A long-standing conjecture in spectral optimization is whether the critical buckling load (the first eigenvalue of the bilaplacian with respect to the laplacian) under area constraint is minimal on a disk. It is known from an argument of Willms and Weinberger that a sufficiently smooth optimal set must be a disk.
In this talk, I will explain a recent result on the regularity of a fourth order free boundary problem that applies to this question. A special feature of this free boundary problem is that the boundary may present cusp points and angular points of opening 1.43pi.
This is a joint work with Jimmy Lamboley
