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The capillary $L_p$-Minkowski problem

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Pith's one-line read The capillary $L_p$-Minkowski problem is solved in the smooth category for every $p>1$: unique solutions for $p>n+1$, uniqueness up to scaling at $p=n+1$, and existence under capillary evenness for $1<p<n+1$.

desk verdict A genuinely new capillary L_p-Minkowski theory with solid a priori estimates, but the borderline case p=n+1 leans on a terse logarithmic gradient lemma whose p-independence needs to be checked before the result is fully trusted. read the letter →

arxiv 2505.07746 v3 pith:UUKNKE3Q submitted 2025-05-12 math.DG math.AP

Xinqun Mei , Guofang Wang , Liangjun Weng This is my paper · ORCID
classification math.DGmath.AP MSC 52A3935J2558J0535B65
keywords capillaryLp-surfaceareameasureLp-MinkowskiproblemMonge-AmpèreequationRobinboundaryconditionconvexbodyGaussmaplogarithmicgradientestimateBrunn-Minkowskitheory
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 introduces a capillary counterpart of the classical $L_p$-Minkowski problem: for convex bodies in the upper half-space that meet the boundary hyperplane at a fixed contact angle $\theta$, one prescribes the weighted surface-area measure $\ell h^{1-p}\det(\nabla^2 h+h\sigma)\,d\sigma$ on the spherical cap $C_\theta$. It shows that, for $p>1$ and $\theta\in(0,\pi/2)$, this geometric prescription is equivalent to solving the Monge-Ampère equation $\det(\nabla^2 h+h\sigma)=f h^{p-1}$ with the Robin boundary condition $\nabla_\mu h=\cot\theta\,h$, and it solves that equation in the smooth category. The result yields existence and uniqueness statements for all $p>1$, with the critical exponent $p=n+1$ handled by a limiting argument and the supercritical range $p>n+1$ by direct a priori estimates. A reader should care because this extends a central problem of convex geometry to a boundary-value setting and supplies uniform estimates for a Robin-boundary Monge-Ampère equation.

What carries the argument

The load-bearing objects are the capillary support function $\ell(\xi)=\sin^2\theta+\cos\theta\langle\xi,e\rangle$ of the reference spherical cap, and $u=h/\ell$, whose substitution converts the Robin problem into a Monge-Ampère equation with homogeneous Neumann boundary condition (3.5); this conversion is what permits the maximum-principle $C^0$ estimates. For the critical exponent $p=n+1$, the decisive mechanism is the logarithmic gradient estimate of Lemma 3.5, proved with the test function $\Phi=\log|\nabla(\log h-\frac{\cot\theta}{2\theta}d_N^2)|^2+e^{s d_N^2/(2\theta)}$, where $d_N$ is the geodesic distance on $C_\theta$ to its north pole $N=(0,\dots,0,1-\cos\theta)$; the exponential term balances boundary and interior contributions so the maximum principle can be applied. The $C^2$ estimates are then reduced to the boundary double-normal estimate through the auxiliary function $\zeta=e^{-d_{\partial C_\theta}}-1$, following the Neumann-boundary theory for Monge-Ampère equations. The continuity method is closed by a kernel-triviality lemma that uses integration by parts and the Robin boundary condition to show the linearized operator is invertible when $p\ne n+1$.

What would settle it

Compute the normalized approximating solutions from equation (4.9) for a sequence $\varepsilon\to 0$ and measure $\sup_{C_\theta}|\nabla\log(h_\varepsilon/\min h_\varepsilon)|$; this quantity must stay bounded for every fixed smooth $f$, and any divergence would refute the $p$-independence of Lemma 3.5 on which Theorem 1.1(2) rests.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.1. For any positive smooth $f$ on $C_\theta$, with $\theta\in(0,\pi/2)$: when $p>n+1$, equation (1.3) has a unique smooth convex solution $h$, and its $C^2$ norm is controlled by $n$, $p$, $\min f$, and $\|f\|_{C^2}$; when $p=n+1$, dilation invariance forces a unique-up-to-scaling solution $h$ together with a positive constant $\gamma$ solving $\det(\nabla^2 h+h\sigma)=\gamma f h^n$; and when $1<p<n+1$, a capillary even $f$, meaning $f(\xi_1,\dots,\xi_n,\xi_{n+1})=f(-\xi_1,\dots,-\xi_n,\xi_{n+1})$, admits a capillary even solution. Theorem 1.2 translates these analytic results into geometric language: the prescribed capillary $L_p$-surface area measure $\ell f\,d\sigma$ is realized by a capillary convex body, uniquely in the supercritical case and uniquely up to dilation in the critical case. The proof is a continuity method whose decisive new input is a logarithmic gradient estimate for the critical case, obtained with a test function that balances the Robin boundary term against interior terms so that the maximum principle applies.

