Geometric structure of singular free boundary points for the logarithmic obstacle problem

In the previous work [Interfaces Free Bound., 19, 351--369, 2017], de Queiroz and Shahgholian established the optimal $C^{1,\log}_{\mathrm{loc}}$ regularity of solutions for the obstacle problem with singular logarithmic forcing term $$-\Delta u = \log u\,\chi_{\{u>0\}} \quad \text{in } \Omega,$$ where $\Omega\subset\mathbb{R}^d$ ($d\geq 2$) is a smooth bounded domain. In our earlier work [arXiv:2408.08104, 2024], we proved the $C^{1,\alpha}$ regularity of the free boundary $\Omega\cap\partial\{u>0\}$ near regular points. In this paper, we investigate the more delicate structure of the \emph{singular} free boundary. Since the nonlinearity $-\log u$ is singular near the free boundary and destroys the scaling invariance, so that neither the classical blow-up arguments nor the standard epiperimetric inequality [Weiss, Invent.\ Math., 138, 23--50, 1999] apply directly; moreover, the Weiss type monotonicity formula requires a variable-parameter correction that introduces non-integrable remainder terms into the energy estimates. Motivated by Colombo--Spolaor--Velichkov [Geom.\ Funct.\ Anal., 28, 1029--1061, 2018], we develop a new \emph{log-epiperimetric inequality} for the modified Weiss energy, also proved by the direct method. A key novelty is the introduction of an auxiliary correction term $T$ that absorbs the non-integrable errors. As consequences, we establish a logarithmic energy decay, uniqueness of blow-ups at singular points, and a $C^{1,\log}$-type geometric description of the singular strata. In dimension two, the logarithmic modulus improves to a H\"older modulus.