Speaker
Description
Consider the following functional
$$u\in H^2(D,\mathbb{R})\mapsto\int_D(|\nabla^2 u|^2+\chi_{u\neq 0})$$
where $D\subset \mathbb{R}^2$. This is a higher order analogue of the Alt-Caffarelli problem, and its local minimizers are linked to several shape optimization question: primarily with the minimization (under area constraint) of the critical buckling load of a clamped plate $\Omega\subset\mathbb{R}^2$, defined as
$$\Lambda(\Omega):=\inf_{u\in H^2_0(\Omega,\mathbb{R})\setminus\{0\}}\frac{\int_{\Omega}|\nabla^2 u|^2}{\int_{\Omega}|\nabla u|^2},$$
as well as the minimization of the drag of an obstacle with fixed measure in a Stokes fluid.
I will give a description of the free boundary, which is expected to be a union of regular curves joined with an angle of $102.5°$.
This is a joint work with Jimmy Lamboley.