Positivity and Almost Positivity of Biharmonic Green’s Functions under Dirichlet Boundary Conditions

Grunau, Hans-Christoph; Robert, Frédéric

doi:10.1007/s00205-009-0230-0

Cited by 39 publications

(51 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…But the difficult part is to gain uniformity when the pole approaches the boundary ∂Ω because here, the well established quantitative methods just mentioned do not yield any longer the desired information. The proof is performed by means of a rescaling and blow up technique which is sketched in [GR1] and explained in detail in [GR2]. Both are joint works with Frédéric Robert.…”

Section: Almost Positivity In Biharmonic Green's Functionsmentioning

confidence: 99%

“…The cases n = 3, 4 are special and technically more involved. For details one may see [GR2], where also more general questions are studied.…”

Section: Here We Denote D(x) = Dist(x ∂B) Moreover ∼ Means That Umentioning

confidence: 99%

“…, ξ n ). For details we refer to [GR2,Lemma 7]. The identity for G H,∆ 2 is due to Boggio's formula (1.6) transformed to the half space H by means of a suitable Möbius transform (or verified directly).…”

Section: Here We Denote D(x) = Dist(x ∂B) Moreover ∼ Means That Umentioning

confidence: 99%

“…Originally, due to a much more general setting, these results require more smoothness on ∂Ω. However, as can be seen from [GR2,Theorem 4] and [GGS,Chapter 3], C 4,γ -smoothness is enough. If n ≥ 3, the estimate from below follows immediately from the previous Theorem 2.5.…”

Section: Here We Denote D(x) = Dist(x ∂B) Moreover ∼ Means That Umentioning

confidence: 99%

“…I am grateful to my coauthors Anna Dall'Acqua (Magdeburg), Klaus Deckelnick (Magdeburg), Steffen Fröhlich (Berlin), Filippo Gazzola (Milan), Frédéric Robert (Nice), Friedhelm Schieweck (Magdeburg) and Guido Sweers (Cologne) for the very enjoyable collaboration in the articles [DDG,DFGS,DG,GR1,GR2,GS1,GS2,GS3] and the forthcmoing book [GGS] which are basis for the present survey article. I am indebted to Klaus Deckelnick, Friedhelm Schieweck and Guido Sweers for allowing me to use their numerically calculated pictures, see Figures 2, 10, 11, 12 and 13.…”

mentioning

confidence: 99%

See 4 more Smart Citations

Nonlinear Questions in Clamped Plate Models

Grunau

2009

Milan J. Math.

Self Cite

View full text Add to dashboard Cite

Abstract. The linear clamped plate boundary value problem is a classical model in mechanics. The underlying differential equation is elliptic and of fourth order. The latter is a peculiar feature with respect to which this equation differs from numerous equations in physics and engineering which are of second order. Concerning the clamped plate boundary value problem, "linear questions" may be considered as well understood. This changes completely as soon as one poses the simplest "nonlinear question": What can be said about positivity preserving? Does a plate bend upwards when being pushed upwards? It is known that the answer is "no" in general. However, there are many positivity issues as e.g. "almost positivity" to be discussed.Boundary value problems for the "Willmore equation" are nonlinear counterparts for the linear clamped plate equation. The corresponding energy functional involves curvature integrals over the unknown surface. The Willmore equation is of interest in mechanics, membrane physics and, in particular, in differential geometry. Quite far reaching results were achieved concerning closed surfaces. As for boundary value problems, by far less is known. These will be discussed in symmetric situations.

show abstract

Section: Almost Positivity In Biharmonic Green's Functionsmentioning

confidence: 99%

“…The cases n = 3, 4 are special and technically more involved. For details one may see [GR2], where also more general questions are studied.…”

Section: Here We Denote D(x) = Dist(x ∂B) Moreover ∼ Means That Umentioning

confidence: 99%

Section: Here We Denote D(x) = Dist(x ∂B) Moreover ∼ Means That Umentioning

confidence: 99%

Section: Here We Denote D(x) = Dist(x ∂B) Moreover ∼ Means That Umentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

Nonlinear Questions in Clamped Plate Models

Grunau

2009

Milan J. Math.

Self Cite

View full text Add to dashboard Cite

show abstract

Positivity properties for the clamped plate boundary problem on the ellipse and strip

Render

Ghergu

2012

Mathematische Nachrichten

View full text Add to dashboard Cite

show abstract

Polyharmonic Boundary Value Problems

Gazzola

Grunau

Sweers

2010

Lecture Notes in Mathematics

Self Cite

456

309

View full text Add to dashboard Cite

The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: SPi Publisher ServicesPrinted on acid-free paper springer.com Dedicated to our wives Chiara, Brigitte and Barbara.The cover figure displays the solution of Δ 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.x Preface Preface xi in bounded domains is to understand whether the results available in the simplest case m = 1 can also be proved for any m, or whether the results for m = 1 are special, in particular as far as positivity and the use of maximum principles are concerned. The differential equation ( * ) is complemented with suitable boundary conditions. As already mentioned above, if m = n = 2, equation ( * ) may be considered as a nonlinear plate equation for plates subject to nonlinear feedback forces, one may think e.g. of suspension bridges. In this case, ( * ) may also be interpreted as a reactiondiffusion equation, where the diffusion operator Δ 2 refers to (linearised) surface diffusion.The first part of Chapter 7 is devoted to the proof of symmetry results for positive solutions to ( * ) in the ball under Dirichlet boundary conditions. As already mentioned, truncation and reflection methods do not apply to higher order problems so that a suitable generalisation of the moving planes technique is needed here. Equation ( * ) deserves a particular attention when f (u) has a power-type behaviour. In this case, a crucial role is played by the critical power s = (n + 2m)/(n − 2m) which corresponds to the critical (Sobolev) exponent which appears whenever n > 2m. Indeed, subcritical problems in bounded domains enjoy compactness properties as a consequence of the Rellich-Kondrachov embedding theorem. But compactness is lacking when the critical growth is attained and by means of Pohožaev-type identities, this gives rise to many interesting phenomena. The existence theory can be developed similarly to the second order case m = 1 while it becomes immediately quite difficult to prove positivity or nonexistence of certain solutions. Nonexistence phenomena are related to so-called critical dimensions introduced by Pucci-Serrin [348,349]. They formulated an interesting conjecture concerning these critical dimensions. We give a proof of a relaxed form of it in Chapter 7. We also give a functional analytic interpretation of these nonexistence results, which is reflected in the possibility of adding L 2 -remainder terms in Sobolev inequalities with critical exponent and optimal constants. Moreover, the influence of topological and geometrical properties of Ω on the solvability of the equation is investigated. Also applications to conformal geometry, such as the Paneitz-Branson equation, i...

show abstract

Positivity and Almost Positivity of Biharmonic Green’s Functions under Dirichlet Boundary Conditions

Cited by 39 publications

References 28 publications

Nonlinear Questions in Clamped Plate Models

Nonlinear Questions in Clamped Plate Models

Positivity properties for the clamped plate boundary problem on the ellipse and strip

Polyharmonic Boundary Value Problems

Contact Info

Product

Resources

About