^{1}Murach, AA^{1}Chepurukhina, IS^{1}Institute of Mathematics of the NAS of Ukraine, Kyiv |

Dopov. Nac. akad. nauk Ukr. 2020, 8:3-10 |

https://doi.org/10.15407/dopovidi2020.08.003 |

Section: Mathematics |

Language: Ukrainian |

Abstract: We consider elliptic problems with nonclassical boundary conditions that contain additional unknown functions on the border of the domain of definition of the elliptic equation and also contain boundary operators of higher orders with respect to the order of this equation. We investigate the solvability of the indicated problems and properties of their solutions in an extended Sobolev scale. It consists of Hilbert generalized Sobolev spaces for which the order of regularity is a general radial function RO-varying in the sense of Avakumović at infinity. We establish a theorem on the Fredholm property of the indicated problems on appropriate pairs of these spaces and theorems on the regularity and the a priori estimate of generalized solutions to the problems. We obtain exact sufficient conditions for components of these solutions to be continuously differentiable. |

Keywords: a priori estimate, elliptic boundary-value problem, Fredholm operator, generalized Sobolev space, regularity of a solution |

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