|1Anop, AV, 1Murach, AA |
1Institute of Mathematics of the NAS of Ukraine, Kyiv
|Dopov. Nac. akad. nauk Ukr. 2018, 3:3-11|
In an extended Sobolev scale, we investigate homogeneous elliptic differential equations, whose solutions satisfy general enough boundary conditions. This scale consists of isotropic Hilbertian Hörmander spaces for which the regularity index is an arbitrary function RO-varying at infinity in the sense of Avakumović. We establish theorems on the character of solvability of these equations and the local regularity (up to the boundary of the domain) of their solutions in the scale indicated. We give an explicit description of all Hilbert spaces that are interpolation ones for pairs of subspaces of Hilbert Sobolev spaces formed by solutions of a homogeneous elliptic equation.
|Keywords: elliptic equation, Fredholm operator, Hörmander space, interpolation space, regularity of a solution, Sobolev space|
- Agranovich, M. S. (1997). Elliptic boundary problems. Encycl. Math. Sci. Vol. 79. Partial differential equations, IX. Berlin: Springer. doi: https://doi.org/10.1007/978-3-662-06721-5_1
- Lions, J.-L. & Magenes, E. (1972). Non-Homogeneous boundary-value problems and applications. Vol. 1. New York, Heidelberg: Springer.
- Mikhailets, V. A. & Murach, A. A. (2014). Hörmander spaces, interpolation, and elliptic problems. Berlin, Boston: De Gruyter. doi: https://doi.org/10.1515/9783110296891
- Mikhailets, V.A. & Murach, A.A. (2013). Extended Sobolev scale and elliptic operators. Ukr. Math. J., 65, No. 3, pp. 435-447. doi: https://doi.org/10.1007/s11253-013-0787-5
- Hörmander, L. (1963). Linear partial differential operators. Berlin: Springer. doi: https://doi.org/10.1007/978-3-642-46175-0
- Seneta, E. (1976). Regularly varying functions. Berlin: Springer. doi: https://doi.org/10.1007/BFb0079658
- Mikhailets, V. A. & Murach, A. A. (2006). Refined scales of spaces and elliptic boundary-value problems. II. Ukr. Math. J., 58, No. 3, pp. 398-417. doi: https://doi.org/10.1007/s11253-006-0074-9
- Anop, A. V. & Kasirenko, T. M. (2016). Elliptic boundary-value problems in Hörmander spaces. Methods Funct. Anal. Topology, 22, No. 4, pp. 295-310.
- Mikhailets, V. A. & Murach, A. A. (2006). Regular elliptic boundary-value problem for homogeneous equation in two-sided refined scale of spaces. Ukr. Math. J., 58, No. 11, pp. 1748-1767. doi: https://doi.org/10.1007/s11253-006-0166-6
- Chepurukhina, I. S. (2015). A semihomogeneous elliptic problem with additional unknown functions in boundary conditions. Dopov. Nac. akad. nauk. Ukr., No. 7, pp. 20-28 (in Ukrainian). doi: https://doi.org/10.15407/dopovidi2015.07.020
- Mikhailets, V. A. & Murach, A. A. (2012). The refined Sobolev scale, interpolation, and elliptic problems. Banach J. Math. Anal., 6, No. 2., pp. 211-281. doi: https://doi.org/10.15352/bjma/1342210171
- Kozlov, V. A., Maz'ya, V. G. & Rossmann, J. (1997). Elliptic boundary value problems in domains with point singularities. Providence: Amer. Math. Soc.
- Hörmander, L. (1983). The analysis of linear partial differential operators, vol. II, Differential operators with constant coefficients. Berlin: Springer.
- Volevich, L. R. & Paneah B. P. (1965). Certain spaces of generalized functions and embedding theorems. Russ. Math. Surv., 20, No. 1, pp. 1-73. doi: https://doi.org/10.1070/RM1965v020n01ABEH004139
- Mikhailets, V. A. & Murach, A. A. (2015). Interpolation Hilbert spaces between Sobolev spaces. Results Math. 67, No. 1, pp. 135-152. doi: https://doi.org/10.1007/s00025-014-0399-x