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On a general class of free boundary Monge-Amp\`ere equations

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Pith's one-line read The paper proves existence of solutions to a broad class of free boundary Monge-Ampère equations with general data, unifying known special cases and opening applications in optimal transport, geometry, and eigenvalue problems.

desk verdict Strong general existence theory for free-boundary Monge–Ampère, but the C^{1,α} regularity claim rests on an unproved linear-vanishing assertion and the hemispherical Minkowski application has a barycenter-condition mismatch. read the letter →

arxiv 2508.05551 v1 pith:R22A76JX submitted 2025-08-07 math.AP math.DG

Tristan C. Collins , Benjy Firester This is my paper
classification math.APmath.DG MSC 35J9635R3549Q20
keywords freeboundaryMonge-AmpèreequationoptimaltransportdegeneratedensitieseigenvalueMinkowskiproblemKähler-RiccisolitonsLegendretransformvariationalmethod
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 proves the existence of solutions to a broad family of free boundary Monge-Ampère equations: $\det D^2 u = \lambda \frac{f(-u)}{g(u^\star)h(\nabla u)}\,\chi_{\{u<0\}}$ on $\mathbb{R}^n$, with the gradient image $\nabla u(\mathbb{R}^n)$ prescribed to be a bounded convex set $P$ containing the origin. Equivalently, the Legendre transform (convex conjugate) $v = u^\star$ solves $\det D^2 v = \lambda^{-1} \frac{g(v)h(y)}{f(-v^\star)}$ on $P$ with $v^\star=0$ on $\partial P$, where $\lambda$ is determined implicitly. The authors show that for any pair of antiderivatives $(F,G)$ satisfying one of three structural hypotheses — either $G(s)=s$ with $\log F(t)=O(t)$, or a concave increasing $G$ and convex $F$ with prescribed growth — together with mild conditions on $h$ (doubling measure with a vanishing order), solutions exist for all large $\Lambda$ (and for all $\Lambda>0$ under the strong versions). This unifies previously known special cases and opens the way to applications in degenerate optimal transport, Monge-Ampère eigenvalue problems, a hemispherical Minkowski problem, and free boundary Kähler–Ricci solitons.

What carries the argument

The engine is a variational principle whose critical points are exactly the solutions: for $v\in \mathcal{C}_+$, the space of positive convex functions on $P$, the energy is $E(v) = -\log I(u) + \Lambda J(v)$, where $u=v^\star$, $I(u)=\int_{\{u<0\}} F(-u)\,dx$, $J(v)=G^{-1}(H^{-1}\int_P G(v)\,h\,dy)$, and $H=\int_P h\,dy$. The structural hypotheses (H1)–(H3) are tailored so that the functional is bounded below on the normalized class $\mathcal{C}_P = \{v : \int_P g(v(y))\,y\,h(y)\,dy = 0\}$ and that its sublevel sets are compact, using sharp lower bounds for $G(J(v))$ obtained via the John ellipsoid and the comparability $v(0)/(n+2)\le \inf_P v$. A min-max argument then produces critical poi

What would settle it

Examine the one-dimensional radial case for the non-example $F(s)=s$, $G(s)=-s^{-(n+1)}$. Integrating the ODE (5.11) with $u'(0)=0$, $u(0)<0$ and attempting to impose $u^\star|_{\partial B_1}=0$ should show that the solution degenerates ($v(0)\to 0$) before the boundary condition is met, confirming that the structural hypotheses cannot be relaxed. Alternatively, for $F(s)=s^2$, $G(s)=-s^{-(n+5)}+s$, plot the sublevel sets of the energy in the $(x,D)$ plane: a non-compact component persisting for all $\Lambda$ would falsify the compactness principle for that data.

