New forms of the Lopatynsky condition

Authors

  • V. Il’kiv National University "Lviv Polytechnic"

DOI:

https://doi.org/10.3842/umzh.v77i2.8534

Keywords:

еліптичні системи, еліптичні задачі, умова Лопатинського, розклад матричного многочлена

Abstract

UDC 517.956.223+512.64

We present known forms of the Lopatynsky condition of ellipticity of a problem for a linear regularly elliptic system of equations and determine new forms of this condition.

Several examples of its application to various elliptic systems of equations are considered.

References

1. S. Agmon, A. Douglis, L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I, Commun. Pure and Appl. Math., 12, 623–727 (1959); DOI.org/10.1002/ cpa.3160120405. DOI: https://doi.org/10.1002/cpa.3160120405

2. S. Agmon, A. Douglis, L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, II, Commun. Pure and Appl. Math., 17, 35–92 (1964); DOI.org/10.1002/cpa.3160170104. DOI: https://doi.org/10.1002/cpa.3160170104

3. M. S. Agranovich, Elliptic boundary problems, Encyclopedia Math. Sci., 79, Partial Differential Equations, IX, Springer, Berlin, 1–144 (1997). DOI: https://doi.org/10.1007/978-3-662-06721-5_1

4. M. S. Agranovich, M. I. Vishik, Elliptic problems with a parameter and parabolic problems of general type, Uspekhi Mat. Nauk, 19, № 3 (117), 53–161 (1964); English translation: Russian Math. Surveys, 19, № 3, 53–157 (1964).

5. Ju. M. Berezanskii, Expansions in eigenfunctions of selfadjoint operators, Transl. Math. Monogr., 17, AMS, Providence, R.I. (1968).

6. A. V. Bitsadze, Boundary value problems for elliptic equations of second order, Nauka, Moscow (1966); English translation: North-Holland, Amsterdam (1968).

7. S. D. Eidel’man, N. V. Zhitarashu, Parabolic boundary-value problems, Basel, Birkhäuser (Oper. Theory Adv. and Appl., 101) (1998). DOI: https://doi.org/10.1007/978-3-0348-8767-0

8. G. Eskin, Boundary value problems for elliptic pseudodifferential equations, Transl. Math. Monogr., 52, AMS, Providence, R.I. (1981).

9. S. K. Godunov, Ordinary differential equations with constant coefficients, AMS, Providence, R.I. (1997). DOI: https://doi.org/10.1090/mmono/169

10. L. Hörmander, Linear partial differential operators, Springer, Heidelberg (1969). DOI: https://doi.org/10.1007/978-3-662-30722-9

11. L. Hörmander, The analysis of linear partial differential operators, vols. 1–4, Springer-Verlag, Berlin (1983–1985).

12. P. S. Kazimirs'kii, To the decomposition of a polynomial matrix into linear factors, Dop. AN URSR, № 4, 446–448 (1964).

13. K. Krupchyk, J. Tuomela, The Shapiro–Lopatinskij condition for elliptic boundary value problems, LMS J. Comput. and Math., 9, 287–329 (2006); DOI:10.1112/S1461157000001285. DOI: https://doi.org/10.1112/S1461157000001285

14. J. L. Lions, E. Magenes, Problèmes aux limites non homogènes et applications, 1, Dunod, Paris (1968).

15. Ya. B. Lopatinskij, On a method of reducing boundary problems for a system of differential equations of elliptic type to regular integral equations, Ukr. Math. J., 5, 123–151 (1953); English translation: Amer. Math. Transl. Ser. 2, 89, 149–183 (1970).

16. Ya. B. Lopatinskij, Decomposition of a polynomial matrix into a product, Nauchn. Zap. Polytechn. Inst., L'vov, Ser. Fiz. Mat., 2, 3–7 (1956).

17. Ya. B. Lopatinskij, On some properties of polynomial matrices, Boundary Value Probl. Math. Phys., Kiev, Inst. Math. AN USSR, 108–146 (1979).

18. V. A. Mikhailets, A. A. Murach, Hörmander spaces, interpolation, and elliptic problems, De Gruyter Stud. Math., 60, De Gruyter, Berlin (2014). DOI: https://doi.org/10.1515/9783110296891

19. C. Miranda, Equazioni alle Derivate Parziale di Tipo Elliptico, Springer-Verlag, Berlin (1955).

20. O. I. Panich, Introduction to the general theory of elliptic boundary-value problems} (in Russian), Vyshcha Shkola, Kiev (1986).

21. V. M. Petrychkovych, Generalized equivalence of matrices and its collections and factorization of matrices over rings, Pidstryhach Inst. Appl. Probl. Mech. and Math. NAS of Ukraine, L'viv (2015).

22. Ya. Roitberg, Elliptic boundary value problems in the spaces of distributions, Kluwer Academic Publishing (1996). DOI: https://doi.org/10.1007/978-94-011-5410-9

23. Z. Ya. Shapiro, On general boundary value problems for equations of elliptic type, Izv. AN SSSR, 17, № 6, 539–562 (1953).

24. I. V. Skrypnik, Methods of analysis of nonlinear elliptic boundary value problems, Transl. Math. Monogr., 139, AMS, Providence (1991); Russian edition (1990).

25. V. P. Shchedryk, Arithmetic of matrices over rings, Akademperiodyka, Kyiv (2021); DOI: https://doi.org/10.15407/akademperiodika.430.278. DOI: https://doi.org/10.15407/akademperiodika.430.278

26. M. Z. Solomyak, On linear elliptic first order systems, Dokl. Akad. Nauk SSSR, 150, № 1, 48–51 (1963); English translation: Sov. Math. Dokl., 4, 604–607 (1963).

27. V. A. Solonnikov, On boundary value problems for linear parabolic systems of differential equations of general form} (in Russian), Trudy Mat. Inst. Steklova, 83, 3–163 (1965).

28. V. A. Solonnikov, On general boundary value problems for systems elliptic in the sense of Douglis–Nirenberg. I, II, Izv. Akad. Nauk SSSR, Ser. Mat., 28, 665–706 (1964) and Tr. Mat. Inst. Steklova, 92, 233–297 (1966); English translation: Amer. Math. Soc. Transl., Ser. II, 56, 193–232 (1964) and Proc. Steklov Inst. Math., 92, 3–32 (1968).

29. L. R. Volevich, On the theory of boundary value problems for general elliptic systems, Dokl. Akad. Nauk SSSR, 148, 489–492 (1963).

30. L. R. Volevich, Solvability of boundary value problems for general elliptic systems, Mat. Sb., 68(110), № 3, 373–416 (1968); English translation: Amer. Math. Soc. Transl., Ser. II, 61, 182–225 (1968).

31. Jo. T. Wloka, B. Rowley, B. Lawruk, Boundary value problems for elliptic systems, Cambridge University Press (1995). DOI: https://doi.org/10.1017/CBO9780511662850

Published

28.03.2025

Issue

Section

Research articles

How to Cite

Il’kiv, V. “New Forms of the Lopatynsky Condition”. Ukrains’kyi Matematychnyi Zhurnal, vol. 77, no. 2, Mar. 2025, pp. 107–122, https://doi.org/10.3842/umzh.v77i2.8534.