Polyharmonic boundary value problems

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xiv Contents 2.4.3 Inhomogeneous boundary value problems. . . . . . . . . . . . . . . . 40 2.5 Regularity results and a priori estimates . . . . . . . . . . . . . . . . . . . . . . . . 42 2.5.1 Schauder theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.5.2 Lp-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.5.3 The Miranda-Agmon maximum modulus estimates . . . . . . . . 46 2.6 Green’s function and Boggio’s formula . . . . . . . . . . . . . . . . . . . . . . . . 47 2.7 The space H2 ∩ H1 2.8 0 and the Sapondˇzyan-Babuˇska paradoxes . . . . . . . Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 57 3 Eigenvalue problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.1 Dirichlet eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.1.1 A generalised Kreĭn-Rutman result . . . . . . . . . . . . . . . . . . . . . 60 3.1.2 Decomposition with respect to dual cones . . . . . . . . . . . . . . . . 61 3.1.3 Positivity of the first eigenfunction . . . . . . . . . . . . . . . . . . . . . . 66 3.1.4 Symmetrisation and Talenti’s comparison principle . . . . . . . . 69 3.1.5 The Rayleigh conjecture for the clamped plate . . . . . . . . . . . . 71 3.2 Buckling load of a clamped plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.3 Steklov eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.3.1 The Steklov spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.3.2 Minimisation of the first eigenvalue . . . . . . . . . . . . . . . . . . . . . 87 3.4 Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4 Kernel estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.1 Consequences of Boggio’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.2 Kernel estimates in the ball . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.2.1 Direct Green function estimates . . . . . . . . . . . . . . . . . . . . . . . . 99 4.2.2 A 3-G-type theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.3 Estimates for the Steklov problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 4.4 General properties of the Green functions . . . . . . . . . . . . . . . . . . . . . . 119 4.4.1 Regularity of the biharmonic Green function . . . . . . . . . . . . . 120 4.4.2 Preliminary estimates for the Green function . . . . . . . . . . . . . 120 4.5 Uniform Green functions estimates in C 4,γ -families of domains . . . . 123 4.5.1 Uniform global estimates without boundary terms . . . . . . . . . 123 4.5.2 Uniform global estimates including boundary terms . . . . . . . 132 4.5.3 Convergence of the Green function in domain approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 4.6 Weighted estimates for the Dirichlet problem . . . . . . . . . . . . . . . . . . . 139 4.7 Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 5 Positivity and lower order perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . 145 5.1 A positivity result for Dirichlet problems in the ball . . . . . . . . . . . . . . 146 5.2 The role of positive boundary data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 5.2.1 The highest order Dirichlet datum . . . . . . . . . . . . . . . . . . . . . . 151 5.2.2 Also nonzero lower order boundary terms . . . . . . . . . . . . . . . . 155 5.3 Local maximum principles for higher order differential inequalities . 162
Contents xv 5.4 Steklov boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 5.4.1 Positivity preserving. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 5.4.2 Positivity of the operators involved in the Steklov problem. . 172 5.4.3 Relation between Hilbert and Schauder setting . . . . . . . . . . . . 175 5.5 Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 6 Dominance of positivity in linear equations. . . . . . . . . . . . . . . . . . . . . . . . 183 6.1 Highest order perturbations in two dimensions . . . . . . . . . . . . . . . . . . 184 6.1.1 Domain perturbations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 6.1.2 Perturbations of the principal part . . . . . . . . . . . . . . . . . . . . . . 189 6.2 Small negative part of biharmonic Green’s functions in two dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 6.2.1 The biharmonic Green function on the limaçons de Pascal . . 193 6.2.2 Filling smooth domains with perturbed limaçons . . . . . . . . . . 197 6.3 Regions of positivity in arbitrary domains in higher dimensions . . . . 204 6.3.1 The biharmonic operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 6.3.2 Extensions to polyharmonic operators . . . . . . . . . . . . . . . . . . . 210 6.4 Small negative part of biharmonic Green’s functions in higher dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 6.4.1 Bounds for the negative part . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 6.4.2 A blow-up procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 6.5 Domain perturbations in higher dimensions . . . . . . . . . . . . . . . . . . . . . 219 6.6 Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 7 Semilinear problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 7.1 A Gidas-Ni-Nirenberg type symmetry result . . . . . . . . . . . . . . . . . . . . 225 7.1.1 Green function inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 7.1.2 The moving plane argument . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 7.2 A brief overview of subcritical problems . . . . . . . . . . . . . . . . . . . . . . . 234 7.2.1 Regularity for at most critical growth problems . . . . . . . . . . . 234 7.2.2 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 7.2.3 Positivity and symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 7.3 The Hilbertian critical embedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 7.4 The Pohoˇzaev identity for critical growth problems . . . . . . . . . . . . . . 249 7.5 Critical growth Dirichlet problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 7.5.1 Nonexistence results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 7.5.2 Existence results for linearly perturbed equations . . . . . . . . . 258 7.5.3 Nontrivial solutions beyond the first eigenvalue . . . . . . . . . . . 266 7.6 Critical growth Navier problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 7.7 Critical growth Steklov problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 7.8 Optimal Sobolev inequalities with remainder terms . . . . . . . . . . . . . . 290 7.9 Critical growth problems in geometrically complicated domains . . . 294 7.9.1 Existence results in domains with nontrivial topology . . . . . . 295 7.9.2 Existence results in contractible domains . . . . . . . . . . . . . . . . 296 7.9.3 Energy of nodal solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
