Abstract
We describe the statements of boundary value problems for multidimensional higher order equations of mixed type and propose a method for constructing coercive boundary value problems. Also, we prove the generalized and regular solvability of these boundary value problems in Sobolev spaces.
Similar content being viewed by others
References
Oleinik O. A. and Radkevič E. V.,Second-Order Equations with Nonnegative Characteristic Form, Amer. Math. Soc., Providence (2012).
Vragov V. N., “On the statement and solvability of the boundary value problems for mixed-composite type equations,” in: Mathematical Analysis and Related Problems of Mathematics [Russian], Sobolev Inst. Math., Novosibirsk (1978), 5–13.
Vragov V. N.,Boundary Value Problems for Nonclassical Equations of Mathematical Physics [Russian], Novosibirsk Univ., Novosibirsk (1983).
Terekhov A. N., “Nonlocal boundary value problems for equations of variable type,” in: Nonclassical Equations of Mathematical Physics [Russian], Inst. Mat., Novosibirsk (1985), 148–158.
Glazatov S. N., “Nonlocal boundary value problems for certain equations of mixed type in a rectangle,” Sib. Mat. Zh., vol. 26, no. 6, 162–164 (1985).
Dzhamalov S. Z., “On the well-posedness of nonlocal boundary value problems for the multidimensional equation of mixed type,” in: Application of Methods of Functional Analysis to Nonclassical Equations of Mathematical Physics [Russian], Inst. Mat., Novosibirsk (1989), 63–70.
Karatoproklieva M. G., “On a nonlocal boundary value problem for the equation of mixed type,” Differ. Uravn., vol. 27, no. 1, 68–79 (1991).
Jamalov S. Z., “On correctness of nonlocal edge problem with constant coefficient for the multidimensional second order equation of mixed type,” Math. Notes of NEFU, vol. 24, no. 4, 17–28 (2017).
Dzhamalov S. Z., “The nonlocal boundary value problem with constant coefficients for the multidimensional equation of the mixed type of the second kind, the second order,” J. Siber. Federal University, vol. 11, no. 4, 472–461 (2018).
Djamalov S. Z., “On the smoothness of the solution of a nonlocal boundary value problem for the multidimensional second-order equation of the mixed type of the second kind in Sobolev space,” Middle Volga Mathematical Society Journal, vol. 21, no. 1, 24–33 (2019).
Egorov I. E. and Fedorov V. E.,Higher-Order Nonclassical Equations of Mathematical Physics [Russian], Computer Center of the SB RAS, Novosibirsk (1995).
Chueshev A. V., “About one high-order linear equation of mixed type,” Sib. Math. J., vol. 43, no. 2, 363–378 (2002).
Dzhamalov S. Z., “On a nonlocal boundary value problem for the mixed-type equation of higher order,” in: Abstracts: 7 International Conference of Mathematical Modeling, NEFU Publ., Yakutsk (2014), 37–38.
Lions J. L. and Magenes E.,Non-Homogeneous Boundary Value Problems and Applications. Vol. 1, Springer-Verlag, Berlin, Heildelberg, and New York (1972).
Triebel H.,Interpolation Theory. Function Spaces. Differential Operators, Berlin, Deutscher Verlag des Wissenschaften (1978).
Grisvard P., “Équations différentielles abstraites,” Ann. Sci. Éc. Norm. Supér. (4), vol. 2, no. 3, 311–395 (1969).
Denk R., Hieber M., and Prüss J.,R-Boundedness, Fourier Multipliers and Problems of Elliptic and Parabolic Type, Amer. Math. Soc., Providence (2003) (Mem. Amer. Math. Soc.; No. 166).
Naimark M. A.,Linear Differential Operators. Pts. 1 and 2, Frederick Ungar Publishing Co., New York (1968, 1969).
Grisvard P., “Caractérisation de quelques espaces d’interpolation,” Arch. Rat. Mech., vol. 25, no. 1, 40–63 (1967).
Grisvard P., “Commutativité de deux foncteurs d’interpolation at applications,” J. Math. Pures Appl., vol. 45, 143–206 (1966).
Chueshev A. V., “Estimates of the resolvent for ordinary differential operators of mixed type,” Mat. Tr., vol. 3, no. 1, 144–196 (2000).
Funding
The authors were supported by the Russian Foundation for Basic Research (Grant 18–51–41009).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Dzhamalov, S.Z., Pyatkov, S.G. On Some Boundary Value Problems for Multidimensional Higher Order Equations of Mixed Type. Sib Math J 61, 610–625 (2020). https://doi.org/10.1134/S0037446620040059
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0037446620040059
Keywords
- mixed type equation
- nonlocal boundary value problem
- boundary value problem
- generalized solution
- regular solution
- uniqueness
- existence