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Uniform estimates for polyharmonic Green functions in domains with small holes
We prove a pointwise control for the Green's function of polyharmonic
operators with holes: this control is uniform while holes shrink. For the usual
Laplacian, such a control is given by the maximum principle; the techniques
developed here applies to general polyharmonic operators for which there is no
comparison principle
Optimal estimates from below for biharmonic Green functions
Optimal pointwise estimates are derived for the biharmonic Green function
under Dirichlet boundary conditions in arbitrary -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.Comment: 11 pages. To appear in "Proceedings of the AMS
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