Improved scales of spaces and elliptic boundary-value problems. I

Abstract

We study improved scales of functional Hilbert spaces over R n and smooth manifolds with boundary. The isotropic Hörmander-Volevich-Paneyakh spaces are elements of these scales. The theory of elliptic boundary-value problems in these spaces is developed.

Improved scales of spaces and elliptic boundary-value problems. II

“…For elliptic boundary-value problems, solvability theorems and estimates for solutions in various classes of functional spaces are known (see [2,4,[8][9][10][11][13][14][15] and the survey [12]). We need the statement on the solvability of the elliptic boundary-value problem (4.1) in the spaces of Bessel potentials presented below (see [14, pp.…”

“…4, also by the method of interpolation, we establish a theorem on the Noether property of the operator of the elliptic boundaryvalue problem in the improved scale of spaces of differentiable functions on a manifold. Sections 1 and 2 were published in the first part of the paper (see [2]). …”

“…Suppose that s > m + 12 and ϕ ∈ M. Then mapping(4.2) can be extended by continuity to the bounded Noetherian operatorΛ : H s, ϕ (Ω) → H s, ϕ = H s−2k, ϕ (Ω) × k j=1 H s−m j −1/2, ϕ (Γ) (4.8)with kernel N and cokernel N * , which, by virtue of Proposition 4.1, are independent of s and ϕ and satisfy(4.4). The restriction of operator(4.8) to subspace(4.5) is performed by the topological isomorphism Λ : P ( H s, ϕ (Ω) ) ↔ Q( H s, ϕ ) (4.9)…”

Elliptic operators on refined Sobolev scales on vector bundles

“…Among these spaces, Mikhailets and Murach [11][12][13] selected the class of inner product spaces H s;' WD B 2; parametrized with the function . / D h i s '.h i/, where s 2 R, the function ' W OE1; 1/ !…”

“…It contains the inner product Sobolev spaces H s D H s;1 and is obtained by the interpolation with a function parameter between these spaces. This interpolation property allowed Mikhailets and Murach [11][12][13][16][17][18][19][20][21] to build the theory of solvability of general elliptic systems and elliptic boundary-value problems on the refined Sobolev scale. Their theory [7] is supplemented in [22][23][24][25][26][27][28][29] for a more extensive class of Hörmander inner product spaces.…”