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A computation of Green functions for the weighted biharmonic operators \(\Delta| z| ^{-2\alpha}\Delta\), with \(\alpha > -1\). (English) Zbl 0813.31001
Ayant précisé le domaine de définition de l’opérateur \(\Delta(| z|^{-2 \alpha} \Delta)\) sur le disque-unité \(D\), on montre qu’une fonction appartenant à ce domaine, au noyau de l’opérateur, et à \({\mathcal C}^ 1(\overline D \backslash \{0\})\), qui s’annule ainsi que son gradient à la frontière de \(D\), est la constante 0. On détermine ainsi de façon unique une fonction de Green \(U_ \alpha\), que l’on calcule explicitement pour \(\alpha\) entier; pour \(\alpha\) non entier, \(U_ \alpha\) est donnée par une formule intégrale où figure la fonction de Green de \(\Delta\) sur \(D\), et peut aussi s’exprimer à l’aide de la fonction de Green de \(\Delta (e^{2Im z} \Delta)\) sur le demi-plan \(\text{Im } z>0\).

MSC:
31A30 Biharmonic, polyharmonic functions and equations, Poisson’s equation in two dimensions
35J60 Nonlinear elliptic equations
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