On the full regularity of the free boundary in a class of variational problems

Abstract:

We consider nonnegative minimizers of the functional \[ J_p(u;\Omega )=\int _\Omega |\nabla u|^p+ \lambda _p^p \chi _{\{u>0\}},\qquad 1<p<\infty , \] on open subsets $\Omega \subset \mathbb {R}^n$. There is a critical dimension $k^*$ such that the free boundary $\partial \{u>0\}\cap \Omega$ has no singularities and is a real analytic hypersurface if $p=2$ and $n<k^*$. A corollary of the main result in this note ensures that there exists $\epsilon _0>0$ such that the same result holds if $|p-2|<\epsilon _0$.