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 is small, the corresponding differential operator is positivity preserving, but if it gets larger than a critical constant , it is no longer positivity preserving. G.Sweers and I were able to prove that the canonical conjecture is false. Here is a disc of same area as .

In the second part of my lecture I will address the pseudo-Laplace eigenvalue problem, which comes from minimizing on . The Euler-Lagrange equation reads

and is more degenerate than . If is a ball, the minimizer is not radially symmetric, but one can still say something about symmetry of the level sets. If 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.

EPSRC Gregynog Workshop, 21-26 July 2002