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Extended reading notes

Core claim

The main theorems solve the free boundary problem for general data: if the antiderivatives $(F,G)$ satisfy Property (H1) (i.e. $G(s)=s$, $\log F(t)=O(t)$) or Property (H2)/(H3) (concave increasing $G$ with prescribed singular growth near $0$, convex increasing $F$ with polynomial-type growth), and if $h$ is a positive doubling density with a vanishing order on $P$, then for every $\Lambda>\Lambda_0$ — with $\Lambda_0=0$ in the strong versions — there exists $v\in C^{1,\gamma}(P)$ solving $\det D^2 v = \lambda^{-1} g(v)h(y)/f(-v^\star)$ in $P$, $v^\star=0$ on $\partial P$, with $\lambda=\lambda(\Lambda)$ implicit. Taking $u=v^\star$ and the convex envelope gives the original equation on $\mat

Load-bearing premise

The whole existence proof depends on the sharp lower bounds for $G(J(v))$ in Proposition 3.2: if those estimates fail for some allowed choice of $F$ and $G$, bounded-energy sequences can slip away to infinity and no solution is found.

Editorial extensions

If this is right

  • Theorem 1.1: for any bounded convex $P$ and density $h(y')\sim d(y',\partial P)^\alpha$, there is a homogeneous optimal transport map from the cone $C(P)$ with density $\rho_\alpha$ to the half-space with density $x_{n+1}^\beta$ for every $\beta>\alpha$.
  • Theorem 1.2: the equation $\det D^2 u = (-u)^k\,\chi_{\{u<0\}}$ with $\nabla u(\mathbb{R}^n)=P$ is solvable, uniquely up to translation, if and only if $0$ is the barycenter of $P$; hence the pair $(\Omega,u)$ is reconstructible from $P$ for the classical Monge-Ampère Dirichlet problem.
  • Theorem 5.2: under a weighted barycenter condition, a hemisphere with prescribed Gauss curvature and boundary on a hyperplane exists, with prescribed Gauss image $s^{-1}(P)$ — a hemispherical Minkowski problem.
  • Section 5.2.4: taking $F(s)=e^s-1$, $G(s)=s$ produces free boundary Kähler–Ricci solitons and toric Kähler–Einstein metrics on toric log Fano varieties.
  • Regularity: solutions lie in $C^{1,\gamma}(P)$, and with $f,g,h\in C^{k,\alpha}$ the solution is $C^{k+2,\beta}(P)$ by the interior regularity theory for Monge-Ampère equations.

Reading between the lines

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

  • If the paper is right, the boundary between existence and non-existence is plausibly sharp: the non-example $F(s)=s$, $G(s)=-s^{-(n+1)}$ (related to affine hemispheres) sits just outside the hypotheses (H3), suggesting the growth conditions are close to necessary.
  • A natural extension the authors flag is a localized compactness principle: for pairs like $F(s)=s^2$, $G(s)=-s^{-(n+5)}+s$, the energy sublevel set has a compact component, and a localized version of their argument might extract a solution there.
  • In the (H1) case the barycenter condition on $P$ is necessary; for nonlinear $G$, one might expect an analogous weighted-barycenter obstruction, giving a concrete test for the necessity of the normalization $\mathcal{C}_P$.
  • One could numerically iterate the reconstruction map of Question 5.1 from a non-symmetric starting polytope with barycenter at the origin to test whether the only fixed points are balls, probing the rigidity of the Monge-Ampère eigenvalue problem.
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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 / 6 minor

Summary. The paper develops a variational method for a general class of free-boundary Monge-Ampère equations of the form det D^2 u = λ f(-u)/(g(u^*)h(∇u)) χ_{u<0} with ∇u(R^n)=P, equivalently a dual equation for v=u^* on P. Under structural assumptions (H1)-(H3) on the antiderivatives F,G and assumptions (A1)-(A2) on h, Theorems 1.4 and 1.5 assert existence of v∈C^{1,γ}(P) solving the dual equation, with an implicit parameter λ=λ(Λ), and with Λ_0=0 in the strong cases. The strategy follows the variational framework of Collins-Tong-Yau: define a functional E(v)=-log I(v)+ΛJ(v), prove compactness of normalized minimizing/min-max sequences, pass to a limit, and identify the limit as the solution of an optimal transport problem. Applications are given to homogeneous optimal transport between a cone and a half-space, the Monge-Ampère eigenvalue problem, a hemispherical Minkowski problem, and free-boundary toric Kähler-Ricci solitons. I examined the stress-test concern about the compactness mechanism: Lemma 3.3 is actually valid (indeed v(y)≥v(0) for all y∈P), and I found no concrete error in Proposition 3.2. The serious problems are in Section 4, where the optimal-transport regularity argument contains an unproved and load-bearing assertion, and in the statement of Property (H1), which is weaker than what the proof uses.

