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# Parameter-elliptic problems and interpolation with a function parameter

### Abstract

Parameter-elliptic boundary-value problems are investigated on the extended Sobolev scale. This scale consists of all Hilbert spaces that are interpolation spaces with respect to a Hilbert Sobolev scale. The latter are the Hörmander spaces $B_{2,k}$ for which the smoothness index $k$ is an arbitrary radial function RO-varying at $+\infty$. We prove that the operator corresponding to this problem sets isomorphisms between appropriate Hörmander spaces provided the absolute value of the parameter is large enough. For solutions to the problem, we establish two-sided estimates, in which the constants are independent of the parameter.

Key words: Parameter-elliptic boundary-value problem, interpolation with a function parameter, RO-varying function, Hörmander space, extended Sobolev scale, isomorphism property, a priori estimate for solutions.

### Article Information

 Title Parameter-elliptic problems and interpolation with a function parameter Source Methods Funct. Anal. Topology, Vol. 20 (2014), no. 2, 103-116 MathSciNet MR3242859 zbMATH 1313.35092 Milestones Received 08/12/2013 Copyright The Author(s) 2014 (CC BY-SA)

### Authors Information

Anna V. Anop
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs'ka, Kyiv, 01601, Ukraine Aleksandr

Aleksandr A. Murach
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs'ka, Kyiv, 01601, Ukraine

### Citation Example

Anna V. Anop and Aleksandr A. Murach, Parameter-elliptic problems and interpolation with a function parameter, Methods Funct. Anal. Topology 20 (2014), no. 2, 103-116.

### BibTex

@article {MFAT735,
AUTHOR = {Anop, Anna V. and Murach, Aleksandr A.},
TITLE = {Parameter-elliptic problems and interpolation with a function parameter},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {20},
YEAR = {2014},
NUMBER = {2},
PAGES = {103-116},
ISSN = {1029-3531},
MRNUMBER = {MR3242859},
ZBLNUMBER = {1313.35092},
URL = {http://mfat.imath.kiev.ua/article/?id=735},
}