Polyharmonic boundary value problems

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Dedicated to our wives Chiara, Brigitte and Barbara. The cover figure displays the solution of ∆ 2 u = f in a rectangle with homogeneous Dirichlet boundary condition for a nonnegative function f with its support concentrated near a point on the left hand side. The dark part shows the region where u < 0.
Preface Linear elliptic equations arise in several models describing various phenomena in the applied sciences, the most famous being the second order stationary heat equation or, equivalently, the membrane equation. For this intensively well-studied linear problem there are two main lines of results. The first line consists of existence and regularity results. Usually the solution exists and “gains two orders of differentiation” with respect to the source term. The second line contains comparison type results, namely the property that a positive source term implies that the solution is positive under suitable side constraints such as homogeneous Dirichlet boundary conditions. This property is often also called positivity preserving or, simply, maximum principle. These kinds of results hold for general second order elliptic problems, see the books by Gilbarg-Trudinger [197] and Protter-Weinberger [346]. For linear higher order elliptic problems the existence and regularity type results remain, as one may say, in their full generality whereas comparison type results may fail. Here and in the sequel “higher order” means order at least four. Most interesting models, however, are nonlinear. By now, the theory of second order elliptic problems is quite well developed for semilinear, quasilinear and even for some fully nonlinear problems. If one looks closely at the tools being used in the proofs, then one finds that many results benefit in some way from the positivity preserving property. Techniques based on Harnack’s inequality, De Giorgi-Nash- Moser’s iteration, viscosity solutions etc., all use suitable versions of a maximum principle. This is a crucial distinction from higher order problems for which there is no obvious positivity preserving property. A further crucial tool related to the maximum principle and intensively used for second order problems is the truncation method, introduced by Stampacchia. This method is helpful in regularity theory, in properties of first order Sobolev spaces and in several geometric arguments, such as the moving planes technique which proves symmetry of solutions by reflection. Also the truncation (or reflection) method fails for higher order problems. For instance, the modulus of a function belonging to a second order Sobolev space may not belong to the same space. The failure of maximum principles and of truncation methods, one could say, are the main reasons why the theory of nonlinear higher order elliptic equations is by far less developed than the theory of analogous second v
Page 1: Filippo Gazzola Hans-Christoph Grun
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Page 40 and 41: 26 2 Linear problems We are interes
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38 2 Linear problems the re-ordered
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40 2 Linear problems 2.4.3 Inhomoge
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42 2 Linear problems which proves t
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44 2 Linear problems Roughly speaki
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46 2 Linear problems 2.5.3 The Mira
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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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