On the full regularity of the free boundary for minima of Alt–Caffarelli functionals in Orlicz spaces

Abstract

In this paper, we discuss two issues about the full regularity of the free boundary for overdetermined Bernoulli-type problems in Orlicz spaces. First, we show that in dimension \(n = 2\) there are no singular points on the free boundary \(F(u) := \partial \{ u > 0 \} \cap \Omega\) of minimizers of the Alt-Caffarelli functional
\(J_G(u) := \int_{\Omega} \left( G(\vert \nabla u \vert) + \lambda \chi_{\{ u > 0 \}} \right) dx\)

for suitable N-functions \(G\). Next, we prove as a consequence of our main results that there exist a critical dimension \(5 \leq n_0 \leq 7\) and a universal constant \(\varepsilon_0 \in (0,1)\) such that if \(G(t)\) is "\(\varepsilon_0\)-close" of \(t^2\), then for \(2 \leq n < n_0\), \(F(u)\) is a real analytic hypersurface.