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

We consider nonnegative minimizers of the functional

$$J_p(u;\Omega)=\int_\Omega \vert\nabla u\vert^p+ \lambda_p^p\,\chi_{\{u>0\}},\qquad 1<p<\infty,$$

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 $\vert p-2\vert<\epsilon_0$.