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# Elliptic problems with boundary operators of higher orders in Hörmander–Roitberg spaces

### Abstract

We investigate elliptic boundary-value problems for which the maximum of the orders of the boundary operators is equal to or greater than the order of the elliptic differential equation. We prove that the operator corresponding to an arbitrary problem of this kind is bounded and Fredholm between appropriate Hilbert spaces which form certain two-sided scales and are built on the base of isotropic Hörmander spaces. The differentiation order for these spaces is given by an arbitrary real number and positive function which varies slowly at infinity in the sense of Karamata. We establish a local a priori estimate for the generalized solutions to the problem and investigate their local regularity (up to the boundary) on these scales. As an application, we find sufficient conditions under which the solutions have continuous classical derivatives of a given order.

Key words: Elliptic problem, Hörmander space, slowly varying function, Fredholm property, generalized solution, a priori estimate, local regularity.

### Article Information

 Title Elliptic problems with boundary operators of higher orders in Hörmander–Roitberg spaces Source Methods Funct. Anal. Topology, Vol. 24 (2018), no. 2, 120-142 Milestones Received 09/01/2018 Copyright The Author(s) 2018 (CC BY-SA)

### Authors Information

Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs’ka str., Kyiv, 01601, Ukraine

Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs’ka str., Kyiv, 01601, Ukraine

### Citation Example

Tetiana Kasirenko and Aleksandr Murach, Elliptic problems with boundary operators of higher orders in Hörmander–Roitberg spaces, Methods Funct. Anal. Topology 24 (2018), no. 2, 120-142.

### BibTex

@article {MFAT1053,
AUTHOR = {Kasirenko, Tetiana and Murach, Aleksandr},
TITLE = {Elliptic problems with boundary operators of higher
orders in Hörmander–Roitberg spaces},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {24},
YEAR = {2018},
NUMBER = {2},
PAGES = {120-142},
ISSN = {1029-3531},
URL = {http://mfat.imath.kiev.ua/article/?id=1053},
}

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