asymptotics of solutions for nonlocal elliptic problems in plane angles and in ℝ2 \ {0}. These problems arise in studying nonlocal problems in bounded domains in the case where support of nonlocal terms intersects with a boundary of a domain. We obtain explicit formulas for the asymptotic coefficients in terms of eigenvectors and associated vectors of both adjoint nonlocal operators acting in spaces of distributions and formally adjoint (with respect to the Green formula) nonlocal transmission problems.
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REFERENCES
T. Carleman, “Sur la théorie des equations integrales et ses applications,” Verhandlungen des Internat. Math. Kongr. Zürich, 1, 132–151 (1932).
A. V. Bitsadze and A. A. Samarskii, “On some simple generalizations of linear elliptic boundary value problems,” Dokl. Akad. Nauk SSSR, 185, 739–740 (1969); English transl. in Sov. Math. Dokl., 10 (1969).
A. L. Skubachevskii, Elliptic Functional Differential Equations and Applications, Birkhäuser, Basel-Boston-Berlin (1997).
A. V. Bitsadze, “On some class of conditionally solvable nonlocal boundary value problems for harmonic functions,” Dokl. Akad. Nauk SSSR, 280, 521–524 (1985); English transl. in Sov. Math. Dokl., 31 (1985).
A. L. Skubachevskii, “Elliptic problems with nonlocal conditions near the boundary,” Mat. Sb., 129(171), 279–302 (1986); English transl. in Math. USSR Sb., 57 (1987).
A. L. Skubachevskii, “Model nonlocal problems for elliptic equations in dihedral angles,” Differents. Uravn., 26, 119–131 (1990); English transl. in Differential Equations, 26 (1990).
A. L. Skubachevskii, “Truncation-function method in the theory of nonlocal problems,” Differents. Uravn., 27, 128–139 (1991); English transl. in Differential Equations, 27 (1991).
A. K. Gushchin and V. P. Mikhailov, “On solvability of nonlocal problems for elliptic equations of second order,” Mat. Sb., 185, 121–160 (1994); English transl. in Math. Sb. (1994).
V. A. Kondrat'ev, “Boundary value problems for elliptic equations in domains with conical or angular points,” Trudy Mos. Mat. Ob-va, 16, 209–292 (1967); English transl. in Trans. Moscow Math. Soc., 16 (1967).
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. in Russian Math. Surveys 38 (1964).
A. L. Skubachevskii, “Regularity of solutions for some nonlocal elliptic problem,” Russian J. Math. Phys., 8, 365–374 (2001).
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).
P. L. Gurevich, “Nonlocal elliptic problems in dihedral angles and the Green formula,” Dokl. Ros. Akad. Nauk, 379, 735–738 (2001); English transl. in Russian Acad. Sci. Dokl. Math. (2001).
S. A. Nazarov and B. A. Plamenevskii, Elliptic Problems in Domains with Piecewise Smooth Boundary [in Russian], Nauka, Moscow (1991).
V. G. Maz'ya and B. A. Plamenevskii, “On coefficients in the asymptotics of solutions for elliptic boundary value problems in a cone,” Zapiski Nauchn. Seminara Leningr. Otdel. Mat. Inst. Akad. Nauk SSSR, 58, 110–128 (1975).
J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. 1, Springer, New York-Heidelberg-Berlin (1972).
S. G. Krein and V. P. Trofimov, “On holomorphic operator-functions of several variables,” Func. Anal. Appl., 3, 85–86 (1969).
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Gurevich, P.L. Asymptotics of Solutions for Nonlocal Elliptic Problems in Plane Angles. Journal of Mathematical Sciences 120, 1295–1312 (2004). https://doi.org/10.1023/B:JOTH.0000016050.97763.74
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DOI: https://doi.org/10.1023/B:JOTH.0000016050.97763.74