Geodesic
On the full regularity of the free boundary for minima of Alt–Caffarelli functionals in Orlicz spaces
J. Ederson M. Braga1 ; Patrícia R. P. Regis1
1 Universidade Federal do Ceará, Departamento de Matemática
Annales Fennici Mathematici, Tome 47 (2022) no. 2, pp. 961-977.

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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.  
DOI : 10.54330/afm.120561
Keywords: Full regularity, free boundary problem, Orlicz spaces

J. Ederson M. Braga 1 ; Patrícia R. P. Regis 1

1 Universidade Federal do Ceará, Departamento de Matemática 
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J. Ederson M. Braga; Patrícia R. P. Regis. On the full regularity of the free boundary for minima of Alt–Caffarelli functionals in Orlicz spaces. Annales Fennici Mathematici, Tome 47 (2022) no. 2, pp. 961-977. doi : 10.54330/afm.120561. http://geodesic.mathdoc.fr/articles/10.54330/afm.120561/

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