Optimal estimates from below for Green functions of higher order elliptic operators with variable leading coefficients

Abstract

Estimates from above and below by the same positive prototype function for suitably modified Green functions in bounded smooth domains under Dirichlet boundary conditions for elliptic operators L of higher order 2m ≥ 4 have been shown so far only when the principal part of L is the polyharmonic operator (−Δ)m. In the present note, it is shown that such kind of result still holds when the Laplacian is replaced by any second order uniformly elliptic operator in divergence form with smooth variable coefficients. For general higher order elliptic operators, whose principal part cannot be written as a power of second order operators, it was recently proved that such kind of result becomes false in general.