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.

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