Partial Differential Equations
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.

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 ΩRn, the negative part of the corresponding Green's function is “small” when compared with its singular positive part, provided that n3.

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 Rn, la partie négative de la fonction de Green correspondante est « petite » comparée à la partie positive singulière dès que n3.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2008.12.013
Grunau, Hans-Christoph 1; Robert, Frédéric 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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Grunau, Hans-Christoph; Robert, Frédéric. 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. http://www.numdam.org/articles/10.1016/j.crma.2008.12.013/

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