Geometric structure of singular free boundary points for the logarithmic obstacle problem
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2026
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| _version_ | 1866914516490518528 |
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| author | Du, Lili Tang, Xu Zhou, Yi |
| author_facet | Du, Lili Tang, Xu Zhou, Yi |
| contents | 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 $$-Δu = \log u\,χ_{\{u>0\}} \quad \text{in } Ω,$$ where $Ω\subset\mathbb{R}^d$ ($d\geq 2$) is a smooth bounded domain. In our earlier work [arXiv:2408.08104, 2024], we proved the $C^{1,α}$ regularity of the free boundary $Ω\cap\partial\{u>0\}$ 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 Hölder modulus. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_26485 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Geometric structure of singular free boundary points for the logarithmic obstacle problem Du, Lili Tang, Xu Zhou, Yi Analysis of PDEs 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 $$-Δu = \log u\,χ_{\{u>0\}} \quad \text{in } Ω,$$ where $Ω\subset\mathbb{R}^d$ ($d\geq 2$) is a smooth bounded domain. In our earlier work [arXiv:2408.08104, 2024], we proved the $C^{1,α}$ regularity of the free boundary $Ω\cap\partial\{u>0\}$ 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 Hölder modulus. |
| title | Geometric structure of singular free boundary points for the logarithmic obstacle problem |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2604.26485 |