Load-bearing premise

The proof of the critical case $p=n+1$ depends on the gradient bound for the logarithm of the solution staying uniform as $p$ approaches $n+1$ from above; if that uniformity fails, the normalized solutions need not converge.

Editorial extensions

If this is right

  • For $p>n+1$, every positive smooth $f$ determines exactly one smooth capillary convex body with capillary $L_p$-surface area measure $\ell f\,d\sigma$, with the solution controlled by the $C^2$ estimate.
  • For $p=n+1$, the correct statement is scale-free: the body is unique up to dilation, with a single positive constant $\gamma$ fixed by $f$.
  • For $1<p<n+1$, existence holds for capillary even data, so the capillary-symmetric part of the problem is resolved in the smooth category.
  • The capillary $L_p$-Brunn-Minkowski inequality of Proposition 2.2 holds for $p>1$, giving the capillary $p$-sum a variational foundation.
  • The uniform a priori estimates for $p\ne n+1$ make the continuity method effective and can serve as compactness input for nearby boundary-value problems on the spherical cap.

Reading between the lines

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

  • Editorial inference: the capillary evenness assumption in the intermediate range probably does more than enforce symmetry; it guarantees the origin lies inside the flat part of the boundary, and a different normalization might remove the assumption, paralleling how the classical problem dropped symmetry when $p\ge n+1$.
  • Editorial inference: the $\Phi$ test function for the logarithmic gradient estimate is a transferable device for Robin-boundary Monge-Ampère and Hessian equations, and could be applied to the capillary Christoffel-Minkowski problem the paper mentions as future work.
  • Editorial inference: the obstruction at $\theta>\pi/2$ is probably genuine rather than purely technical, since the strict convexity of $\partial C_\theta$ used in the $C^2$ estimates fails there; testing the same PDE on a cap with opening angle beyond a right angle would reveal whether the sign inconsistency is fatal.
  • Editorial inference: the constant $\gamma$ in the critical case behaves like a nonlinear Robin eigenvalue; studying how it varies with $f$ and $\theta$ could connect to capillary isoperimetric inequalities.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper introduces a capillary analogue of Lutwak's L_p-Minkowski problem for capillary convex bodies in the upper half-space. With the capillary Gauss map, the problem is reduced to the Monge–Ampère equation det(∇²h+hσ)=f h^{p−1} on the spherical cap C_θ with the Robin boundary condition ∇_μ h = cot θ h. Theorem 1.1 claims: for p>n+1 a unique smooth solution; for p=n+1 a solution unique up to dilation together with a positive constant γ; and for 1<p<n+1 an even capillary solution when f is capillary even. Theorem 1.2 translates these into statements about capillary convex bodies with prescribed capillary L_p-surface area measure. The proof is a continuity-method package: Section 3 establishes C^0, C^1, logarithmic-gradient, and C^2 a priori estimates, and Section 4 performs the continuity argument, with the borderline p=n+1 handled by an approximation p=n+1+ε and a limiting argument. The paper is a direct continuation of the authors' earlier work [54, 55] and relies on those papers for several geometric ingredients.

