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Optimal estimates from below for biharmonic Green functions

by Grunau, H.-Ch.; Robert, F.; Sweers, G..

Series: 2010-16, Preprints

35J40 Boundary value problems for higher-order elliptic equations
35B50 Maximum principles
35B45 A priori estimates

Optimal pointwise estimates are derived for the biharmonic Green function 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.

biharmonic Green function, estimate from below, almost a comparison principle