Nonconvex Sublevel Sets For The Planar Translating Mean Curvature Equation

Guohuan Qiu; Jiahuan Li; Xi-Nan Ma; Yilu Liu

Nonconvex Sublevel Sets For The Planar Translating Mean Curvature Equation

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Pith's one-line read A convex domain can host a translator solution with a nonconvex sublevel set.

desk verdict First counterexample to sublevel-set convexity for the planar translating mean curvature equation on a uniformly convex domain; the proof structure is sound, with the main verification burden concentrated in the heavy algebra of Theorem 4.2. read the letter →

arxiv 2608.06293 v1 pith:JSEFXJXF submitted 2026-08-06 math.AP

Jiahuan Li , Yilu Liu , Xi-Nan Ma , Guohuan Qiu This is my paper · ORCID

classification math.AP MSC 35J9335B0653E10

keywords translatingmeancurvatureequationnonconvexsublevelsetgrim-reaperprofilelogarithmicconcavitylevel-setconvexityDirichletproblemuniformlyconvexdomainbarriermethod

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper constructs a smooth bounded uniformly convex domain in the plane for which the unique zero-Dirichlet solution of the translating mean curvature equation has a nonconvex sublevel set, namely $\{x\in\Omega:u(x)<-\log 2\}$. If the construction is correct, it answers in the negative the logarithmic-concavity question for $\log(-u)$ in the bounded zero-Dirichlet setting, a question left open in reference [12]. The proof builds a corrected grim-reaper profile in a long convex channel, closes the channel with strictly convex caps, and uses explicit upper and lower barriers plus the comparison principle to show that the true solution inherits a finite midpoint defect of order $q^{3/2}$ from the model profile. The upshot is that convexity of the domain alone does not force convexity of the sublevel sets for this equation, already in two dimensions on a smooth uniformly convex domain.

What carries the argument

The engine of the proof is the corrected near-critical grim-reaper profile $V_q(x,y)=g(y)-g(\gamma_q(x))+\tau w(\gamma_q(x),y)$, with $g(s)=-\log\cos s$, $\gamma_q(x)=\arccos q+q^{9/4}x-q^7x^2$, $\tau=q^{9/2}$, and the explicit corrector $w(\gamma,y)=A(y)-A(\gamma)+\frac{\tan^2\gamma}{2}(y\tan y-\gamma\tan\gamma)$, $A(s)=s\tan s+\log\cos s$. The corrector cancels the complete term linear in the squared modulation speed, so that the operator $F[u]=(1+u_y^2)u_{xx}-2u_xu_yu_{xy}+(1+u_x^2)u_{yy}-(1+u_x^2+u_y^2)$, which has the same sign as the equation's left-hand side minus one, leaves residual $O(q^3\sec^2 y)$ on the corrected profile. The transverse barrier $\Psi_q(x,y)=A(\gamma_q(x))-A(y)$ is positive inside the channel and vanishes on its lateral boundary; adding or subtracting multiples of $\Psi_q$ with coefficients $a_q=Kq^3$ and $e_q=C_E|\log q|\cosh(\varepsilon_0qx)/\cosh(\varepsilon_0q\Lambda_q)$ produces super- and subsolutions whose signs are controlled by the exact expansion (4.6)--(4.9). Finally, the level-root curve $y=r_q(x)$, defined by $V_q(x,r_q(x))=-\log 2$, is shown to have $r_q''(x)\asymp q^5>0$, which is the local geometric source of the nonconvexity.

