On applications of the Levi method in the theory of parabolic equations

S. D. Ivasyshen, I. P. Medynsky; I. P. Medynsky

doi:10.15330/ms.47.1.33-46

A short review of papers where a fundamental solution was constructed by the Levi’s parametrix method is done. Some details of applications of this method for parabolic equations and systems of parabolic equations, in particular the procedure of constructing a parametrix for equations with different degenerations and singularities, are considered. The basic results of constructing of a fundamental solution for degenerate ultraparabolic equations of the Kolmogorov type by the Levi paramerix method under different assumptions on the coefficients are reviewed. Some new authors’ results on the constructing of a classical fundamental solution and, so-called, L-fundamental solution for the equation with coefficients with two groups of spatial variables are presented. These results are obtained by stepwise using of the Levi parametrix method and special Holder conditions on the coefficients.

1. Levi E.E. Sulle equazioni lineari totalmente ellitiche alle derivate parziali// Rend. Circ. Matem. Palermo. - 1907. - V.24 - P. 275-317.

2. Levi E.E. On linear elliptic partial differential equations// Usp. Mat. Nauk. - 1941. - V.8. - P. 249-292. (in Russian)

3. Shapiro Z.Ya. On elliptic systems of partial differential equations// Soviet Math. Dokl. - 1945. - V.46, №4. - P. 146-149. (in Russian)

4. Lopatynsky Ya.B. Fundamental system of solutions for elliptic system of linear differential equations// Ukr. Mat. J. - 1951. - V.3, №1. - P. 3-38. (in Russian)

5. Ivasyshen S.D. On influence of ideas of Ya.B. Lopatynsky on developent of the theory of parabolic systems// Nonlinear Boundary Problems. - 2010. - V.20. - P. 45-53. (in Ukrainian)

6. Dressel F. The fundamental solution of the parabolic equation// Duke Math. J. - 1940. - V.7, №4. - P. 186-203.

7. Pogorzelski W. Etude de la solution fondamentale de lequation parabolique// Ricerche di Mat. - 1956. - V.5 - P. 25-57.

8. Aronson D.G. The fundamental solution of a linear parabolic equation containing a small parameter// III. J. Math. - 1959. - V.3. - P. 580-619.

9. Eidelman S.D. Parabolic Systems, North-Holland, Amsterdam, 1969 (Russian edition, Nauka, Moscow, 1964).

10. Eidelman S.D., Ivasyshen S.D., Kochubei A.N. Analytic methods in the theory of differential and pseudodifferential equations of parabolic type. . Operator Theory: Adv. and Appl., 2004, V.152, 390 p.

11. Kolmogorov A.N. Zufallige Bewegungen (Zur Theorie der Brownschen Bewegung)// Ann. Math. - 1934. - V.35. - P. 116-117.

12. Weber M. The fundamental solution of a degenerate partial differential equation of parabolic type// Trans. Amer. Math. Soc. - 1951. - V.71. - P. 24-37.

13. Ilin A.M. On a class of ultra-parabolic equations// Soviet Math. Dokl. - 1964. - V.159, №6. - P. 1214-1217. (in Russian)

14. Sonin I.M. On a class of degenerate diffusion processes// Theor. Probab. Appl. - 1967. - V.12, №3. - P. 490-496. (in Russian)

15. Pascucci A. A. Kolmogorov equations in physics and in finance// Progress in Nonlinear Differential Equations and Their Applications. . Basel: Birkhauser. - 2005. - V.63. - P. 313-324.

16. Ivasyshen S.D., Medynsky I.P. The Fokker-Planck-Kolmogorov equations for some degenerate diffusion processes// Theory of stochastic processes. - 2010. - 16(32), №1. - P. 57-66.

