[Borne pour la partie négative de la fonction de Green biharmonique avec condition de Dirichlet au bord d'un domaine arbitraire]
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 , la partie négative de la fonction de Green correspondante est « petite » comparée à la partie positive singulière dès que .
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 , the negative part of the corresponding Green's function is “small” when compared with its singular positive part, provided that .
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@article{CRMATH_2009__347_3-4_163_0,
author = {Grunau, Hans-Christoph and Robert, Fr\'ed\'eric},
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},
number = {3-4},
year = {2009},
doi = {10.1016/j.crma.2008.12.013},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2008.12.013/}
}
TY - JOUR AU - Grunau, Hans-Christoph AU - Robert, Frédéric TI - Boundedness of the negative part of biharmonic Green's functions under Dirichlet boundary conditions in general domains JO - Comptes Rendus. Mathématique PY - 2009 SP - 163 EP - 166 VL - 347 IS - 3-4 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2008.12.013/ DO - 10.1016/j.crma.2008.12.013 LA - en ID - CRMATH_2009__347_3-4_163_0 ER -
%0 Journal Article %A Grunau, Hans-Christoph %A Robert, Frédéric %T Boundedness of the negative part of biharmonic Green's functions under Dirichlet boundary conditions in general domains %J Comptes Rendus. Mathématique %D 2009 %P 163-166 %V 347 %N 3-4 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2008.12.013/ %R 10.1016/j.crma.2008.12.013 %G en %F CRMATH_2009__347_3-4_163_0
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, Tome 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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