Optimal estimates from below for biharmonic Green functions

Grunau, Hans-Christoph; Robert, Frédéric; Sweers, Guido

doi:10.1090/S0002-9939-2010-10740-2

Optimal estimates from below for biharmonic Green functions
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by Hans-Christoph Grunau, Frédéric Robert and Guido Sweers PDF

Proc. Amer. Math. Soc. 139 (2011), 2151-2161 Request permission

Abstract:

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. The lack of a maximum principle 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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