{"id":"6885482f-abc1-46bc-acb0-d54eeb303621","arxiv_id":"2505.07746","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Existence of smooth convex capillary bodies with prescribed capillary L_p-surface area measure is proved for all p>1, with a symmetry condition needed when 1<p<n+1.","lead":"This paper defines a capillary version of the L_p-Minkowski problem for convex bodies in a half-space with a constant contact angle, and proves existence of smooth solutions for all p>1 under natural conditions. It extends the authors' earlier solution of the capillary Minkowski problem at p=1 to the full L_p range, using new PDE estimates for a Monge-Ampere equation with Robin boundary condition.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3.5's p-independent gradient estimate rests on the unverified algebraic claim (3.29); without it the p=n+1 compactness argument in Theorem 3.1(3)/§4 does not close.","rationale":"The stress-test pass confirms the reader's weakest-assumption diagnosis. I read Lemma 3.5 carefully and compared it with the structure of the continuity argument. The p=n+1 existence is not an isolated argument: it is the limit of n+1<p<n+2 estimates, and the only place where p-independence is proved is Lemma 3.5. The algebra around (3.28)-(3.29) is the least documented part of that lemma. The boundary case is clean, and the φ(d) construction is standard; the interior maximum computation is where the proof compresses too much. I do not assert (3.29) is false; I assert it is the condition on which the central borderline case rests, and the printed verification is too terse to qualify as a proof. A concrete re-derivation with p0 tracked would settle it. Secondary issues noticed: the t=0 starting point in §4 is stated as h=ℓ, but ℓ solves (1.3) with f0=ℓ^{1-p} only after multiplying by sin^{2n}θ; this is fixable by using h0=sin^{2n/(p-1-n)}θ ℓ and does not affect the main argument. The uniqueness claims in Theorem 1.1 are also delegated to references, but uniqueness is not needed for the existence part that is the paper's main new contribution. Because the reader's conditional verdict already marks these points, the appropriate recommendation is to keep the verdict unchanged.","tokens_in":21855,"tokens_out":32616,"duration_ms":319274,"concrete_test":"Independently verify (3.29) with explicit p0 bookkeeping. (a) In the case S=∅, start from the maximality relation (3.30), prove B11=|∇v|²+O(|∇v|), expand the left side of (3.29) fully, and check that 4G11v1²(v·w)+2G11v1²|∇v|² equals 6|∇v|²+O(|∇v|) with the O-term having no p0 factor; then check that all terms coming from (logbf)_k=p0v_k+f_k/f are either nonnegative or bounded by C(1+|∇v|) with C independent of p0. (b) Repeat for S≠∅, verifying the displayed sum over i∈S equals 1+O(|∇v|^{-1}) with p0-independent constant. A failure of either check yields a counterexample to (3.29) and invalidates Lemma 3.5 as stated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The p=n+1 conclusion of Theorem 1.1 is obtained by solving the p=n+1+ε problem and passing ε→0; for this one needs Theorem 3.1(3), whose uniform C^{3,α} bound on eh=h/min h depends on Lemma 3.5 giving |∇log h|≤C with C independent of p for n+1≤p≤n+2. The decisive point in the proof of Lemma 3.5 is the inequality labelled (3.29). It is introduced after (3.28) and used to absorb the term -8|∇v|², but the derivation consists of two sketched cases in which all error terms are compressed into O(|∇v|). The equation being differentiated is detB=e^{p0v}f, so (log bf)_k contains p0v_k; whether this term is dropped as nonnegative or absorbed must be checked, and the remainders O(|∇v|) in (3.31)–(3.33) must be shown independent of p0∈[0,1]. The printed text does not track p0 through (3.29). If (3.29) has a hidden p0-dependent constant, or if the coefficient in (3.33) is not exactly 6, then (3.16) is not uniform in p and the compactness step for ehε in §4 fails. This is the load-bearing gap: it is exactly the step that carries the borderline case p=n+1.