What would settle it

Recompute the exact identity (4.26) with the formulas (4.19)--(4.23) at a small concrete value such as $q=10^{-6}$, on the midline $y=0$ and at $y=\gamma_q(0)/2$, and compare $F[V_q]$ with the claimed $Cq^3\sec^2 y$ bound; also evaluate the remainder estimate (4.8) for the barrier perturbation. If the leading $q^3$ term fails its stated size or sign, or the remainder grows faster than $C_M q^{1/4}(|s|S+s^2X^2)$, the comparison gap of order $q^{3/2}$ used in Theorem 4.3 and Corollary 4.4 would collapse.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 1.1: for the equation $\operatorname{div}(Du/\sqrt{1+|Du|^2})=1/\sqrt{1+|Du|^2}$ on a bounded smooth uniformly convex domain $\Omega$, with $u=0$ on $\partial\Omega$, the sublevel set $\{x\in\Omega:u(x)<-\log 2\}$ need not be convex. The proof constructs a one-parameter family of uniformly convex domains $\Omega_q$ whose central part is a long channel of grim-reaper half-width $\gamma_q(x)=\arccos q+q^{9/4}x-q^7x^2$, closed by circular caps joined by a two-moment smoothing that preserves strict concavity. On this domain the authors define the corrected profile $V_q(x,y)=-\log(\cos y/\cos\gamma_q(x))+q^{9/2}w(\gamma_q(x),y)$, with $w$ chosen so that the leading error of the slowly modulated grim reaper cancels exactly, and they prove that $V_q$ satisfies the equation to residual $O(q^3\sec^2 y)$. Explicit barriers of the form $V_q-a_q\Psi_q$ and $V_q+(a_q+e_q)\Psi_q$ lie on opposite sides of the equation, so the comparison principle gives $|u_q-V_q|\le o(q^{3/2})$ near the center. The level curve of $V_q$ at height $-\log 2$ has positive second derivative of order $q^5$, producing a midpoint gap of order $q^{3/2}$; because the comparison error is much smaller than the gap, the real solution has the same defect and its sublevel set is nonconvex.

Load-bearing premise

The argument stands on the long explicit expansion in Theorem 4.2: if the claimed residual bound $F[V_q]=O(q^3\sec^2 y)$ or the barrier remainder bound (4.8) hides a wrong power of $q$ or a wrong sign, the $O(q^{3/2})$ comparison gap that transfers the model's midpoint defect to the true solution is not established.

Editorial extensions

If this is right

If $\log(-u)$ were concave, every sublevel set $\{u<c\}$ would be convex; Theorem 1.1 therefore rules out logarithmic concavity of $\log(-u)$ for zero-Dirichlet translator solutions over convex planar domains.
The failure is stable in the level: for every $c$ with $|c+\log 2|< c_0 q^{3/2}/2$, the sublevel set $\{u_q<c\}$ is also nonconvex, so the counterexample is not confined to a single exceptional height.
Convexity or even uniform convexity of a smooth bounded domain cannot by itself force convexity of the sublevel sets of the zero-Dirichlet solution to the translating mean curvature equation.
The counterexample is produced by a whole one-parameter family: for every sufficiently small $q$, the constructed uniformly convex domain $\Omega_q$ and its unique solution $u_q$ satisfy the nonconvexity conclusion.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the comparison only needs the values of $u_q-V_q$ at three points, the same corrected-profile and barrier scheme should work for any smooth strictly convex cap completion whose junction tangency prevents boundary interference; this geometric flexibility may simplify adaptations to related quasilinear equations.
The corrector cancels only the leading linear term in the squared modulation speed; pushing the modulation scale further or adding a second corrector might produce larger level-set defects or reveal the next obstruction, a check that could be done numerically from the explicit formulas.
A higher-dimensional analogue would require a multidimensional version of the grim-reaper reduction and of the convexity obstruction in Proposition 2.1; the present method suggests where a corrector would need to be inserted, though that extension is not attempted here.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: this paper appears to deliver what it promises. It constructs a smooth, uniformly convex planar domain for which the zero-Dirichlet solution of the translating mean curvature equation has a nonconvex sublevel set, answering a question Wang left open. The structure is rigorous and the construction is explicit: a near-critical grim-reaper profile with a well-designed corrector, a two-moment smoothing lemma to close the channel with convex caps, and explicit barriers that transfer a quantitative midpoint defect to the exact solution. Proposition 2.1 is a nice, clean obstruction that explains why a naive slow-variable construction fails, and the corrected profile is genuinely new. The comparison argument is coherent, and the central estimate (4.42) gives the precise o(q^{3/2}) error needed to preserve the finite gap. The paper is honest about its scope: it concerns bounded Dirichlet solutions, not the entire-solution problem, and says so.

The soft spot is exactly where the stress-test note puts it: Theorem 4.2. The expansion (4.6), the residual bound (4.9), and the remainder table (4.39) are load-bearing. If a sign or a q-power is wrong in that algebra, the sandwich collapses. The paper does provide exact identities (4.26), (4.34), and (4.35), and a term-by-term table for the remainder groups, which is strong evidence. I read through the main steps and found no explicit error; the q-powers appear consistent. But this is not machine-checked, and the verification burden is real. A referee should re-derive at least (4.9) and the E8 estimates independently before trusting the result. That said, the density of the computation is not a flaw by itself. The paper gives enough detail for an expert check, which is the usual standard.

