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2019 | Vol. 39, no. 1 | 109--124
Tytuł artykułu

Pseudo-differential equations and conical potentials: 2-dimensional case

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We consider two-dimensional elliptic pseudo-differential equation in a plane sector. Using a special representation for an elliptic symbol and the formula for a general solution we study the Dirichlet problem for such equation. This problem was reduced to a system of linear integral equations and then after some transformations to a system of linear algebraic equations. The unique solvability for the Dirichlet problem was proved in Sobolev-Slobodetskii spaces and a priori estimate for the solution is given.
Wydawca

Rocznik
Strony
109--124
Opis fizyczny
Bibliogr. 22 poz.
Twórcy
  • Chair of Differential Equations Belgorod National Research State University Studencheskaya 14/1 Belgorod 308007, Russia, vladimir.b.vasilyev@gmail.com
Bibliografia
  • [1] B.V. Bazaliy, N. Vasylyeva, On the solvability of a transmission problem for the Laplace operator with a dynamic boundary condition on a nonregular interface, J. Math. Anal. Appl. 393 (2012), 651-670.
  • [2] Ju.V. Egorov, B.-W. Schulze, Pseudo-Differential Operators, Singularities, Applications, Birkhauser-Verlag, Basel, 1997.
  • [3] G.I. Eskin, The conjugacy problem for equations of principal type with two independent variables, Trans. Moscow Math. Soc. 21 (1970), 263-316.
  • [4] G.I. Eskin, Boundary Value Problems for Elliptic Pseudodifferential Equations, AMS, Providence, 1981.
  • [5] F.D. Gakhov, Boundary Value Problems, Dover Publications, Mineola, 1981.
  • [6] N.I. Muskhelishvili, Singular Integral Equations, North Holland, Amsterdam, 1976.
  • [7] V.E. Nazaikinskii, A.Yu. Savin, B.-W. Schulze, B.Yu. Sternin, Elliptic Theory on Singular Manifolds, Chapman & Hall/CRC, Boca Raton, 2006.
  • [8] S.A. Nazarov, B.A. Plamenevsky, Elliptic Problems in Domains with Pie.ce.wise Smooth Boundaries, Walter de Gruyter, Berlin-New York, 1994.
  • [9] B. Plamenevskii, Algebras of Pseudodifferential Operators, Springer, Netherlands, 1989.
  • [10] B.-W. Schulze, Boundary Value Problems and Singular Pseudo-Differential Operators, J. Wiley, Chichester, 1998.
  • [11] B.-W. Schulze, B. Sternin, V. Shatalov, Differential Equations on Singular Manifolds; Semiclassical Theory and Operator Algebras, Wiley-VCH, Berlin, 1998.
  • [12] V.A. Solonnikov, E.V. Frolova, Investigation of a problem for the Laplace equation with a boundary condition of a special form, in a plane angle, J. Soviet Math. 62 (1992), 2819-2831.
  • [13] E. Titchmarsh, Introduction to the Theory of Fourier Integrals, Oxford Univ. Press, Oxford, 1948.
  • [14] V.B. Vasil'ev, Boundary value problems and integral equations for some classes of pseudodifferential equations in nonsmooth domains, Differ. Equ. 34 (1998), 1188-1199.
  • [15] V.B. Vasil'ev, Wave Factorization of Elliptic Symbols: Theory and Applications, Kluwer Academic Publishers, Dordrecht-Boston-London, 2000.
  • [16] V.B. Vasilyev, Elliptic equations and boundary value problems in non-smooth domains, [in:] L. Rodino et al. (eds.), Pseudo Differential Operators: Analysis, Applications and Computations, Birkhauser, Basel, 2011, pp. 105-121.
  • [17] V.B. Vasilyev, General boundary value problems for pseudo-differential equations and related difference equations, Adv. Difference Equ. 289 (2013), 1-7.
  • [18] V.B. Vasilyev, Pseudo differential equations on manifolds with non-smooth boundaries, [in:] S. Pinelas et al. (eds.), Differential and Difference Equations and Applications, Springer Nature, Switzerland, 2013, pp. 625-637.
  • [19] V.B. Vasilyev, On the Dirichlet and Neumann problems in multi-dimensional cone, Math. Bohem. 139 (2014), 333-340.
  • [20] V.B. Vasilyev, New constructions in the theory of elliptic boundary value problems, [in:] C. Constanda et al. (eds.), Integral Methods in Science and Engineering, Birkhauser, Cham, 2015, pp. 629-641.
  • [21] V.B. Vasilyev, Potentials for elliptic boundary value problems in cones, Sib. Elektron. Mat. Izv. 13 (2016), 1129-1149 [in Russian].
  • [22] V.B. Vasilyev, Elliptic equations, manifolds with non-smooth boundaries, and boundary value problems, [in:] P. Dang et al. (eds.), New Trends in Analysis and Interdisciplinary Applications, Birkhauser, Cham, 2017, pp. 337-344.
Typ dokumentu
Bibliografia
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Identyfikator YADDA
bwmeta1.element.baztech-224a47d9-56b9-45f7-94a6-f6ffa85c85fe
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