DOMAIN GEOMETRY AND EIGENVALUES

B KAWOHL

kawohl@MI.Uni-Koeln.de

Universität zu Köln
Mathematisches Institut
Weyertal 86-90
50931 Köln, Germany

In my lecture I will first present a solution to a question of McKenna and Walter, concerning the positive deformation of hinged plates in a spring bed under positive load. If the spring constant $b$ is small, the corresponding differential operator is positivity preserving, but if it gets larger than a critical constant $b_c(\Omega)$, it is no longer positivity preserving. G.Sweers and I were able to prove that the canonical conjecture $b_c(\Omega)\leq b_c(\Omega^*)$ is false. Here $\Omega^*$ is a disc of same area as $\Omega\subset\ifmmode{I\hskip -3.2pt R} \else{\hbox{$I\hskip -3.2pt R$}}\fi ^2$.

In the second part of my lecture I will address the pseudo-Laplace eigenvalue problem, which comes from minimizing $\sum_{j=1}^p\int_\Omega \vert\partial v/\partial x_j\vert^p\ dx$ on $K:=\{ v\in W^{1,p}_0(\Omega\ \vert\ \vert\vert v\vert\vert _{L^p(\Omega)}=1\ \}$. The Euler-Lagrange equation reads

$$\sum_{j=1}^n{{\partial }\over{\partial x_j}} \left(\left\vert{{\p... ...rt^{p-2}{{\partial v}\over{\partial x_j}}\right)+\lambda\vert u\vert^{p-2} u=0,$$

and is more degenerate than $\Delta_p u+\lambda\vert u\vert^{p-2}u=0$. If $\Omega$ is a ball, the minimizer is not radially symmetric, but one can still say something about symmetry of the level sets. If $\Omega$ is convex, the level sets are convex. These results were obtained jointly with M.Belloni.

If there is time left, I will present results on the symmetry of eigenfunctions in situations where some standard tricks seem to fail.