Задачи типа Дирихле высокого порядка для двумерного оператора Гельмгольца в комплексном кватернионном анализе

Авторы

  • Барух Шнайдер Остравский университет, Чешская Республика, 70103, Острава, 30.дубна, 22

DOI:

https://doi.org/10.21638/11701/spbu01.2019.410

Аннотация

Хорошо известно, что разработка методов решений задач Дирихле важна и актуальна для различных областей математической физики, связанных с уравнением Лапласа, уравнением Гельмгольца, уравнением Стокса, уравнением Максвелла, уравнением Дирака и др. В своих предыдущих работах автор изучал разрешимость краевых задач Дирихле первого и второго порядков в кватернионном анализе. В данной работе изучается краевая задача Дирихле высокого порядка, связанная с двумерным уравнением Гельмгольца с комплексным потенциалом. В данной работе доказывается существование и единственность решения краевой задачи Дирихле в двумерном случае и ищется соответствующее решение этой задачи. Большинство задач Дирихле решается для случая трех переменных. Отметим, что случай двух переменных не является простым следствием трехмерного случая. Для решения поставленной задачи в работе используется метод ортогонального разложения кватернионного пространства Соболева. Данное ортогональное разложение пространства является также инструментом для изучения многих эллиптических граничных задач, которые возникают в различных областях математики и математической физики. В работе также получено ортогональное разложение кватернионного пространства Соболева относительно оператора Дирака высокого порядка.

Ключевые слова:

кватернионный анализ, оператор Гельмгольца, краевые задачи типа Дирихле

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Библиографические ссылки

Литература

Gürlebeck K., Sprössig W. Quaternionic and Clifford calculus for Physicists and Engineers. New York: Wiley, Chichester, 1997.

Kaehler U. On a direct decomposition of the space Lp(Ω) // Z. Anal. Anwend. 1999. Vol. 18, No. 4. P. 839–848. https://doi.org/10.4171/ZAA/917

Le H.T. Hyperholomorphic structures and corresponding explicit orthogonal function systems in 3D and 4D. Ph. D. Thesis, Institute of Applied Analysis, 2014. Freiberg University of Mining and Technology, Germany.

Le H.T., Morais J., Sprössig W. Orthogonal decompositions of the complex quaternion Hilbert space and their applications // 9th International Conference on Clifford algebras and theirs Applications in Mathematical Physics, Weimar, Germany 15–20 July, 2011 / ed. K. Gürlebeck. 2011.

Le H.T., Morais J., Sprössig W. Orthogonal decompositions and their applications // Proceedings of the 19th International Conference on the Application of Computer Science and Mathematics in Architecture and Civil Engineering, Weimar, Germany 04–06 July, 2012 / eds. K. Gürlebeck, T. Lahmer, F.Werner. 2012.

Le H.T., Morais J., Sprössig W. Orthogonal decompositions of the complex quaternion Hilbert space and their applications // 9th International Conference on Mathematical Problems in Engineering, Aerospace and Sciences: ICNPAA 2012. AIP Conf. Proc. 2012. Vol. 1493. P. 595–602.

Sprössig W. On decompositions of the Clifford valued Hilbert space and their applications to boundary value problems // Adv. Appl. Clifford Algebras. 1995. Vol. 5, No. 2. P. 167–186.

Ammari H., Bao G., Wood A.W. An integral equation method for the electromagnetic scattering from cavities // Math. Methods Appl. Sci. 2000. Vol. 23, no. 12. P. 1057–1072. https://doi.org/10.1002/1099-1476(200008)23:12%3C1057::AID-MMA151%3E3.0.CO;2-6

Li D., Mao J.F. A Koch-like sided fractal bow-tie dipole antenna // Antennas and Propagation, IEEE Transactions on. 2012. Vol. 60, no. 5. P. 2242–2251. https://doi.org/10.1109/TAP.2012.2189719

Gürlebeck K., Sprössig W. Quaternionic analysis and elliptic boundary value problems. Basel: Birkhauser, 1990. https://doi.org/10.1007/978-3-0348-7295-9

Kravchenko V. Applied quaternionic analysis. In: Research and Exposition in Mathematics. Vol. 28. Germany: Heldermann Verlag, 2003.

Kravchenko V., Shapiro M. Integral representations for spatial models of mathematical physics. In: Pitman Res. Notes in Math. Ser. Vol. 351. Harlow: Longman, 1996.

Gerus O.F., Shapiro M. On boundary properties of metaharmonic simple and double layer potentials on rectifiable curves in R2 // Zb. Pr. Inst. Mat. NAN Ukr. 2004. Vol. 1, No. 3. P. 67–76.

Gerus O.F., Shapiro M. On a Cauchy-type integral related to the Helmholtz operator in the plane // Bol. Soc. Mat. Mex., III. Ser. 2004. Vol. 10, No. 1. P. 63–82.

