Boundedness of the negative part of biharmonic Green's functions under Dirichlet boundary conditions in general domains

Hans-Christoph Grunau; Frédéric Robert

doi:10.1016/j.crma.2008.12.013

Partial Differential Equations

Boundedness of the negative part of biharmonic Green's functions under Dirichlet boundary conditions in general domains
[Borne

L^{\infty}

pour la partie négative de la fonction de Green biharmonique avec condition de Dirichlet au bord d'un domaine arbitraire]

Présenté par : Haïm Brezis

Hans-Christoph Grunau ¹ ; Frédéric Robert ²

¹ Fakultät für Mathematik, Otto-von-Guericke-Universität, Postfach 4120, 39016 Magdeburg, Germany
² Université de Nice-Sophia Antipolis, Laboratoire J.A. Dieudonné, parc Valrose, 06108 Nice cedex 2, France

Comptes Rendus. Mathématique, Volume 347 (2009) no. 3-4, pp. 163-166.

Résumé
Abstract

De manière générale, les équations elliptiques de grand ordre et les problèmes aux limites correspondant (comme l'équation biharmonique ou bien l'équation des plaques encastrées) ne satisfont ni un principe du maximum, ni un principe de comparaison ou bien, de façon équivalente, une propriété de conservation de la positivité. En revanche, nous montrons que pour des domaines bornés réguliers de $R^{n}$ , la partie négative de la fonction de Green correspondante est « petite » comparée à la partie positive singulière dès que $n ⩾ 3$ .

In general, higher order elliptic equations and boundary value problems like the biharmonic equation or the linear clamped plate boundary value problem do not enjoy neither a maximum principle nor a comparison principle or – equivalently – a positivity preserving property. It is shown that, on the other hand, for bounded smooth domains $Ω \subset R^{n}$ , the negative part of the corresponding Green's function is “small” when compared with its singular positive part, provided that $n ⩾ 3$ .

Reçu le : 2008-04-09
Accepté le : 2008-12-16
Publié le : 2009-01-30

DOI : 10.1016/j.crma.2008.12.013

Affiliations des auteurs :

Hans-Christoph Grunau ¹ ; Frédéric Robert ²

¹ Fakultät für Mathematik, Otto-von-Guericke-Universität, Postfach 4120, 39016 Magdeburg, Germany
² Université de Nice-Sophia Antipolis, Laboratoire J.A. Dieudonné, parc Valrose, 06108 Nice cedex 2, France

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     author = {Hans-Christoph Grunau and Fr\'ed\'eric Robert},
     title = {Boundedness of the negative part of biharmonic {Green's} functions under {Dirichlet} boundary conditions in general domains},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {163--166},
     publisher = {Elsevier},
     volume = {347},
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     year = {2009},
     doi = {10.1016/j.crma.2008.12.013},
     language = {en},
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Hans-Christoph Grunau; Frédéric Robert. Boundedness of the negative part of biharmonic Green's functions under Dirichlet boundary conditions in general domains. Comptes Rendus. Mathématique, Volume 347 (2009) no. 3-4, pp. 163-166. doi : 10.1016/j.crma.2008.12.013. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.12.013/

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