Significance. If the results are correct, this is a substantial and natural extension of the capillary Minkowski problem to the full range p>1. The paper introduces a capillary L_p-surface area measure, proves a capillary L_p Brunn–Minkowski inequality, and gives smooth existence results under conditions that parallel the classical L_p-Minkowski problem. The analytic core is a nontrivial adaptation of the classical a priori estimates to the Robin boundary condition, and the p=n+1 limiting argument is a delicate part of the contribution. The writing is generally clear and the main structure of the proof is transparent. The paper does not rely on fitted parameters or numerical computation; all constants are claimed to depend only on explicit data. However, as it stands, a load-bearing step in the p-uniform logarithmic gradient estimate is only sketched, and the uniqueness claims in the main theorem are stated without proof. For these reasons the central theorem is not yet fully supported.

major comments (3)
  1. [Section 3.2, Lemma 3.5, Eqs. (3.28)–(3.33)] The proof of Lemma 3.5 does not currently establish the p-independence claimed in (3.16), which is exactly what is needed in Theorem 3.1(3) and in the compactness argument for p=n+1 in Section 4. In Eq. (3.28), the constant C is said to depend on n and ||log bf||_{C^1(C_θ)}. Since bf=e^{p0 v}f with p0=p−n−1, the quantity (log bf)_k contains p0 v_k, so ||log bf||_{C^1} depends on |∇v| unless this is explicitly shown to be harmless. The decisive claim (3.29) is introduced to absorb the term −8|∇v|², but its proof is a sketch: in both cases the remainders are compressed into O(|∇v|), and the text does not track p0 through Eqs. (3.31)–(3.33). If the O(|∇v|) terms have constants that depend on p0, or if the coefficient in (3.33) is not exactly 6 after tracking p0, then the final bound |∇w|≤C in (3.16) is not uniform in p, and the passage ε→0 in the proof of Theorem 1.1(2) does not close. Please provide a complete derivation of (3.29) with all constants explicitly independent of p0∈[0,1], and in particular make explicit how the p0 v_k term in (log bf)_k is treated in (3.28).
  2. [Section 4, proof of Theorem 1.1, uniqueness paragraph] The uniqueness assertions in Theorem 1.1(1)–(3) are part of the main theorem, but the proof merely states that uniqueness can be established by arguments analogous to [22], [23], and [51] and omits the details. This is not adequate for a journal proof, especially for the p=n+1 case, where uniqueness is only up to dilation and the additional constant γ must also be shown to be unique. Please include a self-contained proof of the uniqueness statements, or at a minimum state the precise uniqueness results from the cited references and verify that their hypotheses apply to the Robin boundary value problem (1.3) on the spherical cap C_θ. As written, the theorem claims more than the paper proves.
  3. [Section 4, proof of Theorem 1.1(2), limiting argument] In the passage to the limit for p=n+1, the printed text states that Lemma 3.2 implies (min_{C_θ} h_{ε_j})→γ for some positive γ. What is actually needed in the equation for h̃_ε is convergence of the coefficient (min_{C_θ} h_ε)^ε to a positive constant, because the approximating equation is det(∇²h̃+h̃σ)=(min h̃)^ε f h̃^{n+ε}. Lemma 3.2 bounds h_ε^ε, not h_ε itself, so the stated convergence of min h_{ε_j} is not what follows. This is likely a typographical omission of the exponent, and the argument is repairable by passing to a subsequence for which (min h_{ε_j})^{ε_j} converges, but as written the limiting step contains a gap.
minor comments (3)
  1. [Section 1, abstract and introduction] The abstract states the problem for p∈R while the body of the paper treats p≥1 and the main theorem treats p>1. Please harmonize the stated range to avoid ambiguity.
  2. [Section 2, Proposition 2.2] The proofs of the capillary L_p Brunn–Minkowski inequalities are omitted with a reference to the classical treatment in [59]. Since these inequalities are stated as results of the paper and are used in Section 3, please include either the proofs or a precise statement of which classical result is being adapted and why the capillary setting preserves the argument.
  3. [Section 3, Lemma 3.3] In the proof of the lower bound in Lemma 3.3, the step from a uniform positive lower bound on the volume |Σ| to a positive lower bound on the inner radius is stated in one sentence. Please spell out the argument, since this final implication is needed for the lower bound of h.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorem is proved by a continuity method with a priori estimates, and the critical p=n+1 case is obtained by a limiting argument rather than by assuming the conclusion.

