Polyharmonic boundary value problems

Preface......Page 4
Acknowledgements......Page 10
Classical problems from elasticity......Page 16
The static loading of a slender beam......Page 17
The Kirchhoff-Love model for a thin plate......Page 20
Decomposition into second order systems......Page 22
The Boggio-Hadamard conjecture for a clamped plate......Page 24
The first eigenvalue......Page 27
An eigenvalue problem for a buckled plate......Page 28
A Steklov eigenvalue problem......Page 30
Sapondzyan's paradox by concave corners......Page 31
The Babuška paradox......Page 32
Paneitz-Branson type equations......Page 33
Critical growth polyharmonic model problems......Page 35
Qualitative properties of solutions to semilinear problems......Page 37
Willmore surfaces......Page 38
Polyharmonic operators......Page 40
Definitions and basic properties......Page 42
Embedding theorems......Page 45
Boundary conditions......Page 46
Normal boundary conditions and Green's formula......Page 49
Homogeneous boundary value problems......Page 51
Inhomogeneous boundary value problems......Page 55
Schauder theory......Page 57
Lp-theory......Page 59
The Miranda-Agmon maximum modulus estimates......Page 61
Green's function and Boggio's formula......Page 62
The space H2H10 and the Sapondzyan-Babuška paradoxes......Page 64
Bibliographical notes......Page 72
Eigenvalue problems......Page 74
A generalised Krein-Rutman result......Page 75
Decomposition with respect to dual cones......Page 76
Positivity of the first eigenfunction......Page 81
Symmetrisation and Talenti's comparison principle......Page 84
The Rayleigh conjecture for the clamped plate......Page 86
Buckling load of a clamped plate......Page 90
Steklov eigenvalues......Page 95
The Steklov spectrum......Page 96
Minimisation of the first eigenvalue......Page 102
Bibliographical notes......Page 108
Consequences of Boggio's formula......Page 112
Direct Green function estimates......Page 114
A 3-G-type theorem......Page 123
Estimates for the Steklov problem......Page 129
General properties of the Green functions......Page 134
Preliminary estimates for the Green function......Page 135
Uniform global estimates without boundary terms......Page 138
Uniform global estimates including boundary terms......Page 147
Convergence of the Green function in domain approximations......Page 153
Weighted estimates for the Dirichlet problem......Page 154
Bibliographical notes......Page 158
Positivity and lower order perturbations......Page 160
A positivity result for Dirichlet problems in the ball......Page 161
The role of positive boundary data......Page 165
The highest order Dirichlet datum......Page 166
Also nonzero lower order boundary terms......Page 170
Local maximum principles for higher order differential inequalities......Page 177
Positivity preserving......Page 180
Positivity of the operators involved in the Steklov problem......Page 187
Relation between Hilbert and Schauder setting......Page 190
Bibliographical notes......Page 197
Dominance of positivity in linear equations......Page 198
Highest order perturbations in two dimensions......Page 199
Domain perturbations......Page 201
Perturbations of the principal part......Page 204
The biharmonic Green function on the limaçons de Pascal......Page 208
Filling smooth domains with perturbed limaçons......Page 212
Regions of positivity in arbitrary domains in higher dimensions......Page 219
The biharmonic operator......Page 221
Extensions to polyharmonic operators......Page 225
Bounds for the negative part......Page 227
A blow-up procedure......Page 228
Domain perturbations in higher dimensions......Page 234
Bibliographical notes......Page 236
Semilinear problems......Page 238
A Gidas-Ni-Nirenberg type symmetry result......Page 240
Green function inequalities......Page 242
The moving plane argument......Page 245
Regularity for at most critical growth problems......Page 249
Existence......Page 252
Positivity and symmetry......Page 253
The Hilbertian critical embedding......Page 255
The Pohozaev identity for critical growth problems......Page 264
Nonexistence results......Page 270
Existence results for linearly perturbed equations......Page 273
Nontrivial solutions beyond the first eigenvalue......Page 281
Critical growth Navier problems......Page 290
Critical growth Steklov problems......Page 295
Optimal Sobolev inequalities with remainder terms......Page 305
Critical growth problems in geometrically complicated domains......Page 309
Existence results in domains with nontrivial topology......Page 310
Existence results in contractible domains......Page 311
Energy of nodal solutions......Page 313
The deformation argument......Page 316
A Struwe-type compactness result......Page 320
The conformally covariant Paneitz equation in hyperbolic space......Page 327
Infinitely many complete radial conformal metrics with the same Q-curvature......Page 328
Existence and negative scalar curvature......Page 329
Completeness of the conformal metric......Page 334
Fourth order equations with supercritical terms......Page 342
An autonomous system......Page 346
Regular minimal solutions......Page 355
Characterisation of singular solutions......Page 359
Stability of the minimal regular solution......Page 364
Existence and uniqueness of a singular solution......Page 366
Bibliographical notes......Page 373
An existence result......Page 378
Geometric quantities for surfaces of revolution......Page 380
Surfaces of revolution as elastic curves in the hyperbolic half plane......Page 383
Minimisation of the Willmore functional......Page 388
An upper bound for the optimal energy......Page 389
Monotonicity of the optimal energy......Page 390
Properties of minimising sequences......Page 394
Attainment of the minimal energy......Page 395
Bibliographical notes......Page 398
Notations......Page 400
Bibliography......Page 405
Author--Index......Page 422
Subject--Index......Page 425