Orateur
Description
In this talk, we will present recent results obtained with Mickaël Nahon on a free boundary problem involving the bi-laplace operator. This can be seen as a higher-order analogue of the classical Alt–Caffarelli free boundary problem from 1981. This kind of results can be applied to different shape optimization problems, including the minimization of the first buckling eigenvalue under area constraint, which is conjectured to be solved by the ball.
We will focus on regularity properties in dimension two, with particular emphasis on boundary regularity near points of density less than 1. We prove full regularity of the free boundary, which is analytic outside a possible singular set corresponding to angles with opening approximately 1.43π. The analysis relies on the monotonicity formula introduced by Dipierro, Karakhanyan, and Valdinoci, together with a new epiperimetric inequality that strengthens their approach.