Significance. If repaired, the paper would provide a broad unified existence framework for free-boundary Monge-Ampère equations, subsuming several previously special cases and yielding several attractive geometric applications. The sharp lower-bound estimates in Proposition 3.2 and the compactness program in Section 3 are substantive and appear original. The applications to cone-to-halfspace optimal transport, reconstruction from the gradient image, the hemispherical Minkowski problem, and toric free-boundary Kähler-Ricci solitons are all interesting. However, two load-bearing gaps in the current version prevent acceptance as stated: the proof of C^{1,γ} regularity relies on an unproved linear-vanishing property of u at the free boundary, and the hypotheses of Property (H1) do not include the convexity of F that the proof in the (H1) case explicitly uses. Both are repairable in principle, but they affect the main theorems as stated.

major comments (3)
  1. [§4, Proposition 4.3] The proof asserts 'Because F'(s)=f(s)∼s^{γ−1} for γ≥1 and u vanishes to order 1 on the ∂Ω, it follows from [18, Lemma 2.6] that dµ is a doubling measure.' No proof or justification of the linear vanishing is given. This is not a consequence of the preceding compactness arguments: the minimizer v is constructed only as a uniform limit of convex functions, and Question 4.1 explicitly leaves free-boundary regularity open. Since Theorems 1.4 and 1.5 claim v∈C^{1,γ}(P), this missing step is load-bearing for the central regularity assertion. Please either provide a proof of linear vanishing or state the main theorems with only Alexandrov/C^0 regularity and make C^{1,γ} a conditional statement.
  2. [§1.4, Definition 1.1 and §4, Proposition 4.3 (H1 case)] Property (H1) as stated only requires G'=const>0 and log F(t)=O(t); it does not require F to be convex. However, in the (H1) case of Proposition 4.3 the proof explicitly uses convexity of F to derive inequality (4.2): 'Since F is convex and I(u)=I(û), (4.2) holds.' Convexity is also used in Lemma 4.3. Unless convexity (and likely F(0)=0) is added to Definition 1.1, the proof of Theorem 1.4 is incomplete for the full class stated. The listed applications satisfy convexity, so this is a fixable strengthening, but the theorem currently overclaims.
  3. [§4, Lemma 4.4] Lemma 4.4 and Corollary 5.2 also rely on the same unproved assertion that u vanishes to order 1 on ∂Ω. The statement 'Since f(s)∼s^{γ−1} and u vanishes to order 1 on ∂Ω, the measure ... is doubling by [18, Lemma 2.6]' is used to deduce strict convexity of ∂Ω. Without a proof of the linear vanishing, these conclusions are unsupported. This is not an isolated local gap; it is the same regularity issue as in Proposition 4.3.
minor comments (6)
  1. [§1.4, Definition 1.1] If convexity of F is intended as part of Property (H1), it should be stated explicitly in the definition rather than appearing only in the proof. This would also clarify the relation to Corollary 2.1, which assumes F log-concave.
  2. [Theorem 1.4] The 'only if' part of the h-barycenter characterization is asserted informally in Section 1.2 but not proved in the body. Since Theorem 1.4 states an iff, a short argument should be included.
  3. [§5.3.2] Figure 1 is referenced in the discussion of non-examples, but no figure appears in the text. Please include the figure or remove the reference.
  4. [§3.3, Proposition 3.2] The notation |Ω°| is used without definition; presumably it denotes the volume of the polar body, but this should be stated. Also, the bullet summaries in the proof shift between ρ_- and ρ_+ (e.g., max{0,G(ρ_- L_n/(2√n))} versus max{0,G(nρ_+ ε_2 L_n)}); the equivalence follows from the definition of ε_2 but should be made explicit.
  5. [§5.2.4] The free-boundary Kähler-Einstein discussion is quite terse: the metric h_u on -K_X and the relation to e^{-u} should be defined precisely, or a reference with the formula should be given.
  6. [Question 5.1] The sentence 'What are can be said about this dynamical system?' contains a typo; should read 'What can be said about this dynamical system?'