The citations look appropriate: external inputs are Zhou's existence theorem and Wang's question, both used as black boxes or motivation, and prior counterexamples by Hamel-Nadirashvili-Sire, Wang, and Zhang are clearly distinct. No circularity, no fitting to data.

This deserves peer review. It is a significant, concrete advance in a classical topic, and the central argument holds up on close reading. The main recommendation to the editor: send it to a PDE analyst willing to spend time on the algebra, and if possible ask for a supplementary note expanding the verification of Theorem 4.2. If that checks out, this is a clean counterexample that resolves an open question in the negative.

Referee Report

0 major / 5 minor

Summary. The paper proves Theorem 1.1: there exists a smooth bounded uniformly convex domain Ω ⊂ R² such that the unique zero-Dirichlet solution u of the planar translating mean curvature equation (1.1) has a nonconvex sublevel set, specifically {u < −log 2}. The construction starts from a long convex channel whose half-width is the near-critical grim-reaper profile γ_q(x) = arccos q + q^{9/4}x − q^7 x^2, and introduces an explicit corrector w so that the corrected profile V_q = g(y) − g(γ_q) + τ w(γ_q,y), τ = q^{9/2}, has residual O(q^3 sec^2 y). The channel is closed by circular caps with a two-moment smoothing, the Dirichlet solution u_q is obtained from Zhou's existence theorem, and explicit upper and lower barriers V_q ± (a_q + e_q)Ψ_q sandwich u_q with an error that is o(q^{3/2}) on the central channel. A quantitative midpoint defect of order q^{3/2} for V_q is then transferred to u_q, proving the nonconvex sublevel set.

Significance. The result is significant: if correct, it is the first counterexample to level-set convexity, equivalently to logarithmic concavity of log(−u), for the bounded zero-Dirichlet translating mean curvature equation in the plane, and it answers Wang's question in that bounded setting. The construction is genuinely explicit: the corrector w is given in closed form, the cancellation of the τ-linear term is exact, the residual estimates are organized as a finite table, and the final comparison error is quantitative rather than qualitative. The main residual risk is the long algebra in Theorem 4.2; I checked representative cancellations and q-powers and found them consistent, but the proof would be easier to certify if that computation were fully expanded or supplied in machine-checkable form.

minor comments (5)

[Section 2.3] The definition of A(s) is internally inconsistent: the text states A(s) = tan s + log cos s, while immediately afterwards it asserts A'(s) = s sec^2 s and later uses the evenness of A. The correct definition used throughout the paper is A(s) = s tan s + log cos s; please fix the displayed formula.
[Section 3.1] The phrase 'positive concave half-width f_q' should be qualified: f_q is positive in the interior of the interval and vanishes at the terminal points, where the boundary is smooth only as a parametrized circle, not as a graph y = f_q(x). The current wording could mislead a reader about endpoint regularity.
[Section 4.2, Table (4.39)] The bounds for the remainder groups E3–E8 are stated with representative substitutions but not with the full term-by-term verification. I spot-checked the q-powers and the absorptions and found no error; for archival completeness, please include the complete derivation of the table, or provide a supplementary computation file.
[Section 5, Step 2] The notation O_{C^2(I)}(q^4) and o_{C^2(I)}(1) is introduced without definition; please define it explicitly in terms of the C^2(I) norm, since it is used in the crucial expansion (5.9)–(5.12).
[General] There are minor formatting artifacts in the abstract and headings ('TRANSLA TING', 'CUR V A TURE') and a few other typographical slips; these should be cleaned up in the final version.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the construction is explicit and self-contained; residual and barrier estimates are proved rather than fitted or imported.

full rationale

The paper's central claim is derived from an explicit corrected grim-reaper profile V_q, defined in (2.11) with the corrector w solving the exact first-order cancellation equation (2.9). The residual bound (4.9) is obtained by direct expansion (4.26) and term-by-term estimates, not by fitting data to the desired conclusion. The barrier inequalities (4.4) rest on the exact identities (4.6) and (4.34) plus the explicit remainder table (4.39), all derived in the paper. The midpoint defect in Theorem 5.1 is computed from the implicitly defined level root R(γ,τ) with explicit asymptotics (5.12), not assumed. The transfer to the exact Dirichlet solution uses the comparison principle, Lemma 4.1, and the boundary ordering in Theorem 4.3; no parameter is tuned to force the nonconvex sublevel set. The only external existence input is Zhou's theorem [15, Theorem C.2], an independent published result, and Wang's papers [12, 13] are used as motivation and context, not as load-bearing ingredients. The verification burden is concentrated in the long algebra of Theorem 4.2, but that is a correctness risk, not circularity: no step reduces to its own inputs by definition, and no fitted quantity is renamed as a prediction.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No empirical or fitted parameters appear. The construction is analytic; the constants q, Theta, theta, a0, K, epsilon0, and C_E are auxiliary choices fixed by inequalities in the proof. The paper relies on one external PDE existence theorem, Zhou, and standard tools, the maximum principle and the implicit function theorem, both explicitly cited.