Shapiro M., Tovar L. Two-dimensional Helmholtz operator and its hyperholomorphic solutions // J. Natur. Geom. 1997. Vol. 11. P. 77–100.

Shapiro M., Tovar L. On a class of integral representations related to the two-dimensional Helmholtz operator // Contemp. Math. 1998. Vol. 212. P. 229–244. Amer. Math. Soc., Providence, RI. https://doi.org/10.1090/conm/212/02886

Abreu Blaya R., Ávila-Ávila R., Bory Reyes J., Rodríguez-Dagnino R.M. 2D Quaternionic Time-Harmonic Maxwell System in Elliptic Coordinates // Adv. Appl. Clifford Algebras. 2015. Vol. 25. Issue 2. P. 255–270. https://doi.org/10.1007/s00006-014-0485-x

Bory Reyes J., Abreu Blaya R., Rodríguez-Dagnino R.M., Kats B.A. On Riemann boundary value problems for null solutions of two dimensional Helmholtz equation // Anal. Math. Phys. 2019. Vol. 9. Issue 1. P. 483–496. https://doi.org/10.1007/s13324-018-0210-3

Bory Reyes J., Abreu Blaya R., Pérez de la Rosa M.A., Schneider B. On the 2D Quaternionic Metaharmonic Layer Potentials // Mediterranean Journal of Mathematics. 2017. Vol. 14. Issue 4. Art. no. 195. https://doi.org/10.1007/s00009-017-0995-6

Luna-Elizarrarás M.E., Pérez-de la Rosa M.A., Rodríguez-Dagnino R.M., Shapiro M. On quaternionic analysis for the Schrödinger operator with a particular potential and its relation with the Mathieu functions // Math. Methods Appl. Sci. 2013. Vol. 36, No. 9. P. 1080–1094. https://doi.org/10.1002/mma.2665

Bory-Reyes J., Abreu-Blaya R., Hernández-Simon L.M., Schneider B. Dirichlet-Type Problems for the Two-Dimensional Helmholtz Operator in Complex Quaternionic Analysis // Mediterranean Journal of Mathematics. 2016. Vol. 13. P. 4901–4916. https://doi.org/10.1007/s00009-016-0781-x

Wloka J. Partielle Differentialgleichungen [Partial differential equations]. Sobolevräume und Randwertaufgaben [Sobolev spaces and boundary value problems]. Mathematische Leitfäden. [Mathematical Textbooks]. Stuttgart: B.G. Teubner, 1982. https://doi.org/10.1007/978-3-322-96662-9_1

Garnir H.G. Les Problèmes aux Limites de la Physique Mathèmatique, Basel: Birkhaüser, 1958.

Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables / Eds. M. Abramowitz, I.A. Stegun. New York etc.: John Wiley & Sons, 1972.

Arfken G.B., Weber H.J., Harris F.H. Mathematical Methods for Physicists. 7th edn. Waltham: Academic Press, Elsevier, 2013.

Watson G.N. A Treatise on the Theory of Bessel Functions. 2nd edn. Cambridge: Cambridge University Press, 1995.


References

Gürlebeck K., Sprössig W., Quaternionic and Clifford calculus for Physicists and Engineers (Wiley, Chichester, New York, 1997).

Kaehler U., “On a direct decomposition of the space Lp(Ω)”, Z. Anal. Anwend. 18(4), 839–848 (1999). https://doi.org/10.4171/ZAA/917

Le H.T., Hyperholomorphic structures and corresponding explicit orthogonal function systems in 3D and 4D (Ph. D. Thesis, Institute of Applied Analysis, 2014. Freiberg University of Mining and Technology, Germany).

Le H.T., Morais J., “Sprössig W. Orthogonal decompositions of the complex quaternion Hilbert space and their applications”, 9th International Conference on Clifford algebras and theirs Applications in Mathematical Physics, Weimar, Germany 15-20 July, 2011 (ed. K.Gürlebeck, 2011).

Le H.T., Morais J., SprössigW., “Orthogonal decompositions and their applications”, Proceedings of the 19th International Conference on the Application of Computer Science and Mathematics in Architecture and Civil Engineering, Weimar, Germany 04–06 July, 2012 (eds. Gürlebeck K., Lahmer T., Werner F., 2012).

Le H.T., Morais J., Sprössig W., “Orthogonal decompositions of the complex quaternion Hilbert space and their applications”, 9th International Conference on Mathematical Problems in Engineering, Aerospace and Sciences: ICNPAA 2012. AIP Conf. Proc. 1493, 595–602 (2012).

Sprössig W., “On decompositions of the Clifford valued Hilbert space and their applications to boundary value problems”, Adv. Appl. Clifford Algebras 5(2), 167–186 (1995).

