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arxiv: 2604.26485 · v1 · submitted 2026-04-29 · 🧮 math.AP

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

Lili Du , Xu Tang , Yi Zhou This is my paper

Pith reviewed 2026-05-07 12:59 UTC · model grok-4.3

classification 🧮 math.AP
keywords logarithmic obstacle problemfree boundarysingular pointslog-epiperimetric inequalityWeiss energyblow-up analysisC^{1,log} regularity
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The pith

A new log-epiperimetric inequality yields uniqueness of blow-ups and C^{1,log} structure for singular free boundary points in the logarithmic obstacle problem.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper focuses on the singular free boundary points of the obstacle problem with right-hand side -log u. Because the logarithm destroys scaling and introduces non-integrable errors into the Weiss monotonicity formula, the authors introduce an auxiliary correction term T into a modified Weiss energy and prove a log-epiperimetric inequality for that energy by the direct method. This inequality produces logarithmic decay of the energy at singular points, which in turn gives uniqueness of blow-up limits and a C^{1,log} geometric description of the singular strata. In two dimensions the modulus improves from logarithmic to Hölder.

Core claim

After adding an auxiliary correction term T that absorbs the non-integrable remainder terms arising from the variable-parameter Weiss formula, a log-epiperimetric inequality holds for the modified energy; the inequality implies logarithmic decay, unique blow-ups at singular points, and therefore a C^{1,log} geometric structure for the singular free boundary, improving to Hölder regularity in dimension two.

What carries the argument

The log-epiperimetric inequality proved directly for the modified Weiss energy with auxiliary correction term T.

If this is right

  • Logarithmic energy decay holds at every singular point.
  • Blow-up limits are unique at singular points.
  • The singular free boundary admits a C^{1,log} geometric description in every dimension.
  • In two dimensions the modulus of continuity improves from logarithmic to Hölder.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same correction technique may extend to other singular nonlinearities that destroy scaling invariance in obstacle-type problems.
  • The C^{1,log} structure supplies a concrete modulus that could be used to design adaptive numerical schemes near singular points.
  • The result isolates the precise effect of the logarithmic singularity on the free-boundary regularity, separating it from the effect of the obstacle itself.

Load-bearing premise

The auxiliary term T must cancel the non-integrable errors sufficiently well for the direct-method proof of the log-epiperimetric inequality to succeed.

What would settle it

Exhibiting a singular free boundary point at which two distinct blow-up limits exist, or at which the free boundary fails to be C^{1,log}, would contradict the conclusions.

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

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper studies the structure of singular free boundary points for the obstacle problem -Δu = log u χ_{{u>0}} in R^d (d≥2). Building on prior C^{1,log} regularity of solutions and C^{1,α} regularity at regular free boundary points, the authors introduce a log-epiperimetric inequality for a modified Weiss energy, proved via the direct method. A key device is an auxiliary correction term T that absorbs non-integrable remainder terms generated by the variable-parameter correction needed for the Weiss monotonicity formula. From this they deduce logarithmic energy decay, uniqueness of blow-up limits at singular points, and a C^{1,log}-type geometric description of the singular strata; the modulus improves to Hölder in dimension two.

Significance. If the central inequality holds, the work supplies the first detailed geometric information on singular free boundary points in a setting where scaling invariance is lost and classical Weiss monotonicity plus epiperimetric arguments do not apply directly. The direct-method proof of the log-epiperimetric inequality together with the auxiliary term T constitutes a technical advance that may be useful for other singular free-boundary problems. The results complete the local regularity picture begun in the authors' earlier papers and give concrete, falsifiable predictions for the structure of the singular set.

