Optimal estimates from below for biharmonic Green functions

Hans-Christoph Grunau (1) , Frédéric Robert (2) , Guido Sweers (3)

1 OVGU - Otto-von-Guericke-Universität Magdeburg = Otto-von-Guericke University [Magdeburg]
2 IECN - Institut Élie Cartan de Nancy
3 Mathematisches Institut

Optimal pointwise estimates are derived for the biharmonic Green function under Dirichlet boundary conditions in arbitrary C 4,γ-smooth domains. Maximum principles do not exist for fourth order elliptic equations and the Green function may change sign. It prevents using a Harnack inequality as for second order problems and hence complicates the derivation of optimal estimates. The present estimate is obtained by an asymptotic analysis. The estimate shows that this Green function is positive near the singularity and that a possible negative part is small in the sense that it is bounded by the product of the squared distances to the boundary.

Domaines: Equations aux dérivées partielles [math.AP]

Hans-Christoph Grunau, Frédéric Robert, Guido Sweers. Optimal estimates from below for biharmonic Green functions. Proceedings of the American Mathematical Society, 2011, 139 (6), pp.2151-2161. ⟨10.1090/S0002-9939-2010-10740-2⟩. ⟨hal-01279342⟩