Singular critical points for variational free boundary problem from isoparametric hypersurfaces

Singular critical points for variational free boundary problem from isoparametric hypersurfaces

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Lizhou Wang , Visiting Fellow
Fine Hall 314

Consider the following variational free boundary problem:  \min J(u)=\int_B (|\nabla u|^2+\chi_{\{u>0\}})dx, where $B$ is the unit ball, and the boundary value of $u$ on $\partial B$ is prescribed. The main problem is the regularity for the free boundary $F(u)=\partial\{u>0\}$ for  a minimizer $u$.  $J$ has a trivial minimizer $u(x)=x_n, x_n>0$, $u(x)=0,x_n\leq 0$. As in the theory of minimal surface, regularity of free boundary  is equivalent to existence of singular minimal cone, that is, nontrivial minimizer of $J$ of degree 1: $u(rx)=ru(x)$.   We construct three families of  singular critical points  for $J$, which are homogeneous of degree 1. The level sets of these solutions on the unit sphere are isoparametric surfaces and their focal submanifolds. This provides us some candidates for singular minimal cones of free boundary problems.