Comptes Rendus
Partial Differential Equations
Boundedness of the negative part of biharmonic Green's functions under Dirichlet boundary conditions in general domains
[Borne L pour la partie négative de la fonction de Green biharmonique avec condition de Dirichlet au bord d'un domaine arbitraire]
Comptes Rendus. Mathématique, Volume 347 (2009) no. 3-4, pp. 163-166.

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.

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.

Reçu le :
Accepté le :
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DOI : 10.1016/j.crma.2008.12.013

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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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/

[1] T. Boggio Sull'equilibrio delle piastre elastiche incastrate, Rend. Acc. Lincei, Volume 10 (1901), pp. 197-205

[2] T. Boggio Sulle funzioni di Green d'ordine m, Rend. Circ. Mat. Palermo, Volume 20 (1905), pp. 97-135

[3] A. Dall'Acqua; Ch. Meister; G. Sweers Separating positivity and regularity for fourth order Dirichlet problems in 2d-domains, Analysis, Volume 25 (2005), pp. 205-261

[4] A. Dall'Acqua; G. Sweers Estimates for Green function and Poisson kernels of higher order Dirichlet boundary value problems, J. Differential Equations, Volume 205 (2004), pp. 466-487

[5] E.B. Davies Pointwise lower bounds on the heat kernels of higher-order elliptic operators, Math. Proc. Cambridge Philos. Soc., Volume 125 (1999), pp. 105-111

[6] R.J. Duffin On a question of Hadamard concerning super-biharmonic functions, J. Math. Phys., Volume 27 (1949), pp. 253-258

[7] R.J. Duffin Continuation of biharmonic functions by reflection, Duke Math. J., Volume 22 (1955), pp. 313-324

[8] A. Ferrero; F. Gazzola; H.-Ch. Grunau Decay and eventual local positivity for biharmonic parabolic equations, Discrete Contin. Dyn. Syst., Volume 21 (2008), pp. 1129-1157

[9] P.R. Garabedian A partial differential equation arising in conformal mapping, Pacific J. Math., Volume 1 (1951), pp. 485-524

[10] H.-Ch. Grunau; F. Robert Positivity issues of biharmonic Green's functions under Dirichlet boundary conditions (preprint) | arXiv

[11] H.-Ch. Grunau; G. Sweers Positivity for perturbations of polyharmonic operators with Dirichlet boundary conditions in two dimensions, Math. Nachr., Volume 179 (1996), pp. 89-102

[12] H.-Ch. Grunau; G. Sweers Positivity for equations involving polyharmonic operators with Dirichlet boundary conditions, Math. Ann., Volume 307 (1997), pp. 589-626

[13] H.-Ch. Grunau; G. Sweers Regions of positivity for polyharmonic Green functions in arbitrary domains, Proc. Amer. Math. Soc., Volume 135 (2007), pp. 3537-3546

[14] J. Hadamard Sur certains cas intéressants du problème biharmonique, Œuvres de Jacques Hadamard, Tome III, CNRS, Paris, 1968, pp. 1297-1299

[15] Ju.P. Krasovskiĭ Isolation of singularities of the Green's function, Izv. Akad. Nauk SSSR Ser. Mat., Volume 31 (1967), pp. 977-1010 (in Russian). English translation in: Math. USSR Izv., 1, 1967, pp. 935-966

[16] Z. Nehari On the biharmonic Green's function, Studies Math. Mech., Academic Press, New York, 1954, pp. 111-117 (Presented to Richard von Mises)

[17] M. Nicolesco Les fonctions polyharmoniques, Hermann, Paris, 1936

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