On Some Elliptic Boundary Value Problems in Conic Domains

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Abstract

A model elliptic pseudodifferential equation in a polyhedral cone is considered, and the situation when some of the parameters of the cone tend to their limiting values is investigated. In Sobolev–Slobodetskii spaces, a solution of the equation in the cone is constructed in the case of a special wave factorization of the elliptic symbol. It is shown that a limit solution of the boundary value problem with an additional integral condition can exist only under additional constraints on the boundary function.

About the authors

V. B. Vasilyev

Belgorod National Research University

Author for correspondence.
Email: vbv57@inbox.ru
308015, Belgorod, Russia

References

  1. Ильинский А.С., Смирнов Ю.Г. Дифракция электромагнитных волн на проводящих тонких экранах (Псевдодифференциальные операторы в задачах дифракции). М.: ИПРЖР, 1996.
  2. Speck F.-O. From Sommerfeld diffraction problems to operator factorisation // Constr. Math. Anal. 2019. V. 2. № 4. P. 183–216.
  3. Castro L. Duduchava R., Speck F.-O. Mixed impedance boundary value problems for the Laplace-Beltrami equation // J. Integral Equations Appl. 2020. V. 32. № 3. P. 275–292.
  4. Эскин Г.И. Краевые задачи для эллиптических псевдодифференциальных уравнений. М.: Наука, 1973.
  5. Гахов Ф.Д. Краевые задачи. М.: Наука, 1977.
  6. Мусхелишвили Н.И. Сингулярные интегральные уравнения. М.: Наука, 1968.
  7. Milkhin S.G., Prößdorf S. Singular Integral Operators. Akademie-Verlag, Berlin, 1986.
  8. Владимиров В.С. Методы теории функций многих комплексных переменных. М.: Наука, 1964.
  9. Vasilyev V.B. Pseudo-differential equations on manifolds with non-smooth boundaries. In: Pinelas S. editor. Differential and Difference Equations (and Applications. Springer Proc. Math. & Stat. 47). Berlin: Springer, 2013. P. 625–637.
  10. Vasilyev V.B. Elliptic equations, manifolds with non-smooth boundaries, and boundary value problems. In: Dang P., Ku M., Qian T., Rodino L. eds. New Trends in Analysis and Interdisciplinary Applications. Basel: Birkhäuser, 2017. P. 337–344.
  11. Vasilyev V.B. Pseudo-differential equations, wave factorization, and related problems // Math. Meth. Appl. Sci. 2018. V. 41. № 18. P. 9252–9263.
  12. Vasilyev V.B. Pseudo-differential equations and conical potentials: 2-dimensional case // Opusc. Math. 2019. V. 39. № 1. P. 109–124.
  13. Vasilyev V.B. On certain 3-dimensional limit boundary value problems // Lobachevskii J. Math. 2020. V. 41. № 5. P. 917–925.
  14. 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: Kluwer Acad. Publ., 2000.
  15. Kutaiba Sh., Vasilyev V. On limit behavior of a solution to boundary value problem in a plane sector // Math. Meth. Appl. Sci. 2021. V. 44. № 15. P. 11904–11912.

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