Ammari H., Bao G., Wood A.W., “An integral equation method for the electromagnetic scattering from cavities”, Math. Methods Appl. Sci. 23(12), 1057–1072 (2000). https://doi.org/10.1002/1099-1476(200008)23:12%3C1057::AID-MMA151%3E3.0.CO;2-6

Li D., Mao J.F., “A Koch-like sided fractal bow-tie dipole antenna”, Antennas and Propagation, IEEE Transactions on 60(5), 2242–2251 (2012). https://doi.org/10.1109/TAP.2012.2189719

Gürlebeck K., Sprössig W., Quaternionic analysis and elliptic boundary value problems (Birkhauser, Basel, 1990). https://doi.org/10.1007/978-3-0348-7295-9

Kravchenko V., Applied quaternionic analysis, in: Research and Exposition in Mathematics 28 (Heldermann Verlag, Germany, 2003).

Kravchenko V., Shapiro M., Integral representations for spatial models of mathematical physics, in: Pitman Res. Notes in Math. Ser. 351 (Longman, Harlow, 1996).

Gerus O.F., Shapiro M., “On boundary properties of metaharmonic simple and double layer potentials on rectifiable curves in R2”, Zb. Pr. Inst. Mat. NAN Ukr. 1(3), 67–76 (2004).

Gerus O.F., Shapiro M., “On a Cauchy-type integral related to the Helmholtz operator in the plane”, Bol. Soc. Mat. Mex., III. Ser. 10(1), 63–82 (2004).

Shapiro M., Tovar L., “Two-dimensional Helmholtz operator and its hyperholomorphic solutions”, J. Natur. Geom. 11, 77–100 (1997).

Shapiro M., Tovar L., “On a class of integral representations related to the two-dimensional Helmholtz operator”, Contemp. Math. 212, 229–244 (Amer. Math. Soc., Providence, RI, 1998). https://doi.org/10.1090/conm/212/02886

Abreu Blaya R., Ávila-Ávila R., Bory Reyes J., Rodríguez-Dagnino R.M., “2D Quaternionic Time-Harmonic Maxwell System in Elliptic Coordinates”, Adv. Appl. Clifford Algebras 25, issue 2, 255–270 (2015). https://doi.org/10.1007/s00006-014-0485-x

Bory Reyes, J., Abreu Blaya, R., Rodríguez-Dagnino R.M., Kats B.A., “On Riemann boundary value problems for null solutions of two dimensional Helmholtz equation”, Anal. Math. Phys. 9, issue 1, 483–496 (2019). https://doi.org/10.1007/s13324-018-0210-3

Bory Reyes J., Abreu Blaya R., Pérez de la Rosa M.A., Schneider B., “On the 2D Quaternionic Metaharmonic Layer Potentials”, Mediterranean Journal of Mathematics 14, issue 4, 195 (2017). https://doi.org/10.1007/s00009-017-0995-6

Luna-Elizarrarás M.E., Pérez-de la Rosa M.A., Rodríguez-Dagnino R.M., Shapiro M., “On quaternionic analysis for the Schrödinger operator with a particular potential and its relation with the Mathieu functions”, Math. Methods Appl. Sci. 36(9), 1080–1094 (2013). https://doi.org/10.1002/mma.2665

Bory-Reyes J., Abreu-Blaya R., Hernández-Simon L.M., Schneider B., “Dirichlet-Type Problems for the Two-Dimensional Helmholtz Operator in Complex Quaternionic Analysis”, Mediterranean Journal of Mathematics 13, 4901–4916 (2016). https://doi.org/10.1007/s00009-016-0781-x

Wloka J., Partielle Differentialgleichungen [Partial differential equations]. Sobolevräume und Randwertaufgaben [Sobolev spaces and boundary value problems] (Mathematische Leitfäden, B. G. Teubner, Stuttgart, 1982). https://doi.org/10.1007/978-3-322-96662-9_1

Garnir H.G., Les Problèmes aux Limites de la Physique Mathèmatique (Birkhaüser, Basel, 1958).

Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Eds. M. Abramowitz, I.A. Stegun, John Wiley & Sons, New York etc., 1972).

Arfken G.B., Weber H. J., Harris F.H., Mathematical Methods for Physicists (7th edn., Academic Press, Elsevier, Waltham, 2013).

Watson G.N., A Treatise on the Theory of Bessel Functions (2nd ed., Cambridge University Press, Cambridge, 1995).

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Опубликован

28.11.2019

Как цитировать

Шнайдер, Б. (2019). Задачи типа Дирихле высокого порядка для двумерного оператора Гельмгольца в комплексном кватернионном анализе. Вестник Санкт-Петербургского университета. Математика. Механика. Астрономия, 6(4), 646–658. https://doi.org/10.21638/11701/spbu01.2019.410

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Математика