full rationale

The derivation chain is not circular. Theorem 1.1 is an existence theorem for the Robin Monge–Ampère problem (1.3), and Theorem 1.2 is merely the translation of that PDE statement into the language of the capillary Lp-surface area measure via the identity dS^c_p = ℓ h^{1-p} det(∇^2 h + h σ)dσ; this is a direct reformulation, not a forced prediction. The p=n+1 case is handled by solving the approximating problem det(∇^2 h + hσ) = f h^{n+ε} and passing ε→0: the paper obtains uniform C^{3,α} bounds on eh_ε = h_ε/min h_ε from Theorem 3.1 and Lemma 3.5, then extracts a subsequence and defines γ = lim (min h_{ε_j})^ε. The limit equation det(∇^2 h + hσ) = γ f h^n is derived from the approximating equations, not assumed. The algebraic step (3.29) inside Lemma 3.5, and the claimed p-independence of the constant in (3.16), are the most delicate parts of the proof and could be a correctness or rigor issue if wrong, but they are not circular: no target conclusion is used as an input. The paper's citations to the authors' prior works [54] and [55] supply geometric setup facts, the Robin boundary condition parametrization, and a Poincaré-type inequality used in Lemma 4.1; these are independent published results with stated assumptions that do not include Theorem 1.1, and thus they are real evidence rather than a circular self-citation chain. No parameter is fitted to the prescribed data, and no quantity called a prediction is defined in terms of the answer it is supposed to determine.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new entities or fitted parameters. The central claim rests on prior results from the authors' earlier papers for the geometric setup and on standard elliptic regularity theory; these are external inputs, not consequences of the main theorem.

assumptions (5)
  • domain assumption The capillary Gauss map tilde_nu: Sigma -> C_theta is a diffeomorphism, so convex capillary hypersurfaces can be parametrized by support functions on C_theta.
    Invoked in Section 2 and Eqs. (2.1)-(2.2), citing [55, Section 2].
  • domain assumption Support functions of capillary convex bodies satisfy the Robin condition nabla_mu h = cot(theta) h on the boundary of C_theta.
    Used throughout, e.g., Eq. (1.3), citing [55, Lemma 2.4].
  • domain assumption Capillary isoperimetric inequality for convex capillary bodies in the half-space.
    Used in Lemma 3.3 to bound outer radius and volume; cited from [49, Corollary 1.3, Eq. (1.7)] and [48, Chapter 19].
  • standard math Lieberman-Trudinger regularity and Schauder theory for fully nonlinear elliptic equations with oblique boundary conditions.
    Used in Theorem 3.1 to upgrade C^2 estimates to C^{3,alpha}; cited [37, Theorem 1.1] and [38].
  • domain assumption The inequality from [55, Corollary 3.3] used to prove kernel triviality in Lemma 4.1.
    A weighted Poincare-type inequality for capillary convex functions; cited as a special form in Lemma 4.1.

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Cite this review

Pith. "Pith review of The capillary $L_p$-Minkowski problem." pith.science (2026). https://pith.science/paper/UUKNKE3Q

@misc{pith2026250507746,
  author       = {Pith},
  title        = {Pith review of: The capillary $L_p$-Minkowski problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UUKNKE3Q}},
  note         = {Machine review of arXiv:2505.07746}
}
abstract

This paper is a continuation of our recent work [Adv. Math. 469 (2025), Paper No. 110230] concerning the capillary Minkowski problem. We propose, in this paper, a capillary $L_p$-Minkowski problem for $p\in \mathbb{R}$, which seeks to find a capillary convex body with a prescribed capillary $L_p$-surface area measure in the Euclidean half-space. This formulation provides a natural Robin boundary analogue of the classical $L_p$-Minkowski problem introduced by Lutwak [J. Differential Geom. 38 (1993), no. 1, 131--150]. For $p>1$, we resolve the capillary $L_p$-Minkowski problem in the smooth category by reducing it to a Monge--Amp\`ere equation with a Robin boundary condition on the unit spherical cap.

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Forward citations

Cited by 3 Pith papers

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