major comments (1)
  1. [log-epiperimetric inequality and modified Weiss energy] The log-epiperimetric inequality (abstract and the section developing the modified Weiss energy): the direct-method argument must produce a uniform contraction constant strictly less than 1 after the auxiliary term T is inserted. Explicit estimates are required showing that the non-integrable remainder terms generated by the variable-parameter correction remain controlled uniformly near singular points (where log u becomes singular); if these bounds fail to close, the logarithmic energy decay and all subsequent uniqueness and stratification statements do not follow.
minor comments (2)
  1. [Introduction / Notation] The notation for the modified Weiss energy functional and the auxiliary term T should be introduced with a self-contained definition before the statement of the log-epiperimetric inequality.
  2. [Dimension-two case] In the two-dimensional improvement to Hölder modulus, the dependence of the Hölder exponent on the constants appearing in the log-epiperimetric inequality should be tracked explicitly.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation of the significance of our results and for the detailed comment on the log-epiperimetric inequality. We address this point below.

read point-by-point responses

  1. Referee: The log-epiperimetric inequality (abstract and the section developing the modified Weiss energy): the direct-method argument must produce a uniform contraction constant strictly less than 1 after the auxiliary term T is inserted. Explicit estimates are required showing that the non-integrable remainder terms generated by the variable-parameter correction remain controlled uniformly near singular points (where log u becomes singular); if these bounds fail to close, the logarithmic energy decay and all subsequent uniqueness and stratification statements do not follow.

    Authors: We appreciate the referee highlighting the necessity of a uniform contraction factor and uniform control of remainders. In Section 3, the log-epiperimetric inequality is proved by the direct method applied to the modified Weiss energy that incorporates the auxiliary correction T. The term T is constructed precisely to cancel the leading non-integrable contributions arising from the variable-parameter correction in the Weiss monotonicity formula. The resulting functional satisfies a strict contraction with factor θ < 1, where θ depends only on the dimension d and is independent of the center point. Uniformity near singular points follows from the C^{1,log} regularity of u (established in our prior work), which ensures that log u remains controlled in a manner that renders the error integrals bounded by quantities absorbed by T. These estimates appear explicitly after the definition of T (Lemmas 3.2–3.4 and the proof of Theorem 3.1), yielding the logarithmic energy decay E(r) ≤ θ E(2r) + o(1) with the o(1) term uniform on the singular stratum. This decay directly implies uniqueness of blow-ups and the C^{1,log} stratification. If the referee finds the uniformity insufficiently highlighted, we are prepared to add a dedicated remark or corollary making the d-dependent bounds fully explicit. revision: partial

Circularity Check

0 steps flagged

No circularity: central claims rest on direct proof of new inequality

full rationale

The paper's derivation chain for logarithmic Weiss-energy decay, blow-up uniqueness, and C^{1,log} stratification of singular free-boundary points proceeds by constructing a modified Weiss energy, introducing an auxiliary correction functional T to absorb non-integrable remainder terms arising from the variable-parameter correction, and then proving a new log-epiperimetric inequality for this energy by the direct method. This inequality is not obtained by fitting parameters to data, by renaming a prior result, or by reducing to a self-citation; it is established independently. The self-citation to the authors' earlier arXiv:2408.08104 concerns only the C^{1,α} regularity at regular points and is not load-bearing for the singular-strata analysis. No equation or step in the provided chain reduces a claimed prediction to an input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The proof rests on constructing a new inequality and correction term for this specific singular problem; no free parameters are fitted to data, and the axioms are standard background from free boundary theory.

axioms (2)
  • domain assumption The domain Ω is smooth and bounded in R^d for d ≥ 2
    Stated in the problem formulation and inherited from prior regularity results.
  • domain assumption Solutions to the obstacle problem exist and satisfy the given equation with the logarithmic right-hand side
    Established in the cited 2017 work of de Queiroz and Shahgholian.
invented entities (2)
  • log-epiperimetric inequality no independent evidence
    purpose: To obtain logarithmic decay of the modified Weiss energy at singular points
    Newly developed via the direct method to replace the classical epiperimetric inequality that does not apply directly.
  • auxiliary correction term T no independent evidence
    purpose: To absorb non-integrable remainder terms arising from the variable-parameter correction in the Weiss monotonicity formula
    Key technical device introduced to restore control over the energy estimates despite the logarithmic singularity.

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Reference graph

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