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

We study elliptic boundary-value problems in improved scales of functional Hilbert spaces on smooth manifolds with boundary. The isotropic Hörmander-Volevich-Paneyakh spaces are elements of these scales. The local smoothness of a solution of an elliptic problem in an improved scale is investigated. We establish a sufficient condition under which this solution is classical. Elliptic boundary-value problems with parameter are also studied.     

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

We study improved scales of functional Hilbert spaces over ?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

We study improved scales of functional Hilbert spaces over ?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.     

Elliptic pseudodifferential operators in the improved scale of spaces on a closed manifold

We study linear elliptic pseudodifferential operators in the improved scale of functional Hilbert spaces on a smooth closed manifold. Elements of this scale are isotropic Hörmander-Volevich-Paneyakh spaces. We investigate the local smoothness of a solution of an elliptic equation in the improved scale. We also study elliptic pseudodifferential operators with parameter.     

Approximation Properties of Solutions to Multipoint Boundary-Value Problems

Ukrainian Mathematical Journal - We consider a broad class of linear boundary-value problems for systems of m ordinary differential equations of order r known as general boundary-value problems....     

Protein synthesis in murine organs during postimplantation development detected by two-dimensional gel electrophoresis.

Mouse embryos were isolated from the uterus on days 10 to 11 of gestation and incubated in Dulbecco's modified Eagle's medium (DMEM) with [35S]methionine for 4 h. Subsequently, their hearts and the brains were dissected. The brain was divided into three parts, containing the telencephalon, mesencephalon, and myelencephalon. These tissues were then processed for two-dimensional (2-D) gel electrophoresis. Protein synthesis of the isolated tissues was analyzed for organ-and cell lineage-specific patterns. We studied proteins with isoelectric points (pI) ranging from 4 to 10 and relative molecular weights (M(r)) varying from 10000 to 200000 and found several significant quantitative and qualitative differences between the tissues and the developmental stages analyzed. In particular, we were able to distinguish between protein spots that we now attribute putatively to the corresponding embryonic organs. These differences may reflect some of the organ- and cell lineage-specific changes in protein synthesis and gene expression during early mammalian differentiation.     

Petrovskii elliptic systems in the extended Sobolev scale

Petrovskii elliptic systems of linear differential equations given on a closed smooth manifold are investigated in the extended Sobolev scale. This scale consists of all Hilbert spaces that are interpolation spaces with respect to the Hilbert Sobolev scale. Theorems of solvability of the elliptic systems in the extended Sobolev scale are proved. An a priori estimate for solutions is obtained, and their local regularity is studied.     

Elliptic Operators in a Refined Scale of Functional Spaces

We study the theory of elliptic boundary-value problems in a refined two-sided scale of the Hormander spaces H ^s , ^ϕ, where s ∈ R and ϕ is a functional parameter slowly varying at +∞. For the Sobolev spaces H ^s , the function ϕ(|ξ|) ≡ 1. We establish that the considered operators possess the Fredholm property, and solutions are globally and locally regular. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 5, pp. 689–696, May, 2005.