Elliptic problems with nonclassical coundary conditions in an extended Sobolev scale

TitleElliptic problems with nonclassical coundary conditions in an extended Sobolev scale
Publication TypeJournal Article
Year of Publication2020
AuthorsMurach, AA, Chepurukhina, IS
Abbreviated Key TitleDopov. Nac. akad. nauk Ukr.
Date Published8/2020

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.

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

1. Lions, J.-L. & Magenes, E. (1972). Non-homogeneous boundary-value problems and applications, vol. I. Berlin: Springer.
2. Lawruk, B. (1963). Parametric boundary-value problems for elliptic systems of linear differential equations. I. Construction of conjugate problems. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 11, No. 5, pp. 257-267 (in Russian).
3. Kozlov, V.A., Maz’ya, V.G. & Rossmann, J. (1997). Elliptic boundary value problems in domains with point singularities. Providence: Amer. Math. Soc.
4. Roitberg, Ya.A. (1999). Elliptic boundary value problems in the spaces of distributions. Dordrecht: Kluwer.
5. Mikhailets, V.A. & Murach, A.A. (2013). Extended Sobolev scale and elliptic operators. Ukr. Math. J., 65, No. 3, pp. 435-447.
6. Mikhailets, V.A. & Murach, A.A. (2014). Hörmander spaces, interpolation, and elliptic problems. Berlin, Boston: De Gruyter.
7. Seneta, E. (1976). Regularly varying functions. Berlin: Springer.
8. Bingham, N.H., Goldie, C.M. & Teugels, J.L. (1989). Regular variation. Cambridge: Cambridge University Press.
9. Hörmander, L. (1963). Linear partial differential operators. Berlin: Springer.
10. Volevich, L.R. & Paneah, B.P. (1965). Certain spaces of generalized functions and embedding theorems. Russian Math. Surveys, 20, No. 1, pp. 1-73.
11. Mikhailets, V.A. & Murach, A.A. (2015). Interpolation Hilbert spaces between Sobolev spaces. Results Math., 67, No. 1, pp. 135-152. https://doi.org/10.1007/s00025-014-0399-x
12. Kasirenko, T.M. & Chepurukhina, I.S. (2017). Elliptic problems in the sense of Lawruk with boundary operators of higher orders in refined Sobolev scale. Zbirnyk Prats Institutu Matematyky NAN Ukrainy, 14, No. 3, pp. 161-203 (in Ukrainian).
13. Kasirenko, T.M. & Murach, O.O. (2018). Elliptic problems with boundary conditions of higher orders in Hörmander spaces. Ukr. Math. J., 69, No. 11, pp. 1727-1748.
14. Chepurukhina, I.S. (2015). Elliptic boundary-value problems in the sense of Lawruk in an extended Sobolev scale. Zbirnyk Prats Institutu Matematyky NAN Ukrainy, 12, No. 2, pp. 338-374 (in Ukrainian).
15. Anop, A.V. (2019). Lawruk elliptic boundary-value problems for homogeneous differential equations. Dopov. Nac. akad. nauk Ukr., No. 2, pp. 3-11 (in Ukrainian). https://doi.org/10.15407/dopovidi2019.02.003