The present book is a detailed exposition of the author and his collaborators’ work on boundedness, continuity, and differentiability properties of solutions to elliptic equations in general domains, that is, in domains that are not a priori restricted by assumptions such as “piecewise smoothness” or being a “Lipschitz graph”. The description of the boundary behavior of such solutions is one of the most difficult problems in the theory of partial differential equations. After the famous Wiener test, the main contributions to this area were made by the author. In particular, necessary and sufficient conditions for the validity of imbedding theorems are given, which provide criteria for the unique solvability of boundary value problems of second and higher order elliptic equations. Another striking result is a test for the regularity of a boundary point for polyharmonic equations.
The book will be interesting and useful for a wide audience. It is intended for specialists and graduate students working in the theory of partial differential equations.
Keywords: Wiener test, higher order elliptic equations, elasticity systems, Zaremba problem, weighted positivity, capacity
Author(s): Vladimir G. Maz’ya
Series: EMS Tracts in Mathematics Vol. 30
Publisher: European Mathematical Society
Year: 2018
Language: English
Pages: 443
Introduction......Page 12
Capacitary modulus of continuity of a harmonic function......Page 18
Notation and lemmas......Page 22
Estimates of the solution with finite energy integral......Page 25
Estimates for solutions with unbounded Dirichlet integral and the Phragmen–Lindelöf principle......Page 28
Nonhomogeneous boundary condition......Page 29
Nonhomogeneous equation......Page 31
Refined estimate for the modulus of continuity of a harmonic function......Page 32
Improvement of previous estimates for L-harmonic functions......Page 43
More notations and preliminaries......Page 45
L-harmonic functions vanishing on a part of the boundary......Page 47
Behaviour of L-harmonic functions at infinity and near a singular point......Page 57
Phragmén–Lindelöf type theorems......Page 60
L-harmonic measure and non-homogeneous Dirichlet data......Page 62
The Green function and solutions of the non-homogeneous equation......Page 67
Continuity modulus of solutions and criterion of Hölder regularity of a point......Page 71
Sufficient conditions for Hölder regularity......Page 74
Comments to Chapter 1......Page 76
Formulation of the Zaremba problem......Page 78
Auxiliary assertions......Page 80
Estimates for solutions of the Zaremba problem......Page 82
Regularity criterion for the point at infinity......Page 85
Estimates for the Green function and for the harmonic measure of the Zaremba problem......Page 92
Comments to Chapter 2......Page 96
Introduction......Page 98
Weighted function spaces and weak solutions......Page 100
Change of variables......Page 104
Regularity test......Page 108
The capacity cap......Page 113
The capacity capK......Page 117
Comments to Chapter 3......Page 123
Preliminaries......Page 124
Main result......Page 136
Comments to Chapter 4......Page 141
Construction of a special solution......Page 144
Asymptotic formula for the Hölder exponent......Page 151
Absence of Hölder continuity......Page 158
Absence of continuity......Page 159
Comments to Chapter 5......Page 162
Introduction......Page 164
Capacities and the L-capacitary potential......Page 166
Weighted positivity of L(partial)......Page 174
Further properties of the L-capacitary potential......Page 177
Poincaré inequality with m-harmonic capacity......Page 178
Proof of sufficiency in Theorem 6.1.2......Page 180
Equivalence of two definitions of regularity......Page 183
Regularity as a local property......Page 184
Proof of necessity in Theorem 6.1.2......Page 185
Proof of sufficiency in Theorem 6.1.1......Page 187
Proof of necessity in Theorem 6.1.1......Page 190
The biharmonic equation in a domain with inner cusp (n >= 8)......Page 198
Comments to Chapter 6......Page 201
Weighted positivity of (-Delta)m......Page 202
Local estimates......Page 210
Pointwise estimates for the Green function......Page 212
Comments to Chapter 7......Page 214
Introduction......Page 216
Notations and preliminaries......Page 217
Weighted positivity of (-Delta)^mu......Page 219
Proof of Lemma 8.3.2......Page 220
Non-positivity......Page 225
Local estimates......Page 229
Regularity of a boundary point......Page 235
Comments to Chapter 8......Page 237
Statement of results......Page 238
Proof of Theorem 9.1.1......Page 239
Proof of Theorem 9.1.2......Page 251
Comments to Chapter 9......Page 256
Introduction......Page 258
Integral identity and global estimate......Page 261
Local energy and L2 estimates......Page 264
Estimates for the Green function......Page 269
The capacity Cap_P......Page 274
1-Regularity of a boundary point......Page 279
Sufficient condition for 1-regularity......Page 281
Necessary condition for 1-regularity......Page 287
Examples and further properties of Cap_P and Cap......Page 297
Comments to Chapter 10......Page 306
Introduction......Page 308
Integral inequalities and global estimate: the case of odd dimension. Part I: power weight......Page 311
Preservation of positivity for solutions of ordinary differential equations......Page 320
Integral inequalities and global estimate: the case of odd dimension. Part II: weight g......Page 326
Integral identity and global estimate: the case of even dimension. Part I: power-logarithmic weight......Page 335
Integral identity and global estimate: the case of even dimension. Part II: weight g......Page 338
Pointwise and local L2 estimates for solutions to the polyharmonic equation......Page 349
Green function estimates......Page 355
Estimates for solutions of the Dirichlet problem......Page 364
Comments to Chapter 11......Page 366
Introduction......Page 368
Regularity of solutions to the polyharmonic equation......Page 372
Higher-order regularity of a boundary point as a local property......Page 379
The new notion of polyharmonic capacity......Page 384
Poincaré-type inequalities......Page 394
Odd dimensions......Page 401
Even dimensions......Page 407
Fine estimates on the quadratic forms......Page 408
Scheme of the proof......Page 413
Main estimates. Bounds for auxiliary functions T and W related to polyharmonic potentials on the spherical shells......Page 414
Conclusion of the proof......Page 427
Comments to Chapter 12......Page 429
Bibliography......Page 430
General Index......Page 440
Index of Mathematicians......Page 442
