Banach Journal of Mathematical Analysis

The refined Sobolev scale, interpolation and elliptic problems

Vladimir A. Mikhailets and Aleksandr A. Murach

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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

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

First available in Project Euclid: 13 July 2012

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Sobolev scale Hörmander interpolation with function parameter elliptic operator elliptic boundary-value problem


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.

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