| Abstrakt: |
Abstract: We survey recent results concerning estimates of the principal eigenvalue of the Dirichlet p-Laplacian and the Navier p-biharmonic operator on a ball of radius R in \mathbb R^N and its asymptotics for p approaching 1 and \infty. Let p tend to \infty. There is a critical radius R_C of the ball such that the principal eigenvalue goes to \infty for 0R_C. The critical radius is R_C=1 for any N\in\mathbb N for the p-Laplacian and R_C=\sqrt{2N} in the case of the p-biharmonic operator. When p approaches 1, the principal eigenvalue of the Dirichlet p-Laplacian is NR^{-1}\*(1-(p-1)\log R(p-1))+o(p-1) while the asymptotics for the principal eigenvalue of the Navier p-biharmonic operator reads 2N/R^2+O(-(p-1)\log(p-1)). |
