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 Hormander 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.
Preface v
Preface to the English edition vi
Acknowledgements vii
Introduction 1 (8)
1 Interpolation and Hormander spaces 9 (50)
1.1 Interpolation with function parameter 9 (20)
1.1.1 Definition of interpolation 9 (2)
1.1.2 Embeddings of spaces 11 (2)
1.1.3 Reiteration property 13 (2)
1.1.4 Interpolation of dual spaces 15 (3)
1.1.5 Interpolation of orthogonal sums 18 (2)
of spaces
1.1.6 Interpolation of subspaces and 20 (1)
factor spaces
1.1.7 Interpolation of Fredholm 21 (2)
operators
1.1.8 Estimate of the operator norm in 23 (2)
interpolation spaces
1.1.9 Criterion for a function to be an 25 (4)
interpolation parameter
1.2 Regularly varying functions and their 29 (9)
generalization
1.2.1 Regularly varying functions 29 (2)
1.2.2 Quasiregularly varying functions 31 (5)
1.2.3 Auxiliary results 36 (2)
1.3 Hormander spaces and the refined 38 (9)
Sobolev scale
1.3.1 Preliminary information and 38 (2)
notation
1.3.2 Hormander spaces 40 (2)
1.3.3 Refined Sobolev scale 42 (2)
1.3.4 Properties of the refined scale 44 (3)
1.4 Uniformly elliptic operators on the 47 (8)
refined scale
1.4.1 Pseudodifferential operators 47 (3)
1.4.2 A priori estimate of the solutions 50 (1)
1.4.3 Smoothness of the solutions 51 (4)
1.5 Remarks and comments 55 (4)
2 Hormander spaces on closed manifolds and 59 (52)
their applications
2.1 Hormander spaces on closed manifolds 59 (19)
2.1.1 Equivalent definitions 59 (2)
2.1.2 Interpolation properties 61 (7)
2.1.3 Equivalent norms 68 (9)
2.1.4 Embedding theorem 77 (1)
2.2 Elliptic operators on closed manifolds 78 (15)
2.2.1 Pseudodifferential operators on 79 (2)
closed manifolds
2.2.2 Elliptic operators on the refined 81 (3)
scale
2.2.3 Smoothness of solutions to the 84 (2)
elliptic equation
2.2.4 Parameter-elliptic operators 86 (7)
2.3 Convergence of spectral expansions 93 (5)
2.3.1 Convergence almost everywhere for 93 (2)
general orthogonal series
2.3.2 Convergence almost everywhere for 95 (2)
spectral expansions
2.3.3 Convergence of spectral 97 (1)
expansions in the metric of the space Ck
2.4 RO-varying functions and Hormander 98 (10)
spaces
2.4.1 RO-varying functions in the sense 98 (2)
of Avakumovic
2.4.2 Interpolation spaces for a pair 100(7)
of Sobolev spaces
2.4.3 Applications to elliptic operators 107(1)
2.5 Remarks and comments 108(3)
3 Semihomogeneous elliptic boundary-value 111(54)
problems
3.1 Regular elliptic boundary-value 111(3)
problems
3.1.1 Definition of the problem 111(2)
3.1.2 Formally adjoint problem 113(1)
3.2 Hormander spaces for Euclidean domains 114(12)
3.2.1 Spaces for open domains 115(5)
3.2.2 Spaces for closed domains 120(3)
3.2.3 Rigging of L2(ω) with 123(3)
Hormander spaces
3.3 Boundary-value problems for 126(16)
homogeneous elliptic equations
3.3.1 Main result: boundedness and 126(1)
Fredholm property of the operator
3.3.2 A theorem on interpolation of 127(4)
subspaces
3.3.3 Elliptic boundary-value problem 131(3)
in Sobolev spaces
3.3.4 Proof of the main result 134(5)
3.3.5 Properties of solutions to the 139(3)
homogeneous elliptic equation
3.4 Elliptic problems with homogeneous 142(16)
boundary conditions
3.4.1 Theorem on isomorphisms for
elliptic operators .142
3.4.2 Interpolation and homogeneous 146(6)
boundary conditions
3.4.3 Proofs of theorems on 152(4)
isomorphisms and the Fredholm property
3.4.4 Local increase in smoothness of 156(2)
solutions up to the boundary
3.5 Some properties of Hormander spaces 158(4)
3.5.1 Space H0S,φ(ω) and its 158(2)
properties
3.5.2 Equivalent description of 160(2)
HS,φ(ω)
3.6 Remarks and comments 162(3)
4 Inhomogeneous elliptic boundary-value 165(86)
problems
4.1 Elliptic boundary-value problems in 165(23)
the positive one-sided scale
4.1.1 Theorems on Fredholm property and 165(4)
isomorphisms
4.1.2 Smoothness of the solutions up to 169(4)
the boundary
4.1.3 Nonregular elliptic 173(2)
boundary-value problems
4.1.4 Parameter-elliptic boundary-value 175(11)
problems
4.1.5 Formally mixed elliptic 186(2)
boundary-value problem
4.2 Elliptic boundary-value problems in 188(22)
the two-sided scale
4.2.1 Preliminary remarks 188(1)
4.2.2 The refined scale modified in the 189(10)
sense of Roitberg
4.2.3 Roitberg-type theorems on 199(5)
solvability. The complete collection of
isomorphisms
4.2.4 Smoothness of generalized 204(3)
solutions up to the boundary
4.2.5 Interpolation in the modified 207(3)
refined scale
4.3 Some properties of the modified 210(16)
refined scale
4.3.1 Statement of results 210(2)
4.3.2 Proof of results 212(14)
4.4 Generalization of the Lions-Magenes 226(17)
theorems
4.4.1 Lions-Magenes theorems 227(3)
4.4.2 Key individual theorem 230(6)
4.4.3 Individual theorem for Sobolev 236(2)
spaces
4.4.4 Individual theorem for weight 238(5)
spaces
4.5 Hormander spaces and individual 243(4)
theorems on solvability
4.5.1 Key individual theorem for the 243(1)
refined scale
4.5.2 Other individual theorems 244(3)
4.6 Remarks and Comments 247(4)
5 Elliptic systems 251(24)
5.1 Uniformly elliptic systems in the 251(6)
refined Sobolev scale
5.1.1 Uniformly elliptic systems 251(1)
5.1.2 A priori estimate for the 252(1)
solutions of the system
5.1.3 Smoothness of solutions 253(4)
5.2 Elliptic systems on a closed manifold 257(11)
5.2.1 Elliptic Systems 257(1)
5.2.2 Operator of the elliptic system 258(4)
on the refined scale
5.2.3 Local smoothness of solutions 262(2)
5.2.4 Parameter-elliptic systems 264(4)
5.3 Elliptic boundary-value problems for 268(4)
systems of equations
5.3.1 Statement of the problem 269(2)
5.3.2 Theorem on solvability 271(1)
5.4 Remarks and comments 272(3)
Bibliography 275(16)
Index 291
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