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Pseudodifferential equations in cones with conjugation points on the boundary. (English. Russian original) Zbl 1334.35459

Differ. Equ. 51, No. 9, 1113-1125 (2015); translation from Differ. Uravn. 51, No. 9, 1123-1135 (2015).
The paper deals with the solvability of pseudo-differential equations in Sobolev spaces over certain models for non-Lipschitz singular spaces. The models are non-convex cones \(C_+\subset\mathbb{R}^m\) obtained as finite unions of disjoint convex cones \(C_j\) with common apex. A Wiener-Hopf-type factorization method is presented and the solution decompositions for \(C_+\) are assembled from associated solutions for the \(C_j\)’s.

MSC:

35S15 Boundary value problems for PDEs with pseudodifferential operators
35S05 Pseudodifferential operators as generalizations of partial differential operators
58J40 Pseudodifferential and Fourier integral operators on manifolds
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[1] Eskin, G.I., Kraevye zadachi dlya ellipticheskikh psevdodifferentsial’nykh uravnenii (Boundary Value Problems for Elliptic Pseudodifferential Equations), Moscow: Nauka, 1973.
[2] Vasilyev, V.B., Elliptic Equations and Boundary Value Problems in Non-Smooth Domains, in Pseudo Differential Operators: Analysis, Applications and Computations, Rodino, L., Wong, M.W., and Zhu, H., Eds., Operator Theory: Advances and Applications, Basel, 2011, vol. 213, pp. 105-121. · Zbl 1247.35215
[3] Vasilyev, V.B., Pseudo Differential Equations on Manifolds with Non-Smooth Boundaries, Differential Difference Equ. Appl., 2013, vol. 47, pp. 625-637. · Zbl 1319.35320 · doi:10.1007/978-1-4614-7333-6_58
[4] Vasil’ev, V. B., No article title, Matem. Forum (Itogi Nauki. Yug Rossii), 8, 87-92 (2014)
[5] Vasilyev, V.B., On the Dirichlet and Neumann Problems in Multi-Dimensional Cone, Math. Bohem., 2014, vol. 139, no. 2, pp. 333-340. · Zbl 1340.35029
[6] Gakhov, F.D., Kraevye zadachi (Boundary Value Problems), Moscow: Nauka, 1977. · Zbl 0449.30030
[7] Muskhelishvili, N.I., Singulyarnye integral’nye uravneniya (Singular Integral Equations), Moscow: Nauka, 1968. · Zbl 0174.16202
[8] Karapetyants, N. K.; Samko, S. G., Uravneniya s involyutivnymi operatorami i ikh prilozheniya (Equations with Involutive Operators and Their Applications) (1988) · Zbl 0782.47011
[9] Kakichev, V.A., Boundary Value Problems of Linear Conjugacy for Functions Holomorphic in Bicylindrical Regions, Teor. Funk. Funk. Anal. Prilozen., 1967, vol. 5, pp. 37-58. · Zbl 0179.11901
[10] Vladimirov, V.S., The Linear Conjugation Problem for Holomorphic Functions of Several Variables, Izv. Akad. Nauk SSSR Ser. Mat., 1965, vol. 29, no. 4, pp. 807-834. · Zbl 0166.33704
[11] Vasil’ev, V.B., Regularization of Multidimensional Singular Integral Equations in Nonsmooth Domains, Tr. Mosk. Mat. Obs., 1998, vol. 59, pp. 73-105. · Zbl 0912.45005
[12] Vasil’ev, V.B., Wave Factorization of Elliptic Symbols: Theory and Applications. Introduction to the Theory of Boundary Value Problems in Non-Smooth Domains, Dordrecht; Boston; London, 2000. · Zbl 0961.35193 · doi:10.1007/978-94-015-9448-6
[13] Vasil’ev, V.B., Wave Factorization of Elliptic Symbols, Mat. Zametki, 2000, vol. 68, no. 5, pp. 653-667. · Zbl 0992.35127 · doi:10.4213/mzm987
[14] Vasil’ev, V. B., Mul’tiprikatory integralov Fur’e (2010)
[15] Vladimirov, V.S., Metody teorii funktsii mnogikh kompleksnykh peremennykh (Methods of the Theory of Functions of Several Complex Variables), Moscow: Nauka, 1964.
[16] Vasilyev, V. B.; Constanda, C. (ed.); Harris, P. J. (ed.), Asymptotical Analysis of Singularities for Pseudo Differential Equations in Canonical Non-Smooth Domains (2011)
[17] Vishik, M.I. and Eskin, G.M., Convolution Equations in a Bounded Region, Uspekhi Mat. Nauk, 1965, vol. 20, no. 3, pp. 89-152. · Zbl 0152.34202
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