Circularity Check

1 steps flagged · score 4.0 of 10

Central existence proof is largely self-contained, but a load-bearing boundary-normalization lemma is imported by self-citation from the authors' own preprint [13], and the C^{1,γ} regularity step asserts an unproved linear vanishing of u.

  1. self citation load bearing [Section 4, Lemma 4.1, used in Proposition 4.2]
    "Proof. This is just [13, Lemma 4], and the proof follows from the argument therein, using Lemma 1.1 in place of [13, Proposition 3.1]."

    Lemma 4.1 is the step that normalizes the minimizer to satisfy v >= phi_Omega and v = phi_Omega on partial P, which in turn gives the free boundary condition v* = 0 on partial P in the main theorems. The proof is not given in the present paper; it is deferred to [13], a preprint by the first author and coauthors. Thus a load-bearing part of the derivation chain is justified by self-citation rather than by a self-contained argument. This does not reduce the whole theorem to its inputs, but it is a circular reliance on the authors' own prior work at a central point.

full rationale

The central existence mechanism is not circular: Claim 2.1 derives the Euler-Lagrange equation from the energy functional, Section 3 supplies self-contained compactness estimates, and the h-barycenter condition is a genuine necessary condition rather than a hidden definition. Theorems 1.4 and 1.5 do not reduce to their inputs by construction. However, the proof of Proposition 4.2 uses Lemma 4.1 to impose the boundary normalization v = phi_Omega on partial P, and Lemma 4.1 is not proved: it is quoted as '[13, Lemma 4]' from a preprint by the first author and coauthors, with only a hint about replacing [13, Proposition 3.1] by the present Lemma 1.1. This is a load-bearing self-citation at the point where the free boundary condition v* = 0 enters. A separate, non-circular gap is in Proposition 4.3: the assertion that 'u vanishes to order 1 on the boundary' is stated without proof and is exactly the nondegeneracy needed to apply [18, Lemma 2.6]; this affects the correctness of the C^{1,gamma} conclusion but is not an equivalence-by-construction. Weighing these, the paper has some self-citation but the central claim retains independent content, giving score 4.

Assumptions & free parameters 3 free parameters · 9 assumptions · 0 invented entities

The paper's results rest on: (a) standard convex-geometry and analysis theorems (John, Prekopa, Gangbo-McCann, Caffarelli, Savin, Jhaveri-Savin); (b) modeling assumptions on the data h (doubling, vanishing order) and the barycenter condition; (c) structural conditions on (F,G) that are purpose-built for the compactness proof. There are no fitted constants: the only hand-chosen knob is Lambda, with a universal threshold, and the strong versions of (H1)-(H3) even give Lambda_0 = 0. No new entities (particles, forces, dimensions) are postulated. The heaviest external weight is on the authors' own preprint [13], which supplies the variational framework and Lemmas 4.1-4.2.

