Hörmander spaces on manifolds, and their application to elliptic boundaryvalue problems

1Kasirenko, TM, 1Murach, AA, 1Chepurukhina, IS
1Institute of Mathematics of the NAS of Ukraine, Kyiv
Dopov. Nac. akad. nauk Ukr. 2019, 3:9-16
https://doi.org/10.15407/dopovidi2019.03.009
Section: Mathematics
Language: Ukrainian
Abstract: 

We introduce an extended Sobolev scale on a smooth compact manifold with boundary. The scale is formed by innerproduct Hörmander spaces, for which a radial function ROvarying in the sense of Avakumovic serves as a regularity index. These spaces do not depend on a choice of local charts on the manifold. The scale consists of all Hilbert spaces that are interpolation ones for pairs of innerproduct Sobolev spaces, is obtained by the interpolation with a function parameter of these pairs, and is closed with respect to this interpolation. As an application of the scale introduced, we give a theorem on the Fredholm property of a general elliptic bounda ryvalue problem on appropriate Hörmander spaces and find sufficient conditions, under which its generalized solutions belong to the space of p…0 times continuously differentiable functions.

Keywords: elliptic boundaryvalue problem., extended Sobolev scale, Hörmander space, interpolation between spaces, interpolation space
References: 

1. Hörmander, L. (1963). Linear partial differential operators. Berlin: Springer. doi: https://doi.org/10.1007/978-3-642-46175-0
2. Hörmander, L. (1983). The analysis of linear partial differential operators. (Vol. 2). Differential operators with constant coefficients. Berlin: Springer.
3. Jacob, N. (2001, 2002, 2005). Pseudodifferential operators and Markov processes (in 3 vols). London: Imperial College Press.
4. Nicola, F. & Rodino, L. (2010). Global Pseudodifferential Calculus on Euclidean spaces. Basel: Birkhäuser. doi: https://doi.org/10.1007/978-3-7643-8512-5
5. Mikhailets, V. A. & Murach, A. A. (2014). Hörmander spaces, interpolation, and elliptic problems. Berlin, Boston: De Gruyter.
6. Mikhailets, V. A. & Murach, A. A. (2009). Elliptic operators on a closed compact manifold. Dopov. Nac. akad. nauk. Ukr., No. 3, pp. 13-19 (in Russian).
7. 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
8. 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
9. Avakumovic, V. G. (1936). O jednom Oinverznom stavu. Rad. Jugoslovenske Akad. Znatn. Umjetnosti, 254, pp. 167-186.
10. Seneta, E. (1976). Regularly varying functions. Berlin: Springer. doi: https://doi.org/10.1007/BFb0079658
11. Matuszewska, W. (1964). On a generalization of regularly increasing functions. Studia Math., 24, pp. 271-279. doi: https://doi.org/10.4064/sm-24-3-271-279
12. 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
13. 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
14. Foias, C. & Lions, J.L. (1961). Sur certains théorèmes d’interpolation. Acta Sci. Math. (Szeged), 22, No. 34, pp. 269-282.
15. Agranovich, M.S. (1997). Elliptic boundary problems. In Encyclopaedia of Mathematical Sciences (Vol. 79). Partial differential equations, IX (pp. 1-144). Berlin: Springer. doi: https://doi.org/10.1007/978-3-662-06721-5_1