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

Abstract: 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.

• The paper establishes a modified log-epiperimetric inequality that quantifies energy decay and ensures uniqueness of blow-up limits at singular free boundaries.
• The paper adapts blow-up analysis and Weiss-type monotonicity techniques to overcome the challenges posed by the logarithmic nonlinearity.
• The paper provides a geometric stratification of singular sets, demonstrating C^(1,log) regularity in higher dimensions and improved C^(1,β) regularity in two dimensions.

Geometric Structure of Singular Free Boundary Points for the Logarithmic Obstacle Problem

Introduction and Problem Setting

This work rigorously investigates the singular set in the free boundary of minimizers to the singular obstacle problem with a logarithmic nonlinearity, characterized by the elliptic PDE

$$-\Delta u = \log u \, \chi_{\{u>0\}} \quad \text{in } \Omega,$$

where $\Omega \subset \mathbb{R}^d$ $(d \geq 2)$ is a smooth bounded domain. The associated variational functional is

$$\mathcal{L}(u; \Omega) = \int_\Omega \left( \frac{1}{2} |\nabla u|^2 - u^+ (\log u - 1) \right) dx,$$

minimized over $u \geq 0$ with fixed boundary values. Key structural questions pertain to the regularity and geometry of the free boundary $\mathscr{F}(u) = \Omega \cap \partial \{ u > 0 \}$, especially near singular points.

While classical obstacle problems ($-\Delta u = \chi_{\{u>0\}}$) admit strong regularity and geometric results for both solutions and free boundaries due to homogeneity and scaling properties, the introduction of the singular, non-homogeneous $\log u$ term obliterates these advantages. In particular, the scaling covariance of standard Weiss-type monotonicity and epiperimetric arguments is destroyed. This paper produces sharp regularity, uniqueness, and geometric stratification theorems for the singular set via the development of a modified log-epiperimetric inequality adapted to the anisotropic, logarithmic framework.

Analytical Framework and Technical Contributions

Blow-up Analysis and Energy Scaling

In the setting of the logarithmic obstacle, minimizers exhibit super-quadratic growth near the free boundary, specifically

$u(x^0 + r x) \sim r^2 |\log r|$

as $r \to 0$ at any free boundary point $\Omega \subset \mathbb{R}^d$0. This determines the natural rescaling for blow-up limits and the form of the relevant Weiss-type energy functional. Notably, the correct normalization becomes

$\Omega \subset \mathbb{R}^d$1

with blow-up limits defined as weak $\Omega \subset \mathbb{R}^d$2 limits. As $\Omega \subset \mathbb{R}^d$3, the rescaled equations revert to the classical obstacle problem.

Weiss Monotonicity and Log-Epiperimetric Inequality

The loss of scaling symmetry is manifested at the monotonicity level: the Weiss energy here requires a scale-dependent parameter $\Omega \subset \mathbb{R}^d$4 and accrues non-integrable error terms. The main technical advance is a log-epiperimetric inequality for a corrected Weiss-type functional: $\Omega \subset \mathbb{R}^d$5 where $\Omega \subset \mathbb{R}^d$6 is an auxiliary term that absorbs remaining non-integrable errors, and $\Omega \subset \mathbb{R}^d$7 for $\Omega \subset \mathbb{R}^d$8, $\Omega \subset \mathbb{R}^d$9 for $(d \geq 2)$0. This innovation is essential: classical epiperimetric techniques fail due to the singularity and non-homogeneity of $(d \geq 2)$1. The proof combines direct construction of competitors (in the style of Colombo-Spolaor-Velichkov [csv18]) and a sophisticated analysis of Fourier mode decompositions on the sphere.

Logarithmic Contraction and Energy Decay

Unlike the classical setting, the epiperimetric "contraction" achieved is logarithmic in nature: $(d \geq 2)$2 resulting not in Hölder but logarithmic decay rates for the excess energy and convergence of blow-up sequences. Correspondingly, the uniqueness of blow-ups at singular points is established, and geometric consequences for the singular strata are deduced.

Main Results

1. Logarithmic Energy Decay and Uniqueness of Singular Blow-Ups

The corrected Weiss energy decays at singular points according to

$(d \geq 2)$3

for all $(d \geq 2)$4, ensuring uniqueness of blow-up limits at singular points and providing explicit quantitative convergence rates.