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":22167,"tokens_out":12116,"duration_ms":109373,"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":[{"comment":"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).","section":"Section 3.2, Lemma 3.5, Eqs. (3.28)–(3.33)"},{"comment":"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.","section":"Section 4, proof of Theorem 1.1, uniqueness paragraph"},{"comment":"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.","section":"Section 4, proof of Theorem 1.1(2), limiting argument"}],"minor_comments":[{"comment":"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.","section":"Section 1, abstract and introduction"},{"comment":"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.","section":"Section 2, Proposition 2.2"},{"comment":"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.","section":"Section 3, Lemma 3.3"}],"recommendation":"major_revision","confidential_remarks":"This is a serious paper by experienced authors and the overall strategy is appropriate for the journal. The main obstacle is the p-uniform logarithmic gradient estimate in Lemma 3.5; the sketched algebra around Eq. (3.29) is the kind of step that often hides a real dependence on p, and the theorem's borderline case depends on it. The uniqueness omissions are also worth fixing but are less fundamental. I would be willing to look at a revision that supplies the missing tracking of p0 in Lemma 3.5 and gives complete uniqueness proofs."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe short version: this is a solid, genuinely new extension of the L_p-Minkowski problem to a capillary Robin setting, and the main existence theorem for p>1 is believable and mostly well argued. The part I would send a referee at is Lemma 3.5, the logarithmic gradient estimate with constants independent of p, because that is exactly what carries the borderline case p=n+1.\n\nWhat is new and good: the paper formulates the capillary L_p-surface area measure and the associated Minkowski problem, reduces it to a Monge-Ampère equation with Robin boundary condition, and solves the smooth existence problem for all p>1. The p=n+1 case is handled by approximation, and the new test function (1.5) is a reasonable device for taming the boundary. The C^0, C^1, and C^2 estimates are laid out in detail, the continuity method is standard, and there is no fitting or circular normalization. The authors lean on their earlier work [54], [55], but that is honest and appropriate: the geometric setup and the Poincaré-type inequality come from there, and the papers are independent published work.\n\nSoft spots, in proportion. The typo in Eq. (3.4) is minor. The uniqueness claim in Theorem 1.1 is stated with “we omit the details for brevity,” which is acceptable only if the analogue is genuinely close to known results; I would want a precise citation or a short argument for at least one of the three ranges. The larger concern is the stress-test: Lemma 3.5 has to give |∇log h| bounded independently of p for n+1 ≤ p ≤ n+2, and the printed derivation is compressed. In (3.28) the constant is said to depend on ||log bf||_{C^1}, where bf = e^{p0 v}f, so the term p0∇v is sitting there and has to be absorbed in a way that is not fully shown. Claim (3.29) and the surrounding algebra are sketched in two cases, and the text does not track p0 through them. I think the estimate is probably repairable—the structure is plausible and p0∈[0,1] gives some room—but as printed it is not a line-by-line verification. Because Theorem 1.1(2) and Theorem 3.1(3) depend on this uniformity, this is not a cosmetic gap.\n\nWho is this for: convex geometers and PDE people working on Monge-Ampère equations with boundary conditions. It deserves a serious referee. My recommendation: send it out, and require the authors to expand Lemma 3.5, track p0 through (3.28)–(3.33), and either supply the uniqueness arguments or cite precise statements. If the p-independence checks out, this is a valuable contribution to the capillary Brunn-Minkowski theory.","headline":"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.","tokens_in":22700,"tokens_out":6218,"would_cite":true,"duration_ms":58931,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["52A39","35J25","58J05","35B65"],"pacs":[],"model":"deepseek-v4-flash","headline":"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 $1<p<n+1$.","keywords":["capillary Lp-surface area measure","Lp-Minkowski