Optimal estimates from below for biharmonic Green functions

Details Optimal estimates from below for biharmonic Green functions Optimal estimates from below for biharmonic Green functions

2011-03-03

arxiv.org/abs/1103.0651v1

Mathematics

Analysis of PDEs

11 pages. To appear in "Proceedings of the AMS"

Scientific paper

Optimal pointwise estimates are derived for the biharmonic Green function under Dirichlet boundary conditions in arbitrary $C^{4,\gamma}$-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.

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