17. Kuptsov L.P. Fundamental solutions of certain degenerate second-order parabolic equations// Math. Notes. - 1982. - V.31, №4. - P. 283-289. (in Russian)

18. Malitskaya A.P. Fundamental solutions for a class of degenerate parabolic equations// Approximate Methods of Integration of Differential and Integral Equations. - Kiev: Kiev. Ped. Inst, - 1973. - P. 109-130. (in Russian)

19. Eidelman S.D., Tychinska L.M. Construction of fundamental solutions for some degenerate parabolic equations of an arbitrary order// Dopovidi AN Ukr. SSR, Ser. A. - 1979. - №11. - P. 896-899. (in Ukrainian)

20. Malitskaya A.P. Construction of a fundamental solution for a class of high-order degenerate parabolic equations// Ukr. Math. J. - 1980. - V.32, №6. - P. 508-514. (in Russian)

21. Malitskaya A.P. Structure of the fundamental solutions of some ultraparabolic equations of high order// Ukr. Math. J. - 1985. - V.37, №6. - P. 582-587.

22. Ivasyshen S.D., Tychynskaya L.M., Eidelman S.D. Fundamental solutions of the Cauchi problem for a class of second order ultraparabolic equations// Dopovidi AN Ukr. SSR, Ser. A. - 1990. - №5. - P. 6-8. (in Russian)

23. Ivasyshen S.D., Androsova L.N. Integral representation of solutions of a class of degenerate parabolic Kolmogorov equations// Diff. Equations. - 1991. - V.27, №3. - P. 479-487. (in Russian)

24. Dron V.S., Ivasyshen S.D. Properties of fundamental solutions of the Cauchi problem for a class of ultraparabolic equations// Ukr. Mat. J. - 1998. - V.50, №11. - P. 1482-1496. (in Ukrainian)

25. Eidelman S.D., Ivasyshen S.D., Malytska H.P. A modified Levi method: development and application// Dopovidi NAN Ukrainy. – 1998. – №5. – P. 14–19. (in Ukrainian)

26. Dron’ V.S. On correct solvability of the Cauchy problem for the ultraparabolic equatin of Kolmogorov type// Mat. Metody Fiz.-Mech. Polya. – 1999. – V.42, №3. – P. 52–55. (in Ukrainian)

27. Ivasyshen S.D., Layuk V.V. Fundamental solutions of the Cauchy problem for some degenerate parabolic equations of Kolmogorov type// Ukr. Mat. J. – 2011. – V.63, №11. – P. 1469–1500. (in Ukrainian)

28. Lanconelli E., Polidoro S. On a class hypoelliptic evolution operators// Rend. Sem. Mat. Univ. Pol. Torino. – 1994. – V.52, №1. – P. 29–63.

29. Polidoro S. On a class of ultraparabolic operators of Kolmogorov-Fokker-Planck type// Le Matematiche. – 1994. – V.49, №1. – P. 53–105.

30. Polidoro S. Uniqueness and representation theorems for solution of Komogorov-Fokker-Planck equations// Arch. Rational Mech. Anal. – 1997. – V.137, №4. – P. 321–340.

31. Di Francesco M., Pascucci А. On a class of degenerate parabolic equations of Kolmogorov type// AMRX Appl. Math. Res. Express. – 2005. – №3. – P. 77–116.

32. Ivasyshen S.D., Medynsky I.P. Classical fundamental solutions of the Cauchy problem for ultraparabolic equations of Kolmogorov type with two groops of spartial variables// Proceedings of Institute of Mathematics NAS of Ukraine. – 2016. – V.13, №1. – P. 108–155. (in Ukrainian)

33. Ivasyshen S.D., Medynsky I. P. The classical fundamental solution of a degenerate Kolmogorov’s equation with coefficients indepedent on variables of degeneration// Bukov. Mat. J. – 2014. – V.2, №2–3. – P. 94–106. (in Ukrainian)

34. Shatyro Ya.I. Inner estimates for solutions of a class of ultra-parabolic equations// Mat. Zapiski Uralsk. Univ. – 1970. – V.7, №4. – P. 131–154. (in Russian)

35. Shatyro Ya.I. On smoothness of solutions of some degenerate second order equations// Mat. Zametki. – 1971. – V.10, №1. – P. 101–111. (in Russian)

