Hörmander Spaces, Interpolation, and Elliptic Problems
1st Edition
Published on 14. July 2014
XII, 297 pages
978-3-11-036906-9 (ISBN)
Description
The monograph gives a detailed exposition of the theory of general elliptic operators (scalar and matrix) and elliptic boundary value problems in Hilbert scales of Hörmander function spaces. This theory was constructed by the authors in a number of papers published in 2005-2009. It is distinguished by a systematic use of the method of interpolation with a functional parameter of abstract Hilbert spaces and Sobolev inner product spaces. This method, the theory and their applications are expounded for the first time in the monographic literature.
The monograph is written in detail and in a reader-friendly style. The complete proofs of theorems are given. This monograph is intended for a wide range of mathematicians whose research interests concern with mathematical analysis and differential equations.
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Vladimir A. Mikhailets | Aleksandr A. Murach
Hörmander Spaces, Interpolation, and Elliptic Problems
Book
07/2014
1st Edition
De Gruyter
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Vladimir A. Mikhailets | Aleksandr A. Murach
Hörmander Spaces, Interpolation, and Elliptic Problems
E-Book
07/2014
1st Edition
De Gruyter
€164.95
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Vladimir A. Mikhailets | Aleksandr A. Murach
Hörmander Spaces, Interpolation, and Elliptic Problems
Book
04/2014
1st Edition
De Gruyter
€164.95
Shipment within 7-9 days
Persons
V. A. Mikhailets and A. A. Murach, Institute of Mathematics of NAS of Ukraine, Kiev, Ukraine.
Content
- Intro
- Preface
- Preface to the English edition
- Acknowledgements
- Introduction
- 1 Interpolation and Hörmander spaces
- 1.1 Interpolation with function parameter
- 1.1.1 Definition of interpolation
- 1.1.2 Embeddings of spaces
- 1.1.3 Reiteration property
- 1.1.4 Interpolation of dual spaces
- 1.1.5 Interpolation of orthogonal sums of spaces
- 1.1.6 Interpolation of subspaces and factor spaces
- 1.1.7 Interpolation of Fredholm operators
- 1.1.8 Estimate of the operator norm in interpolation spaces
- 1.1.9 Criterion for a function to be an interpolation parameter
- 1.2 Regularly varying functions and their generalization
- 1.2.1 Regularly varying functions
- 1.2.2 Quasiregularly varying functions
- 1.2.3 Auxiliary results
- 1.3 Hörmander spaces and the refined Sobolev scale
- 1.3.1 Preliminary information and notation
- 1.3.2 Hörmander spaces
- 1.3.3 Refined Sobolev scale
- 1.3.4 Properties of the refined scale
- 1.4 Uniformly elliptic operators on the refined scale
- 1.4.1 Pseudodifferential operators
- 1.4.2 A priori estimate of the solutions
- 1.4.3 Smoothness of the solutions
- 1.5 Remarks and comments
- 2 Hörmander spaces on closed manifolds and their applications
- 2.1 Hörmander spaces on closed manifolds
- 2.1.1 Equivalent definitions
- 2.1.2 Interpolation properties
- 2.1.3 Equivalent norms
- 2.1.4 Embedding theorem
- 2.2 Elliptic operators on closed manifolds
- 2.2.1 Pseudodifferential operators on closed manifolds
- 2.2.2 Elliptic operators on the refined scale
- 2.2.3 Smoothness of solutions to the elliptic equation
- 2.2.4 Parameter-elliptic operators
- 2.3 Convergence of spectral expansions
- 2.3.1 Convergence almost everywhere for general orthogonal series
- 2.3.2 Convergence almost everywhere for spectral expansions
- 2.3.3 Convergence of spectral expansions in the metric of the space Ck
- 2.4 RO-varying functions and Hörmander spaces
- 2.4.1 RO-varying functions in the sense of Avakumovic
- 2.4.2 Interpolation spaces for a pair of Sobolev spaces
- 2.4.3 Applications to elliptic operators
- 2.5 Remarks and comments
- 3 Semihomogeneous elliptic boundary-value problems
- 3.1 Regular elliptic boundary-value problems
- 3.1.1 Definition of the problem
- 3.1.2 Formally adjoint problem
- 3.2 Hörmander spaces for Euclidean domains
- 3.2.1 Spaces for open domains
- 3.2.2 Spaces for closed domains
- 3.2.3 Rigging of L2(W) with Hörmander spaces
- 3.3 Boundary-value problems for homogeneous elliptic equations
- 3.3.1 Main result: boundedness and Fredholm property of the operator
- 3.3.2 A theorem on interpolation of subspaces
- 3.3.3 Elliptic boundary-value problem in Sobolev spaces
- 3.3.4 Proof of the main result
- 3.3.5 Properties of solutions to the homogeneous elliptic equation
- 3.4 Elliptic problems with homogeneous boundary conditions
- 3.4.1 Theorem on isomorphisms for elliptic operators
- 3.4.2 Interpolation and homogeneous boundary conditions
- 3.4.3 Proofs of theorems on isomorphisms and the Fredholm property
- 3.4.4 Local increase in smoothness of solutions up to the boundary
- 3.5 Some properties of Hörmander spaces
- 3.5.1 Space Hs
- ' 0 (W) and its properties
- 3.5.2 Equivalent description of Hs
- '(W)
- 3.6 Remarks and comments
- 4 Inhomogeneous elliptic boundary-value problems
- 4.1 Elliptic boundary-value problems in the positive one-sided scale
- 4.1.1 Theorems on Fredholm property and isomorphisms
- 4.1.2 Smoothness of the solutions up to the boundary
- 4.1.3 Nonregular elliptic boundary-value problems
- 4.1.4 Parameter-elliptic boundary-value problems
- 4.1.5 Formally mixed elliptic boundary-value problem
- 4.2 Elliptic boundary-value problems in the two-sided scale
- 4.2.1 Preliminary remarks
- 4.2.2 The refined scale modified in the sense of Roitberg
- 4.2.3 Roitberg-type theorems on solvability. The complete collection of isomorphisms
- 4.2.4 Smoothness of generalized solutions up to the boundary
- 4.2.5 Interpolation in the modified refined scale
- 4.3 Some properties of the modified refined scale
- 4.3.1 Statement of results
- 4.3.2 Proof of results
- 4.4 Generalization of the Lions-Magenes theorems
- 4.4.1 Lions-Magenes theorems
- 4.4.2 Key individual theorem
- 4.4.3 Individual theorem for Sobolev spaces
- 4.4.4 Individual theorem for weight spaces
- 4.5 Hörmander spaces and individual theorems on solvability
- 4.5.1 Key individual theorem for the refined scale
- 4.5.2 Other individual theorems
- 4.6 Remarks and Comments
- 5 Elliptic systems
- 5.1 Uniformly elliptic systems in the refined Sobolev scale
- 5.1.1 Uniformly elliptic systems
- 5.1.2 A priori estimate for the solutions of the system
- 5.1.3 Smoothness of solutions
- 5.2 Elliptic systems on a closed manifold
- 5.2.1 Elliptic Systems
- 5.2.2 Operator of the elliptic system on the refined scale
- 5.2.3 Local smoothness of solutions
- 5.2.4 Parameter-elliptic systems
- 5.3 Elliptic boundary-value problems for systems of equations
- 5.3.1 Statement of the problem
- 5.3.2 Theorem on solvability
- 5.4 Remarks and comments
- Bibliography
- Index