free parameters (3)
  • Lambda (energy slope) = Lambda > Lambda_0; Lambda_0 = 0 under (sH1)/(sH2)/(sH3)
    Tuning parameter in E(v) = -log I(u) + Lambda J(v); the PDE constant lambda = H g(G(J(v)))/(Lambda I(v)) is implicit. Chosen by hand, but theorems hold for every Lambda above a universal threshold, so it does not force the conclusion.
  • Structural exponents (nu, gamma, beta_1, beta_2) in (H2)/(H3) = nu >= 0, gamma >= 0, beta_2 >= beta_1 >= 0, nu >= v_o(h), gamma > nu (H3)
    Introduced ad hoc so the lower bounds for G(J(v)) in Proposition 3.2 close. Section 5.3 gives a concrete pair ((F,G) = (s, -s^{-(n+1)})) that fails these conditions and for which existence appears to fail, so the parameters are meaningful but proof-driven.
  • Vanishing order v_o(h) and exponent alpha in h ~ d(y, dP)^alpha = alpha >= 0, v_o(h) = alpha
    Properties of the given density h, not chosen by the authors; they enter the hypothesis nu >= v_o(h) of Theorem 1.5 and the alpha in Theorem 1.1. Listed for completeness.
assumptions (9)
  • standard math John's ellipsoid theorem (existence of John ellipsoid E with E subset Omega subset nE)
    Invoked in Lemma 3.1 and used throughout Section 3 to normalize the free boundary and convert energy estimates into axis-length estimates L_1, ..., L_n.
  • standard math Prekopa's theorem and its equality case (Brascamp-Lieb)
    Justifies convexity of -log I(u_t) (Claim 2.2) and the translation-uniqueness of minimizers in Lemma 5.1.
  • domain assumption Gangbo-McCann optimal transport theory [15]
    Proposition 4.3 uses it to conclude that the minimizer's gradient is the optimal transport map between dmu and dnu, hence solves the equation in the Alexandrov sense.
  • domain assumption Caffarelli's regularity theory and Jhaveri-Savin [18, Theorem 1.1]
    Converts doubling measures into C^{1,alpha} regularity of u and v in Proposition 4.3; depends on the asserted linear vanishing of u on the free boundary.
  • domain assumption Savin's free-boundary theory for the Monge-Ampere obstacle problem [27] (Lemma 4.4)
    Used for strict convexity of the solution domain; cited, not reproduced.
  • domain assumption (A1) dy_h = h dy is doubling; (A2) well-defined vanishing order v_o(h)
    Definitions 1.5-1.6; needed for Lemma 1.1, Proposition 4.1 and the regularity step. Theorem 5.2 does not restate them for h = K(1+|y|^2)^{(n+2)/2}.
  • domain assumption h-barycenter condition: integral over P of (y - y_0) dy_h = 0
    The necessary and sufficient condition in Theorems 1.4, 1.2 and 5.2-5.3; necessity is asserted in Section 1.2, not proved.
  • ad hoc to paper Structural Properties (H1), (H2) and (H3) on (F,G)
    Definitions 1.1-1.3 are designed so the integral estimates of Proposition 3.2 close. Not vacuous: Section 5.3's non-example F(s)=s, G(s)=-s^{-(n+1)} fails (H3) and solutions appear not to exist.
  • domain assumption Berman-Berndtsson [2] and Wang-Zhu [30] correspondence for toric Kaehler-Einstein metrics
    Section 5.2.4's identification of u_Lambda with a free-boundary Kaehler-Ricci soliton relies on this correspondence; the Ric(omega)=omega computation is sketched in one paragraph.

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Pith. "Pith review of On a general class of free boundary Monge-Amp\`ere equations." pith.science (2026). https://pith.science/paper/R22A76JX

@misc{pith2026250805551,
  author       = {Pith},
  title        = {Pith review of: On a general class of free boundary Monge-Amp\`ere equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/R22A76JX}},
  note         = {Machine review of arXiv:2508.05551}
}
abstract

We solve a general class of free boundary Monge-Amp\`ere equations given by \[ \det D^2u = \lambda \dfrac{f(-u)}{g(u^\star)h(\nabla u)}\chi_{\{u<0\}} \; \text{ in } \mathbb{R}^n, \quad \nabla u (\mathbb{R}^n) = P \] where $P$ is a bounded convex set containing the origin, and $h>0$ on $P$. We consider applications to optimal transport with degenerate densities, Monge-Amp\`ere eigenvalue problems, and geometric problems including a hemispherical Minkowski problem and free boundary K\"ahler-Ricci solitons on toric Fano manifolds.

Figures

Figures reproduced from arXiv: 2508.05551 by the authors.

Figure 1
Figure 1. The regions enclosed by the white boundary form the sublevel set of Eˆ(x, D) for the pair F(s) = 1 2 s 2 , and G(s) = −s −(n+5) + s for n = 4. The sublevel set has both a compact part, where we expect a solution to exist, and a non-compact part. It is possible that such solutions may be rigorously proved to exist by developing a suitably “localized” version of our main result. References [1] Alexandroff, A., Existen… view at source ↗

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