A general elliptic boundary-value problem in the extended Sobolev scale
DOI:
https://doi.org/10.15407/dopovidi2014.04.007Keywords:
elliptic boundary-value problem, Sobolev scaleAbstract
A general elliptic boundary-value problem given in a bounded Euclidean domain with smooth boundary is investigated in the extended Sobolev scale. The latter consists of all Hilbert spaces that are interpolation spaces for pairs of inner product Sobolev spaces. We prove that the operator of the problem is bounded and Fredholm in appropriate pairs of Hörmander spaces, which belong to the extended Sobolev scale. A priori estimates for the solutions to the problem are established, and their regularity in H¨ormander spaces is investigated. We find a new sufficient condition, under which a generalized solution to the problem is classical.
Downloads
References
Berezansky Yu. M. Expansion in eigenfunctions of self-adjoint operators. Kyiv: Nauk. dumka, 1965 (in Russian).
Egorov Yu. V. Linear differential equations of the main type. Moscow: Nauka, 1984 (in Russian).
Hormander L. Linear partial differential operators. Moscow: Mir, 1965 (in Russian).
Hormander L. Analysis of linear differential operators with partial derivatives. In 4 vols. Vol. 3. Pseudodifferential operators. Moscow: Mir, 1987 (in Russian).
Mikhailets V. A., Murach A. A. Hörmander spaces, interpolation and elliptic problems. Kyiv: Institute of Mathematics of NAS of Ukr., 2010 Retrieved from arXiv:1106.3214 (in Russian).
Mikhailets V. A., Murach A. A. Banach J. Math. Anal., 2012, 6, No. 2: 211–281. https://doi.org/10.15352/bjma/1342210171
Mikhailets V. A., Murach A. A. Interpolation Hilbert spaces for a couple of Sobolev spaces. Retrieved from arXiv:1106.2049v2.
Mikhailets V. A., Murach A. A. Ukr. mat. zhurn., 2013, 65, No. 3: 368–380 (in Russian).
Ovchinnikov V. I. Math. Rep. Ser. 1., 1984, No. 2: 349–515.
Zinchenko T. N., Murach A. A. Ukr. mat. zhurn., 2012, 64, No. 11: 1477–1491 (in Russian).
Zinchenko T. N. Dopov. Nac. akad. nauk Ukr., 2013, No. 3: 14–20 (in Russian).
Agranovich M. S. Elliptic boundary problems. In: Encycl. Math. Sci. Vol. 79. Partial differential equations, IX. Berlin: Springer, 1997: 1–144. https://doi.org/10.1007/978-3-662-06721-5_1
Seneta E. Correctly changing functions. Moscow: Nauka, 1985 (in Russian).
Volevich L. R., Paneyakh B. P. Uspekhi mat. nauk, 1965, 20, No. 1: 3–74 (in Russian).
Slenzak G. Moscow Univ. Math. Bull., 1974, 29, No. 3–4: 80–88.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2025 Reports of the National Academy of Sciences of Ukraine
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