problem","Monge-Ampère equation","Robin boundary condition","capillary convex body","capillary Gauss map","logarithmic gradient estimate","Brunn-Minkowski theory"],"falsifier":"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.","tokens_in":21664,"feed_emoji":"📐","tokens_out":12453,"duration_ms":102958,"temperature":0.7,"pith_summary":"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.","feed_headline":"Capillary Lp-Minkowski problem solved for all p>1","feed_subtitle":"Existence and uniqueness for convex bodies in a half-space meeting the boundary at a fixed angle, for every p>1.","key_machinery":"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 $p\\ne n+1$.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Sets up capillary convex bodies in the half-space, the capillary area measure, and the reduction from the geometric prescription to the Monge-Ampère equation with Robin boundary condition.","marker":"[54]"},{"why":"Defines the classical $L_p$-surface area measure and $L_p$-Minkowski problem that the present work adapts to the capillary setting.","marker":"[43]"},{"why":"Provides the continuity-method template and the logarithmic gradient estimate for the critical case $p=n+1$ on the sphere.","marker":"[22]"},{"why":"Supplies the Neumann-boundary Monge-Ampère $C^2$ estimate framework used to control the boundary double-normal derivative.","marker":"[38]"},{"why":"Gives the capillary isoperimetric inequality used in the $C^0$ estimates for $1<p<n+1$.","marker":"[49]"},{"why":"Supplies the oblique-boundary regularity theory that upgrades the $C^2$ bounds to the $C^{3,\\alpha}$ estimates.","marker":"[37]"},{"why":"Provides the integration-by-parts kernel-triviality argument proving invertibility of the linearized operator in the continuity method.","marker":"[23]"},{"why":"Introduces the capillary even condition on $C_\\theta$ used for the intermediate range $1<p<n+1$.","marker":"[50]"}],"fun_headline_variants":["Capillary Lp-Minkowski: existence and uniqueness for all p>1","Capillary Lp-Minkowski problem fully solved for p>1","Unique capillary convex bodies for every p>1","Capillary Lp-Minkowski: Robin boundary analogue resolved","Capillary Lp-Minkowski solved: every p>1"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Capillary Lp-Minkowski: existence and uniqueness for all p>1","Capillary Lp-Minkowski problem fully solved for p>1","Unique capillary convex bodies for every p>1","Capillary Lp-Minkowski: Robin boundary analogue resolved","Capillary Lp-Minkowski solved: every p>1"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000866,"raw_usage":{"total_tokens":3771,"prompt_tokens":980,"completion_tokens":2791,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":596,"completion_tokens_details":{"reasoning_tokens":2701}},"tokens_in":596,"tokens_out":2791,"duration_ms":20017,"temperature":1.0,"reasoning_tokens":2701,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T22:10:29.575671+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"The capillary Minkowski problem","cited_arxiv_id":null,"evidence_quote":"Sets up capillary convex bodies in the half-space, the capillary area measure, and the reduction from the geometric prescription to the Monge-Ampère equation with Robin boundary condition."},{"cited_title":"TheBrunn-Minkowski-Fireytheory.I.MixedvolumesandtheMinkowski problem","cited_arxiv_id":null,"evidence_quote":"Defines the classical $L_p$-surface area measure and $L_p$-Minkowski problem that the present work adapts to the capillary setting."},{"cited_title":"On the equationdet(uij +uδij) = upf on Sn","cited_arxiv_id":null,"evidence_quote":"Provides the continuity-method template and the logarithmic gradient estimate for the critical case $p=n+1$ on the sphere."},{"cited_title":"TheNeumannproblemforequations of Monge-Ampère type","cited_arxiv_id":null,"evidence_quote":"Supplies the Neumann-boundary Monge-Ampère $C^2$ estimate framework used to control the boundary double-normal derivative."},{"cited_title":"Lp Christoffel-Minkowski problem: the case1 < p < k+ 1","cited_arxiv_id":null,"evidence_quote":"Provides the integration-by-parts kernel-triviality argument proving invertibility of the linearized operator in the continuity method."}],"review_version":1}