##
**The nonlocal boundary value problem with constant coefficients for the multidimensional mixed type equation of the first kind.**
*(Russian.
English summary)*
Zbl 1413.35324

Summary: In this paper the unique solvability and smoothness of generalized solution of a nonlocal boundary value problem with constant coefficients for the multidimensional mixed type equation of the first kind in Sobolev spaces \(W_{2}^{l }(Q)\), (\(2\leq l \) is integer number), have been proved. First, the unique solvability of the generalized solution from space \(W_{2}^{2 }(Q)\) has been studied. Further, the uniqueness of the generalized solution of nonlocal boundary value problem with constant coefficients for the multidimensional mixed type equation was proved by a priory estimates. For the proof of the existence of the generalized solution, we used method of “\(\epsilon\)-regularization” together with Galerkin method. Precisely, first, we study regular solvability of the nonlocal boundary value problem for the multidimensional mixed type equation by functional analysis methods, i.e. we obtained necessary a priory estimates for the considered problems. Using these estimates we solve composite type equation, then by the theorem on weak compactness, we pass to the limit and deduce to the multidimensional mixed type equation of the first kind. At the end, smoothness of the generalized solution of the considered problems has been discussed.

### MSC:

35M12 | Boundary value problems for PDEs of mixed type |

### Keywords:

multidimensional mixed type equations; nonlocal boundary value problem with constants coefficients; unique solvability; smoothness of the generalized solution; \(\epsilon\)-regularization method; Galerkin method### References:

[1] | [1] Bitsadze A. V., “Incorrectness of Dirichlet’s problem for the mixed type of equations in mixed regions”, Dokl. Akad. Nauk SSSR, 122:2 (1958), 167–170 (In Russian) · Zbl 0145.35403 |

[2] | [2] Kal’menov T. Sh, “The semiperiodic Dirichlet problem for a class of equations of mixed type”, Differ. Uravn., 14:3 (1978), 546–547 (In Russian) · Zbl 0384.35043 |

[3] | [3] Kal’menov T. Sh., Sadybekov M. A., “The Dirichlet problem and nonlocal boundary value problems for the wave equation”, Differ. Equ., 26:1 (1990), 55–59 · Zbl 0729.35074 |

[4] | [4] Dzhamalov S., “On a nonlocal boundary-value problem for a second-order mixed type equation of the second kind”, Uzbek Mathematical Journal, 2014, no. 1, 5–14 (In Russian) |

[5] | [5] Dzhamalov S., “On a nonlocal boundary-value problem with constant coefficients for the Tricomi equation”, Uzbek Mathematical Journal, 2016, no. 2, 51–60 (In Russian) |

[6] | [6] Sabitov K. B., “The Dirichlet problem for equations of mixed type in a rectangular domain”, Dokl. Math., 75:2 (2007), 193-196 · Zbl 1166.35352 · doi:10.1134/S1064562407020056 |

[7] | [7] Tsybikov B N., “Well-posedness of a periodic problem for a multidimensional equation of mixed type”, Nonclassical partial differential equations, Collect. Sci. Works, ed. V. N. Vragov, Novosibirsk, 1986, 201–206 (In Russian) · Zbl 1081.35501 |

[8] | [8] Frankl F. I., “On the problems of Chaplygin for mixed sub- und supersonic flows”, Izv. Akad. Nauk SSSR, Ser. Mat., 9:2 (1945), 121–143 (In Russian) · Zbl 0063.01435 |

[9] | [9] Vragov V. N., Kraevye zadachi dlia neklassicheskikh uravnenii matematicheskoi fiziki [Boundary-Value Problems for Nonclassical Equations of Mathematical Physics], Novosibirsk State Univ., Novosibirsk, 1983, 84 pp. (In Russian) |

[10] | [10] Kozhanov A. I., Kraevye zadachi dlia uravnenii matematicheskoi fiziki nechetnogo poriadka [Boundary value problems for the equations of mathematical physics of odd order], Novosibirsk State Univ., Novosibirsk, 1990, 132 pp. (In Russian) · Zbl 0784.35001 |

[11] | [11] Ladyzhenskaya O. A., The boundary value problems of mathematical physics, Applied Mathematical Sciences, 49, Springer Verlag, New York etc., 1985, xxx+322 pp. · Zbl 0588.35003 · doi:10.1007/978-1-4757-4317-3 |

[12] | [12] Berezanskii Yu. M., Expansion in eigenfunctions of self-adjoint operators, Translations of Mathematical Monographs, 17, American Mathematical Society, Providence RI, 1968, ix+809 pp. · Zbl 0157.16601 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.