On the existence of boundary and initial values for solutions of degenerate parabolic equations in the Lyapunov boundary domains

  • Kapitsyna Tatyana V., kapitsynatv@mpei.ru National Research University “Moscow Power Engineering Institute” 14 Krasnokazarmennaya Street, 111250 Moscow, Russia
Keywords: parabolic equations, function spaces, first mixed problem, boundary and initial values of solutions, a priori estimates

Abstract

This work, being a continuation of [1], establishes the necessary and sufficient conditions for the solution of the second-order parabolic equations with a lateral boundary from the class $C^{1+\lambda}$, $\lambda >0$, degenerating on the boundary of the domain, to have an average limit on the lateral surface of the cylindrical domain and the limit in the mean on its lower base. Also, we study the question of the unique solvability of the first mixed problem for such equations in the case when the boundary and initial functions belong to spaces of the типа $L_2$ type. The closest to the questions under consideration are the theorems of F. Riesz and J. Littlewood and R. Paley, in which criteria are given for the limit values in $L_p$, $p > 1$, of the functions analytic in the unit disk. Further development of this topic for uniformly elliptic equations was obtained in the papers by V. P. Mikhailov and A. K. Gushchin [2–4]. The boundary smoothness condition ($\partial Q \in C^2$) can be weakened (see [5]). Under the weakest restrictions on the smoothness of the boundary (and on the coefficients of the equation), the criteria for the existence of a boundary value were established in [4, 6–8]. In this case, as shown in [7], all directions of the acceptance of boundary values for uniformly elliptic equations turn out to be equal, while the solution has a property similar to the property of continuity with respect to the 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 is more complicated. In this case, the formulation of the first boundary value problem is determined by the type of degeneracy. In the case when the values of the corresponding quadratic form of the degenerate elliptic equation on the normal vector are different from zero (the Tricomi type degeneracy), the Dirichlet problem is correct, and the properties of such degenerate equations are very close to the properties of uniformly elliptic equations; in particular, in this situation analogues of the Riesz [9] and Littlewood–Paley theorems [10, 11] are valid.

References


[1]
Petrushko I. M. and Kapitsyna T. V., “On the first mixed problem $L_p$, p > 1, for degenerate translation equations,” Vestn. MEhI, No. 6, 143–154 (2011).

[2]
Mikhailov V. P., “On the boundary values of solutions of elliptic equations in domains with a smooth boundary [in Russian],” Mat. Sb., N. Ser., 101, 163–188 (1976).

[3]
Mikhailov V. P., “On the existence of limit values of a biharmonic function on the boundary of a domain,” Dokl. Math., 69, No. 2, 228–230 (2004).

[4]
Gushchin A. K. and Mikhailov V. P., “On boundary values of solutions in $L_p$, p > 1, of elliptic equations,” Math. USSR, Sb., 36, No. 1, 1–19 (1980).

[5]
Petrushko I. M., “On boundary values in $L_p$, p > 1, of solutions of elliptic equations in domains with a Lyapunov boundary,” Math. USSR, Sb., 48, No. 2, 565–585 (1984).

[6]
Gushchin A. K. and Mikhailov V. P., “On the existence of boundary values of solutions of an elliptic equation,” Math. USSR, Sb., 73, No. 1, 171–194 (1992).

[7]
Gushchin A. K., “Some properties of the solutions of the Dirichlet problem for a second-order elliptic equation,” Sb. Math., 189, No. 7, 1009–1045 (1998).

[8]
Petrushko I. M., “On the mixed E problem for second-order parabolic equations degenerating at the boundary of the domain [in Russian],” Mat. Zametki SVFU, 26, No. 3, 57–70 (2019).

[9]
Riesz F., “Über die Randwerte einer analytischen Funktion,” Math. Z., 18, 87–95 (1923).

[10]
Littlewood J. and Paley R., “Theorems on Fourier series and power series. II,” Proc. Lond. Math. Soc., II Ser., 42, 52–89 (1936).

[11]
Littlewood J. and Paley R., “Theorems on Fourier series and power series. III,” Proc. Lond. Math. Soc., II Ser., 43, 105–126 (1937).

[12]
Miranda K. Equazioni alle derivate parziali di tipo ellitico. Springer-Verl., Berlin (1970).

[13]
Giraud G. “G´en´eralization des probl`emes sur les operations du type elliptique // Bull. Sci. Math. 56, 248–272; 281–312; 316–352 (1932).

[14]
Kapitsyna T. V., “On the existence of boundary and initial values for degenerate parabolic equations in stellar domains [in Russian],” Mat. Zametki SVFU, 25, No. 4, 15–33 (2018).

[15]
Petrushko I. M., “On boundary and initial conditions in $L_p$, p > 1, of solutions of parabolic equations,” Math. USSR, Sb., 53, No. 2, 489–522 (1986).

[16]
Petrushko I. M., “Existence of boundary values for solutions of degenerate elliptic equations,” Sb. Math., 190, No. 7, 973–1004 (1999).
How to Cite
Kapitsyna, T. (2020) “On the existence of boundary and initial values for solutions of degenerate parabolic equations in the Lyapunov boundary domains”, Mathematical notes of NEFU, 27(2), pp. 21-38. doi: https://doi.org/10.25587/SVFU.2020.67.75.002.
Section
Mathematics