Page 1 and 2: Filippo Gazzola Hans-Christoph Grun
Page 3 and 4: Preface Linear elliptic equations a
Page 5 and 6: Preface vii even for polygonal boun
Page 7 and 8: Preface ix (−∆) m u = f (u) (
Page 9: Acknowledgements The authors are gr
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Page 40 and 41: 26 2 Linear problems We are interes
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48 2 Linear problems by means of a
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50 2 Linear problems Corollary 2.29
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52 2 Linear problems −∆bα = v
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54 2 Linear problems 4. If we exten
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56 2 Linear problems Then existence
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58 2 Linear problems in [320]. Babu
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60 3 Eigenvalue problems 3.1 Dirich
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62 3 Eigenvalue problems It was Mie
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64 3 Eigenvalue problems Therefore,
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66 3 Eigenvalue problems We need to
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68 3 Eigenvalue problems cones also
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70 3 Eigenvalue problems a bounded
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72 3 Eigenvalue problems In 1894, L
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74 3 Eigenvalue problems B b |∆w
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76 3 Eigenvalue problems B = {Ω
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78 3 Eigenvalue problems Note that
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80 3 Eigenvalue problems The functi
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82 3 Eigenvalue problems This prove
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84 3 Eigenvalue problems The vector
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86 3 Eigenvalue problems Writing (3
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88 3 Eigenvalue problems We first g
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90 3 Eigenvalue problems v∆udω
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92 3 Eigenvalue problems (uε) ∂
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94 3 Eigenvalue problems For the or
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Chapter 4 Kernel estimates In Chapt
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4.2 Kernel estimates in the ball 99
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4.3 Estimates for the Steklov probl
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4.3 Estimates for the Steklov probl
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4.4 General properties of the Green
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4.4 General properties of the Green
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4.5 Uniform Green functions estimat
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4.6 Weighted estimates for the Diri
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4.6 Weighted estimates for the Diri
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4.7 Bibliographical notes 143 4.7 B
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146 5 Positivity and lower order pe
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184 6 Dominance of positivity in li
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Chapter 7 Semilinear problems We st
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7.1 A Gidas-Ni-Nirenberg type symme
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7.2 A brief overview of subcritical
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7.3 The Hilbertian critical embeddi
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7.4 The Pohoˇzaev identity for cri
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7.5 Critical growth Dirichlet probl
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7.6 Critical growth Navier problems
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7.7 Critical growth Steklov problem
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7.8 Optimal Sobolev inequalities wi
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7.8 Optimal Sobolev inequalities wi
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7.9 Critical growth problems in geo
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7.10 The conformally covariant Pane
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7.11 Fourth order equations with su
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7.12 Bibliographical notes 359 scri
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7.12 Bibliographical notes 361 For
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364 8 Willmore surfaces of revoluti
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386 Notations en d(x) = dist(x,∂
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388 Notations f (t) g(t) ∃C > 0
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390 Bibliography Bibliography 1. R.
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392 Bibliography 48. M.S. Berger an
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394 Bibliography 96. Y.S. Choi and
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396 Bibliography 147. Z. Djadli, A.
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398 Bibliography 197. D. Gilbarg an
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400 Bibliography 247. C.E. Kenig. R
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402 Bibliography 293. V.G. Maz’ya
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404 Bibliography 343. G. Pólya and
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406 Bibliography 398. R.C.A.M. van
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408 Author-Index Gel’fand, I.M.,
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410 Subject-Index Subject-Index a p
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412 Subject-Index first buckling ei
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414 Subject-Index Poisson ratio, 5
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Polyharmonic boundary value problems

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