 | Mikhailets V.A. Hörmander spaces, interpolation, and elliptic problems / V.A.Mikhailets, A.A.Murach. - Berlin; Boston: de Gruyter, 2014. - xii, 297 p. - (De Gruyter studies in mathematics; 60). - Bibliogr.: p.275-290. - Ind.: p.291-297. - ISBN 978-3-11-029685-3; ISSN 0179-0986
Шифр: (И/В16-M68) 02
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Preface ......................................................... v
Preface to the English edition ................................. vi
Acknowledgements .............................................. vii
Introduction .................................................... 1
1 Interpolation and Hörmander spaces ......................... 9
1.1 Interpolation with function parameter ...................... 9
1.1.1 Definition of interpolation ......................... 9
1.1.2 Embeddings of spaces ............................... 11
1.1.3 Reiteration property ............................... 13
1.1.4 Interpolation of dual spaces ....................... 15
1.1.5 Interpolation of orthogonal sums of spaces ......... 18
1.1.6 Interpolation of subspaces and factor spaces ....... 20
1.1.7 Interpolation of Fredholm operators ................ 21
1.1.8 Estimate of the operator norm in interpolation
spaces ............................................. 23
1.1.9 Criterion for a function to be an interpolation
parameter .......................................... 25
1.2 Regularly varying functions and their generalization ...... 29
1.2.1 Regularly varying functions ........................ 29
1.2.2 Quasiregularly varying functions ................... 31
1.2.3 Auxiliary results .................................. 36
1.3 Hörmander spaces and the refined Sobolev scale ............ 38
1.3.1 Preliminary information and notation ............... 38
1.3.2 Hörmander spaces ................................... 40
1.3.3 Refined Sobolev scale .............................. 42
1.3.4 Properties of the refined scale .................... 44
1.4 Uniformly elliptic operators on the refined scale ......... 47
1.4.1 Pseudodifferential operators ....................... 47
1.4.2 A priori estimate of the solutions ................. 50
1.4.3 Smoothness of the solutions ........................ 51
1.5 Remarks and comments ...................................... 55
2 Hörmander spaces on closed manifolds and their
applications .............................................. 59
2.1 Hörmander spaces on closed manifolds ...................... 59
2.1.1 Equivalent definitions ............................. 59
2.1.2 Interpolation properties ........................... 61
2.1.3 Equivalent norms ................................... 68
2.1.4 Embedding theorem .................................. 77
2.2 Elliptic operators on closed manifolds .................... 78
2.2.1 Pseudodifferential operators on closed manifolds ... 79
2.2.2 Elliptic operators on the refined scale ............ 81
2.2.3 Smoothness of solutions to the elliptic equation ... 84
2.2.4 Parameter-elliptic operators ....................... 86
2.3 Convergence of spectral expansions ........................ 93
2.3.1 Convergence almost everywhere for general
orthogonal series .................................. 93
2.3.2 Convergence almost everywhere for spectral
expansions ......................................... 95
2.3.3 Convergence of spectral expansions in the
metric of the space Ck ............................. 97
2.4 RO-varying functions and Hörmander spaces ................. 98
2.4.1 RO-varying functions in the sense of Avakumović .... 98
2.4.2 Interpolation spaces for a pair of Sobolev
spaces ............................................ 100
2.4.3 Applications to elliptic operators ................ 107
2.5 Remarks and comments ..................................... 108
3 Semihomogeneous elliptic boundary-value problems ......... 111
3.1 Regular elliptic boundary-value problems ................. 111
3.1.1 Definition of the problem ......................... 111
3.1.2 Formally adjoint problem .......................... 113
3.2 Hörmander spaces for Euclidean domains ................... 114
3.2.1 Spaces for open domains ........................... 115
3.2.2 Spaces for closed domains ......................... 120
3.2.3 Rigging of L2(Ω) with Hörmander spaces ............ 123
3.3 Boundary-value problems for homogeneous elliptic
equations ................................................ 126
3.3.1 Main result: boundedness and Fredholm property
of the operator ................................... 126
3.3.2 A theorem on interpolation of subspaces ........... 127
3.3.3 Elliptic boundary-value problem in Sobolev
spaces ............................................ 131
3.3.4 Proof of the main result .......................... 134
3.3.5 Properties of solutions to the homogeneous
elliptic equation ................................. 139
3.4 Elliptic problems with homogeneous boundary conditions ... 142
3.4.1 Theorem on isomorphisms for elliptic operators .... 142
3.4.2 Interpolation and homogeneous boundary
conditions ........................................ 146
3.4.3 Proofs of theorems on isomorphisms аnd the
Fredholm property ................................. 152
3.4.4 Local increase in smoothness of solutions up to
the boundary ...................................... 156
3.5 Some properties of Hörmander spaces ...................... 158
3.5.1 Space Н0s,φ(Ω) and its properties .................. 158
3.5.2 Equivalent description of Hs,φ,(Ω) ................. 160
3.6 Remarks and comments ..................................... 162
4 Inhomogeneous elliptic boundary-value problems ........... 165
4.1 Elliptic boundary-value problems in the positive one-
sided scale .............................................. 165
4.1.1 Theorems on Fredholm property and isomorphisms .... 165
4.1.2 Smoothness of the solutions up to the boundary .... 169
4.1.3 Nonregular elliptic boundary-value problems ....... 173
4.1.4 Parameter-elliptic boundary-value problems ........ 175
4.1.5 Formally mixed elliptic boundary-value problem .... 186
4.2 Elliptic boundary-value problems in the two-sided scale .. 188
4.2.1 Preliminary remarks ............................... 188
4.2.2 The refined scale modified in the sense of
Roitberg .......................................... 189
4.2.3 Roitberg-type theorems on solvability. The
complete collection of isomorphisms ............... 199
4.2.4 Smoothness of generalized solutions up to the
boundary .......................................... 204
4.2.5 Interpolation in the modified refined scale ....... 207
4.3 Some properties of the modified refined scale ............ 210
4.3.1 Statement of results .............................. 210
4.3.2 Proof of results .................................. 212
4.4 Generalization of the Lions-Magenes theorems ............. 226
4.4.1 Lions-Magenes theorems ............................ 227
4.4.2 Key individual theorem ............................ 230
4.4.3 Individual theorem for Sobolev spaces ............. 236
4.4.4 Individual theorem for weight spaces .............. 238
4.5 Hörmander spaces and individual theorems on solvability .. 243
4.5.1 Key individual theorem for the refined scale ...... 243
4.5.2 Other individual theorems ......................... 244
4.6 Remarks and Comments ..................................... 247
5 Elliptic systems ......................................... 251
5.1 Uniformly elliptic systems in the refined Sobolev scale .. 251
5.1.1 Uniformly elliptic systems ........................ 251
5.1.2 A priori estimate for the solutions of the
system ............................................ 252
5.1.3 Smoothness of solutions ........................... 253
5.2 Elliptic systems on a closed manifold .................... 257
5.2.1 Elliptic Systems .................................. 257
5.2.2 Operator of the elliptic system on the refined
scale ............................................. 258
5.2.3 Local smoothness of solutions ..................... 262
5.2.4 Parameter-elliptic systems ........................ 264
5.3 Elliptic boundary-value problems for systems of
equations ................................................ 268
5.3.1 Statement of the problem .......................... 269
5.3.2 Theorem on solvability ............................ 271
5.4 Remarks and comments ..................................... 272
Bibliography .................................................. 275
Index ......................................................... 291
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