## Banach Journal of Mathematical Analysis

### The refined Sobolev scale, interpolation and elliptic problems

#### Abstract

The paper gives a detailed survey of recent results on elliptic problems in Hilbert spaces of generalized smoothness. The latter are the isotropic Hörmander spaces $H^{s,\varphi}:=B_{2,\mu}$, with $\mu(\xi)=\langle\xi\rangle^{s}\varphi(\langle\xi\rangle)$ for $\xi\in\mathbb{R}^{n}$. They are parametrized by both the real number $s$ and the positive function $\varphi$ varying slowly at $+\infty$ in the Karamata sense. These spaces form the refined Sobolev scale, which is much finer than the Sobolev scale $\{H^{s}\}\equiv\{H^{s,1}\}$ and is closed with respect to the interpolation with a function parameter. The Fredholm property of elliptic operators and elliptic boundary-value problems is preserved for this new scale. Theorems of various type about a solvability of elliptic problems are given. A~local refined smoothness is investigated for solutions to elliptic equations. New sufficient conditions for the solutions to have continuous derivatives are found. Some applications to the spectral theory of elliptic operators are given.

#### Article information

Source
Banach J. Math. Anal. Volume 6, Number 2 (2012), 211-281.

Dates
First available in Project Euclid: 13 July 2012

https://projecteuclid.org/euclid.bjma/1342210171

Digital Object Identifier
doi:10.15352/bjma/1342210171

Mathematical Reviews number (MathSciNet)
MR2945999

Zentralblatt MATH identifier
1258.46014

Subjects
Primary: 46E35
Secondary: 35J40: Boundary value problems for higher-order elliptic equations

#### Citation

Mikhailets , Vladimir A.; Murach, Aleksandr A. The refined Sobolev scale, interpolation and elliptic problems. Banach J. Math. Anal. 6 (2012), no. 2, 211--281. doi:10.15352/bjma/1342210171. https://projecteuclid.org/euclid.bjma/1342210171