Parameter-elliptic problems and interpolation with a function parameter. (English) Zbl 1313.35092

The extended Sobolev scale consists of all those Hilbert spaces that are interpolation spaces with respect to the classical Sobolev scale. These spaces are the Hörmander spaces \(B_{2,k}\) for which the smoothness index \(k\) is an arbitrary radial function RO-varying at infinity; see the papers by A. A. Murach [Ukr. Math. J. 61, No. 3, 467–477 (2009; Zbl 1224.35101)]; V. A. Mikhailets and A. A. Murach [Ukr. Math. J. 65, No. 3, 435–447 (2013; Zbl 1294.46036)] devoted to elliptic operators and systems considered on this scale.
In the paper under review, the authors investigate parameter-elliptic boundary value problems on the extended scale. It is proved that the corresponding operators are isomorphisms provided the absolute value of the parameter is large enough. Two-sided estimates of solutions are obtained.


35J40 Boundary value problems for higher-order elliptic equations
46B70 Interpolation between normed linear spaces
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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