The free boundary for the singular obstacle problem with logarithmic forcing term

In the previous work [Interfaces Free Bound., 19, 351-369, 2017], de Queiroz and Shahgholian investigated the regularity of the solution to the obstacle problem with singular logarithmic forcing term \begin{equation*} -\Delta u = \log u \, \chi_{\{u>0\}} \quad \text{in} \quad \Omega, \end{equation*} where $\chi_{\{u>0\}}$ denotes the characteristic function of the set $\{u>0\}$ and $\Omega \subset \mathbb{R}^n$ ($n \geq 2$) is a smooth bounded domain. The solution solves the minimum problem for the following functional, \begin{equation*} \mathscr{J}(u):=\int_{\Omega}\left(\frac{|\nabla u|^2}{2}-u^+ (\log u-1)\right) \, dx, \end{equation*} where $u^+=\max{\{0,u\}}$. In this paper, based on the regularity of the solution, we establish the $C^{1,\alpha}$ regularity of the free boundary $\Omega \cap \partial\{u>0\}$ near the regular points for some $\alpha\in (0,1)$. The logarithmic forcing term becomes singular near the free boundary $\Omega\cap\partial\{u>0\}$ and lacks the scaling properties, which are very crucial in studying the regularity of the free boundary. Despite these challenges, we draw inspiration for our overall strategy from the "epiperimetric inequality" method introduced by Weiss in 1999 [Invent. Math., 138, 23-50, 1999]. Central to our approach is the introduction of a new type of energy contraction. This allows us to achieve energy decay, which in turn ensures the uniqueness of the blow-up limit, and subsequently leads to the regularity of the free boundary.