assumptions (3)

domain assumption Existence and uniqueness of the zero-Dirichlet solution on smooth bounded mean-convex domains for the translating mean curvature equation (Zhou, [15, Theorem C.2]).
Invoked in Proposition 3.2 to obtain u_q; not proved in this paper.
standard math Strong maximum principle and comparison principle for uniformly elliptic quasilinear operators (Lemma 4.1).
Used to sandwich u_q between the explicit barriers in Theorems 4.2 and 4.3.
standard math Parameter-dependent implicit function theorem with uniform C^2 control of the branch.
Used in Step 2 of Theorem 5.1 to analyze the scaled root rho(q, xi, delta).

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Cite this review

Pith. "Pith review of Nonconvex Sublevel Sets For The Planar Translating Mean Curvature Equation." pith.science (2026). https://pith.science/paper/JSEFXJXF

@misc{pith2026260806293,
  author       = {Pith},
  title        = {Pith review of: Nonconvex Sublevel Sets For The Planar Translating Mean Curvature Equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JSEFXJXF}},
  note         = {Machine review of arXiv:2608.06293}
}

abstract

Translating solitons arise as models for type~II singularities of mean-convex mean curvature flow. We construct a smooth bounded uniformly convex domain $\Om\Subset\R^2$ such that the zero-Dirichlet solution of the planar translating mean curvature equation has a nonconvex sublevel set. The construction is based on a corrected near-critical grim-reaper profile and explicit barriers on a long convex channel.

Figures

Figures reproduced from arXiv: 2608.06293 by the authors.

**Figure 1.** Schematic construction of Ωq. 3.2. The zero-Dirichlet solution. The preceding construction gives the domain before any comparison argument is used. We now solve the boundary value problem on that domain. Proposition 3.2. For every sufficiently small q, there is a unique function uq ∈ C ∞(Ωq) ∩ C(Ωq) such that (3.17) ( Q[uq] = 1 in Ωq, uq = 0 on ∂Ωq. Proof. The domain Ωq is bounded and C∞, and its boundary curvature … view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

L. A. Caffarelli and J. Spruck,Convexity properties of solutions to some classical variational problems, Comm. Partial Differential Equations7(1982), no. 11, 1337–1379

work page 1982

[2] [2]

Hamel, N

F. Hamel, N. Nadirashvili, and Y. Sire,Convexity of level sets for elliptic problems in convex domains or convex rings: two counterexamples, Amer. J. Math.138(2016), no. 2, 499–527

work page 2016

[3] [3]

Kawohl,Rearrangements and Convexity of Level Sets in PDE, Lecture Notes in Math., vol

B. Kawohl,Rearrangements and Convexity of Level Sets in PDE, Lecture Notes in Math., vol. 1150, Springer, Berlin, 1985

work page 1985

[4] [4]

A. U. Kennington,Power concavity and boundary value problems, Indiana Univ. Math. J.34(1985), no. 3, 687–704

work page 1985

[5] [5]

N. J. Korevaar,Convex solutions to nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J.32 (1983), no. 4, 603–614

work page 1983

[6] [6]

N. J. Korevaar,Convexity of level sets for solutions to elliptic ring problems, Comm. Partial Differential Equations15(1990), no. 4, 541–556

work page 1990

[7] [7]

Ma,Convexity and the Dirichlet problem of translating mean curvature flows, Kodai Math

L. Ma,Convexity and the Dirichlet problem of translating mean curvature flows, Kodai Math. J.41(2018), no. 2, 348–358. doi:10.2996/kmj/1530496846

work page arXiv 2018

[8] [8]

S. Sakaguchi,Uniqueness of critical point of the solution to the prescribed constant mean curvature equation over convex domain in R2, Recent Topics in Nonlinear PDE IV, North-Holland Math. Stud., vol. 160, North-Holland, Amsterdam, 1989, 129–151

work page 1989