	On applications of the Levi method in the theory of parabolic equations
Author	S. D. Ivasyshen, I. P. Medynsky ivasyshen_sd@mail.ru; i.p.medynsky@gmail.com National Technical University of Ukraine, Kyiv, Ukriane; National University Lviv Polytechnic, Lviv, Ukraine
Abstract	A short review of papers where a fundamental solution was constructed by the Levi’s parametrix method is done. Some details of applications of this method for parabolic equations and systems of parabolic equations, in particular the procedure of constructing a parametrix for equations with different degenerations and singularities, are considered. The basic results of constructing of a fundamental solution for degenerate ultraparabolic equations of the Kolmogorov type by the Levi paramerix method under different assumptions on the coefficients are reviewed. Some new authors’ results on the constructing of a classical fundamental solution and, so-called, L-fundamental solution for the equation with coefficients with two groups of spatial variables are presented. These results are obtained by stepwise using of the Levi parametrix method and special Holder conditions on the coefficients.
Keywords	parabolic equations with degeneration; fundamental solutions; the Levi parametrix method; ultraparabolic equations of the Kolmogorov type
DOI	doi:10.15330/ms.47.1.33-46
Reference	1. Levi E.E. Sulle equazioni lineari totalmente ellitiche alle derivate parziali// Rend. Circ. Matem. Palermo. - 1907. - V.24 - P. 275-317. 2. Levi E.E. On linear elliptic partial differential equations// Usp. Mat. Nauk. - 1941. - V.8. - P. 249-292. (in Russian) 3. Shapiro Z.Ya. On elliptic systems of partial differential equations// Soviet Math. Dokl. - 1945. - V.46, №4. - P. 146-149. (in Russian) 4. Lopatynsky Ya.B. Fundamental system of solutions for elliptic system of linear differential equations// Ukr. Mat. J. - 1951. - V.3, №1. - P. 3-38. (in Russian) 5. Ivasyshen S.D. On influence of ideas of Ya.B. Lopatynsky on developent of the theory of parabolic systems// Nonlinear Boundary Problems. - 2010. - V.20. - P. 45-53. (in Ukrainian) 6. Dressel F. The fundamental solution of the parabolic equation// Duke Math. J. - 1940. - V.7, №4. - P. 186-203. 7. Pogorzelski W. Etude de la solution fondamentale de lequation parabolique// Ricerche di Mat. - 1956. - V.5 - P. 25-57. 8. Aronson D.G. The fundamental solution of a linear parabolic equation containing a small parameter// III. J. Math. - 1959. - V.3. - P. 580-619. 9. Eidelman S.D. Parabolic Systems, North-Holland, Amsterdam, 1969 (Russian edition, Nauka, Moscow, 1964). 10. Eidelman S.D., Ivasyshen S.D., Kochubei A.N. Analytic methods in the theory of differential and pseudodifferential equations of parabolic type. . Operator Theory: Adv. and Appl., 2004, V.152, 390 p. 11. Kolmogorov A.N. Zufallige Bewegungen (Zur Theorie der Brownschen Bewegung)// Ann. Math. - 1934. - V.35. - P. 116-117. 12. Weber M. The fundamental solution of a degenerate partial differential equation of parabolic type// Trans. Amer. Math. Soc. - 1951. - V.71. - P. 24-37. 13. Ilin A.M. On a class of ultra-parabolic equations// Soviet Math. Dokl. - 1964. - V.159, №6. - P. 1214-1217. (in Russian) 14. Sonin I.M. On a class of degenerate diffusion processes// Theor. Probab. Appl. - 1967. - V.12, №3. - P. 490-496. (in Russian) 15. Pascucci A. A. Kolmogorov equations in physics and in finance// Progress in Nonlinear Differential Equations and Their Applications. . Basel: Birkhauser. - 2005. - V.63. - P. 313-324. 16. Ivasyshen S.D., Medynsky I.P. The Fokker-Planck-Kolmogorov equations for some degenerate diffusion processes// Theory of stochastic processes. - 2010. - 16(32), №1. - P. 57-66. 