On the first mixed problem for degenerate parabolic equations in stellar domains with Lyapunov boundary in Banach spaces
https://doi.org/10.25587/SVFU.2023.56.84.002
Abstract
The article is devoted to the study of behavior of the solution to a second-order parabolic equation with Tricomi degeneration on the lateral boundary of a cylindrical domain QT , where Q is a stellar region whose boundary ∂Q is an (n−1)-dimensional closed surface without boundary of class C1+λ, 0 < λ < 1. We study the question of unique solvability of the first mixed problem for the equation with the boundary and initial functions belonging to spaces of type Lp, p > 1. This topic goes back to the classical works of Littlewood–Paley and F. Riesz devoted to the boundary values of analytic functions. All directions of taking boundary values for uniformly elliptic equations turn out to be equal, and the solution has a property similar to the continuity with respect to a set of variables. In the case of degeneracy of the equation on the boundary of the domain when the directions are not equal, the situation becomes more complicated. In this case, the statement of the first boundary value problem is determined by the type of degeneracy.
About the Authors
I. M. Petrushko
National Research University "MPEI"
Russian Federation
Russian Federation
Igor M. Petrushko
14 Kracnokazarmennaya, 111250 Moscow
T. V. Kapitsyna
National Research University "MPEI"
Russian Federation
Russian Federation
Tatyana V. Kapitsyna
14 Kracnokazarmennaya, 111250 Moscow
M. I. Petrushko
National Research University "MPEI"
Russian Federation
Russian Federation
Maksim I. Petrushko
14 Kracnokazarmennaya, 111250 Moscow
References
1. Riesz F., “ ¨Uber die Randwerteeiner analytischen Funktion,” Math. Z., 18, 87–95 (1923).
2. Littlewood J. E. and Paley R. E. A. C., “Theorem on Fourier series and power series,” J. Lond. Math. Soc., 6, 230–233 (1931).
3. Littlewood J. E. and Paley R. E. A. C., “Theorem on Fourier series and power. II,” Proc. Lond. Math. Soc., Ser. 2, 42, 52–89 (1936).
4. Littlewood J. E. and Paley R. E. A. C., “Theorem on Fourier series and power. II,” Proc. Lond. Math. Soc., Ser. 2, 43, 105–126 (1937).
5. Mikhailov V. P., “On boundary properties of solution of elliptic equations,” Sov. Math. Dokl., 17, 274–277 (1976).
6. Mikhailov V. P., “On the boundary value of solutions of elliptic equations in domains with a smooth boundary,” Math. USSR, Sb., 30, No. 2, 143–166 (1976).
7. Gushchin A. K. and Mikhailov V. P., “On boundary values in Lp, p > 1, of solutions of elliptic equations,” Math. USSR, Sb., 36, No. 1, 1–19 (1980).
8. Gushchin A. K. and Mikhailov V. P., “On the existence of boundary values of solutions of elliptic equation,” Math. USSR, Sb., 73, No. 1, 171–194 (1992).
9. Petrushko I. M., “On boundary and initial conditions in Lp, p > 1, of solutions of parabolic equations,” Math. USSR, Sb., 53, No. 2, 489–522 (1986).
10. Petrushko I. M., “On the boundary value in Lp, p > 1, of solutions of elliptic equations in domains with Lyapunov boundary,” Math. USSR, Sb., 48, No. 2, 565–585 (1984).
11. Gushchin A. K., “On the Dirichlet problem for a second-order elliptic equation,” Math. USSR, Sb., 65, No. 1, 19–66 (1990).
12. Gushchin A. K., “On the interior smoothness of solutions to second-order elliptic equation,” Sib. Math. J., 46, 826–840 (2005).
13. Gushchin A. K., “A strengthening of the interior H¨older continuity property for solutions for Dirichlet problem for a second-order elliptic equation,” Theor. Math. Phys., 157, No. 3, 1655–1670 (2008).
14. Gushchin A. K., “The Dirichlet problem for a second-order elliptic equation with an Lp boundary function,” Sb. Math., 203, No. 1, 1–27 (2012).
15. Gushchin A. K., “A criterion for existence of Lp boundary value of solutions to an elliptic equation,” Proc. Steklov Inst. Math., 301, 44–64 (2018).
16. Dumanyan V. Zh., “Solvability of the Dirichlet problem for a general second-order elliptic equation,” Sb. Math., 202, No. 7, 1001–1020 (2011).
17. Tricomi F. G., “Ancora sull’equazione yuxx + uyy = 0,” Rend. Acc. Lincei, Ser. VI, 6 (1927).
18. Keldysh M. V., “Same cases of the degeneracy of equations of elliptic type at the boundary of a domain,” Dokl. Akad. Nauk SSSR, 77, No. 2, 181–183 (1951).
19. Fichera G., “On a unified theory of boundary value problems for elliptic-parabolic equations of second order,” Magyar Tud. Akad., Mat. Fiz. Tud. Oszt. K¨o zl., 13, 375–393 (1963).
20. Oleinik O. A. and Radkevich E. V., “Second order equations with nonnegative characteristic form,” in: Itogi Nauki, Mat. Analiz, pp. 7–252, VINITI, Moscow (1971).
21. Petrushko I. M., “Boundary and initial conditions in Lp, p > 1, for solutions of parabolic equations,” Math USSR, Sb., 125, No. 4, 489–521 (1984).
22. Kapitsyna T. V., “On the existence of boundary and initial values for the degenerating parabolic equations in the stellar domains,” Mat. Zamet. SVFU, 25, No. 4, 15–33 (2018).
23. Kapitsyna T. V., “On the existence of boundary and initial values for the degenerating parabolic equations in the stellar domains. Part II,” Mat. Zamet. SVFU, 27, No. 2, 21–38 (2020).
Review
For citations:
Petrushko I.M., Kapitsyna T.V., Petrushko M.I. On the first mixed problem for degenerate parabolic equations in stellar domains with Lyapunov boundary in Banach spaces. Mathematical notes of NEFU. 2023;30(1):21-39. (In Russ.) https://doi.org/10.25587/SVFU.2023.56.84.002
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