Tobias Lamm and Hans-Christoph Grunau
16:15 Uhr | Tobias Lamm | Branched Willmore Spheres |
17:45 Uhr | Hans-Christoph Grunau | Estimates from above and below for biharmonic Green functions
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:
The lecture is based on joint works with F. Robert (Nancy) and G. Sweers (Cologne).
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