17. Kuptsov L.P. Fundamental solutions of certain degenerate second-order parabolic equations// Math. Notes. - 1982. - V.31, №4. - P. 283-289. (in Russian) 18. Malitskaya A.P. Fundamental solutions for a class of degenerate parabolic equations// Approximate Methods of Integration of Differential and Integral Equations. - Kiev: Kiev. Ped. Inst, - 1973. - P. 109-130. (in Russian) 19. Eidelman S.D., Tychinska L.M. Construction of fundamental solutions for some degenerate parabolic equations of an arbitrary order// Dopovidi AN Ukr. SSR, Ser. A. - 1979. - №11. - P. 896-899. (in Ukrainian) 20. Malitskaya A.P. Construction of a fundamental solution for a class of high-order degenerate parabolic equations// Ukr. Math. J. - 1980. - V.32, №6. - P. 508-514. (in Russian) 21. Malitskaya A.P. Structure of the fundamental solutions of some ultraparabolic equations of high order// Ukr. Math. J. - 1985. - V.37, №6. - P. 582-587. 22. Ivasyshen S.D., Tychynskaya L.M., Eidelman S.D. Fundamental solutions of the Cauchi problem for a class of second order ultraparabolic equations// Dopovidi AN Ukr. SSR, Ser. A. - 1990. - №5. - P. 6-8. (in Russian) 23. Ivasyshen S.D., Androsova L.N. Integral representation of solutions of a class of degenerate parabolic Kolmogorov equations// Diff. Equations. - 1991. - V.27, №3. - P. 479-487. (in Russian) 24. Dron V.S., Ivasyshen S.D. Properties of fundamental solutions of the Cauchi problem for a class of ultraparabolic equations// Ukr. Mat. J. - 1998. - V.50, №11. - P. 1482-1496. (in Ukrainian) 25. Eidelman S.D., Ivasyshen S.D., Malytska H.P. A modified Levi method: development and application// Dopovidi NAN Ukrainy. – 1998. – №5. – P. 14–19. (in Ukrainian) 26. Dron’ V.S. On correct solvability of the Cauchy problem for the ultraparabolic equatin of Kolmogorov type// Mat. Metody Fiz.-Mech. Polya. – 1999. – V.42, №3. – P. 52–55. (in Ukrainian) 27. Ivasyshen S.D., Layuk V.V. Fundamental solutions of the Cauchy problem for some degenerate parabolic equations of Kolmogorov type// Ukr. Mat. J. – 2011. – V.63, №11. – P. 1469–1500. (in Ukrainian) 28. Lanconelli E., Polidoro S. On a class hypoelliptic evolution operators// Rend. Sem. Mat. Univ. Pol. Torino. – 1994. – V.52, №1. – P. 29–63. 29. Polidoro S. On a class of ultraparabolic operators of Kolmogorov-Fokker-Planck type// Le Matematiche. – 1994. – V.49, №1. – P. 53–105. 30. Polidoro S. Uniqueness and representation theorems for solution of Komogorov-Fokker-Planck equations// Arch. Rational Mech. Anal. – 1997. – V.137, №4. – P. 321–340. 31. Di Francesco M., Pascucci А. On a class of degenerate parabolic equations of Kolmogorov type// AMRX Appl. Math. Res. Express. – 2005. – №3. – P. 77–116. 32. Ivasyshen S.D., Medynsky I.P. Classical fundamental solutions of the Cauchy problem for ultraparabolic equations of Kolmogorov type with two groops of spartial variables// Proceedings of Institute of Mathematics NAS of Ukraine. – 2016. – V.13, №1. – P. 108–155. (in Ukrainian) 33. Ivasyshen S.D., Medynsky I. P. The classical fundamental solution of a degenerate Kolmogorov’s equation with coefficients indepedent on variables of degeneration// Bukov. Mat. J. – 2014. – V.2, №2–3. – P. 94–106. (in Ukrainian) 34. Shatyro Ya.I. Inner estimates for solutions of a class of ultra-parabolic equations// Mat. Zapiski Uralsk. Univ. – 1970. – V.7, №4. – P. 131–154. (in Russian) 35. Shatyro Ya.I. On smoothness of solutions of some degenerate second order equations// Mat. Zametki. – 1971. – V.10, №1. – P. 101–111. (in Russian)
Pages	33-46
Volume	47
Issue	1
Year	2017
Journal	Matematychni Studii
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