Optimal estimates from below for biharmonic Green functions

TL;DR

This paper derives optimal pointwise estimates for the biharmonic Green function in smooth domains, overcoming challenges posed by sign-changing behavior and lack of maximum principles for fourth order elliptic equations.

Contribution

It introduces an asymptotic analysis method to obtain sharp estimates for the Green function, including bounds on its negative part.

Findings

Green function is positive near the singularity

Negative part of the Green function is small and bounded by boundary distances

Estimates are valid in arbitrary smooth domains

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. 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.