2. Geometric Stratification of the Singular Set

The structure of singular strata $(d \geq 2)$5, comprised of singular points whose blow-ups vanish in $(d \geq 2)$6 directions, satisfies:

• For each $(d \geq 2)$7 and $(d \geq 2)$8, there exists $(d \geq 2)$9 such that $$\mathcal{L}(u; \Omega) = \int_\Omega \left( \frac{1}{2} |\nabla u|^2 - u^+ (\log u - 1) \right) dx,$$0 is contained in a single $$\mathcal{L}(u; \Omega) = \int_\Omega \left( \frac{1}{2} |\nabla u|^2 - u^+ (\log u - 1) \right) dx,$$1 $$\mathcal{L}(u; \Omega) = \int_\Omega \left( \frac{1}{2} |\nabla u|^2 - u^+ (\log u - 1) \right) dx,$$2-dimensional manifold.
• In dimension $$\mathcal{L}(u; \Omega) = \int_\Omega \left( \frac{1}{2} |\nabla u|^2 - u^+ (\log u - 1) \right) dx,$$3, the modulus improves to Hölder, yielding $$\mathcal{L}(u; \Omega) = \int_\Omega \left( \frac{1}{2} |\nabla u|^2 - u^+ (\log u - 1) \right) dx,$$4 singular strata—$$\mathcal{L}(u; \Omega) = \int_\Omega \left( \frac{1}{2} |\nabla u|^2 - u^+ (\log u - 1) \right) dx,$$5 is discrete, $$\mathcal{L}(u; \Omega) = \int_\Omega \left( \frac{1}{2} |\nabla u|^2 - u^+ (\log u - 1) \right) dx,$$6 is contained in a $$\mathcal{L}(u; \Omega) = \int_\Omega \left( \frac{1}{2} |\nabla u|^2 - u^+ (\log u - 1) \right) dx,$$7 curve.
• For $$\mathcal{L}(u; \Omega) = \int_\Omega \left( \frac{1}{2} |\nabla u|^2 - u^+ (\log u - 1) \right) dx,$$8, the geometric description matches the "logarithmic regime" obtained for classical obstacle problems but is achieved via fundamentally different mechanisms due to the lack of scaling and appearance of new error terms.

3. Regularity and Asymptotics at Singular Points

The mapping $$\mathcal{L}(u; \Omega) = \int_\Omega \left( \frac{1}{2} |\nabla u|^2 - u^+ (\log u - 1) \right) dx,$$9 from singular points to their unique quadratic blow-up is log-continuous: $u \geq 0$0 with the stronger Hölder estimate in $u \geq 0$1.

Implications and Future Directions

This analysis clarifies the intricate geometric and analytic behavior of singular points in free boundaries for variational inequalities with singular logarithmic nonlinearities. Main implications include:

• Extension of deep structural results from the classical obstacle setting to the substantially more complex logarithmic singular forced regime, with adapted monotonicity and contraction principles.
• The methodological shift (introduction of the $u \geq 0$2-correction term and variable-parameter functionals) is likely to inform and inspire future work on obstacle-type problems with other forms of nonlinearity or scaling anomaly, including inhomogeneous, degenerate, or anisotropic settings.
• While the $u \geq 0$3 description matches previous results in general dimensions, the fundamental mechanism is distinct, and achieving higher-order expansions analogous to the $u \geq 0$4-regularity in $u \geq 0$5 for the classical obstacle problems remains open and would likely require innovations beyond the log-epiperimetric approach.

Conclusion

This work constructs a modified log-epiperimetric framework that enables precise quantification and geometric description of singular free boundary points in the logarithmic obstacle problem. By demonstrating energy decay, uniqueness of singular blow-ups, and regularity of the singular strata, the analysis closes a gap in the structural theory for obstacle problems with singular, non-homogeneous nonlinearities. The technical developments here are expected to influence both the study of fine free boundary regularity in singular PDEs and the methodology for energy decay via corrected monotonicity principles in the absence of scaling.

[See (2604.26485) for the complete details.]