The Green function $G_{-\Delta,\Omega}$ for the Laplacian under Dirichlet boundary conditions in a bounded smooth domain $\Omega\subset \mathbb{R}^n$ enjoys in dimensions $n\ge 3$ the estimate: $$ 0\le G_{-\Delta,\Omega}(x,y) \le \frac{1}{n(n-2)e_n}|x-y|^{2-n}. $$ Here, $e_n$ denotes the volume of the unit ball $B=B_1(0)\subset \mathbb{R}^n$. This estimate follows from the maximum principle, the construction of $G_{-\Delta,\Omega}$ and the explicit expression of a suitable fundamental solution.   In higher order elliptic equations the maximum principle fails and deducing Green function estimates becomes an intricate subject. We consider the clamped plate boundary value problem as a prototype: $$ \left\{ \begin{array}{ll} \Delta^2 u=f \quad &\text{ in } \Omega,\\ u=|\nabla u|=0\quad &\text{ on }\partial \Omega . \end{array} \right. $$ I shall discuss estimates for the corresponding Green function $G_{\Delta^2,\Omega}$ focussing on two aspects:

• Keeping $\Omega$ fixed, can one show -- although $G_{\Delta^2,\Omega}$ is in general sign changing -- that it is somehow "almost positive"?
• Removing arbitrarily small holes (with almost infinite curvature) from a fixed domain $\Omega$ prevents uniform constants in classical Green function estimates. Can one nevertheless deduce estimates for this singular family of domains which are uniform with respect to the size of the hole?

The lecture is based on joint works with F. Robert (Nancy) and G. Sweers (Cologne).