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arxiv: 2511.15021 · v2 · pith:46UUFOPNnew · submitted 2025-11-19 · 🧮 math.AP

A Liouville theorem for convex functions with periodic Monge-Amp\`ere measure

Tianling Jin , YanYan Li , Hung V. Tran , Xushan Tu This is my paper

Pith reviewed 2026-05-25 07:34 UTC · model grok-4.3

classification 🧮 math.AP
keywords Liouville theoremMonge-Ampère equationconvex functionsperiodic measureHarnack inequalityDirichlet-Voronoi tilingsglobal solutions
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The pith

Every global convex solution to det D²u = μ with periodic measure μ decomposes uniquely as a quadratic polynomial plus a periodic function.

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

The paper proves a Liouville-type rigidity result for convex functions on all of R^n whose Monge-Ampère measure is nonnegative, locally finite, and periodic. Any such u splits, up to an additive constant, into a quadratic polynomial and a periodic function. The result covers measures that may vanish, concentrate, or become singular, removing the density or bounded-log assumptions required in earlier theorems of Caffarelli-Li and Li-Lu. A key step is a new dichotomous Harnack inequality for the linearized equation that works even when standard doubling properties fail. In the model case of a periodic Dirac measure on the integer lattice the solutions correspond one-to-one with Dirichlet-Voronoi tilings of R^n, up to linear terms.

Core claim

Every global convex solution u of det D²u = μ in R^n, where μ is a nonnegative locally finite periodic Borel measure, admits a unique decomposition, up to an additive constant, as the sum of a quadratic polynomial and a periodic function. The proof relies on a new dichotomous Harnack-type inequality for linearized Monge-Ampère equations with nonnegative periodic measures. When μ is the periodic Dirac measure supported on Z^n, the solutions correspond, up to addition of a linear function, with Dirichlet-Voronoi tilings of R^n.

What carries the argument

Unique decomposition of a global convex solution into a quadratic polynomial plus a periodic function, obtained via a dichotomous Harnack inequality for the linearized Monge-Ampère operator under periodic measures.

If this is right

  • The decomposition classifies all global convex solutions under periodic forcing, including degenerate and singular cases.
  • In the Dirac-lattice example the solutions are in bijection with Dirichlet-Voronoi tilings up to linear functions.
  • The result extends the earlier theorems of Caffarelli-Li and Li-Lu to the full class of periodic Borel measures.
  • The dichotomous Harnack inequality compensates for the lack of doubling and engulfing properties in the degenerate setting.

Where Pith is reading between the lines

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

  • The same decomposition technique may apply to other fully nonlinear elliptic equations whose right-hand side is periodic.
  • Explicit constructions of periodic measures beyond the Dirac case could produce new families of periodic convex solutions.
  • The correspondence with Voronoi tilings suggests possible links to optimal-transport problems with periodic cost functions.

Load-bearing premise

The right-hand side measure must be periodic.

What would settle it

An explicit global convex function u whose Monge-Ampère measure is periodic yet u minus any quadratic polynomial fails to be periodic.

read the original abstract

We study global convex solutions of the Monge-Amp\`ere equation \[ \det D^2 u = \mu \quad \text{in } \mathbb{R}^n, \] where $\mu \not\equiv 0$ is a nonnegative locally finite periodic Borel measure on $\mathbb{R}^n$. We prove a Liouville-type theorem showing that every such solution admits a unique decomposition, up to an additive constant, as the sum of a quadratic polynomial and a periodic function. This extends earlier results of Caffarelli-Li and Li-Lu, which required $\mu$ to have a density with regular or bounded logarithm, to the full generality of periodic measures, allowing degeneracy and singularities. A key ingredient is a new dichotomous Harnack-type inequality for linearized Monge-Amp\`ere equations with nonnegative periodic measures, which compensates for the failure of doubling and engulfing properties in the degenerate setting. In the extremal example where $\mu$ is the periodic Dirac measure supported on the integer lattice, we show that the solutions, up to addition of a linear function, are in one-to-one correspondence with Dirichlet-Voronoi tilings of $\mathbb{R}^n$.

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

0 major / 2 minor

Summary. The manuscript proves a Liouville-type theorem for global convex solutions u to det(D²u) = μ in R^n, where μ is any nonnegative locally finite periodic Borel measure. Every such u decomposes uniquely (up to an additive constant) as a quadratic polynomial plus a periodic function. The argument relies on a new dichotomous Harnack inequality for the linearized Monge-Ampère operator associated to periodic μ, followed by extraction of the quadratic term whose Hessian equals the average mass of μ per period cell, with the remainder shown to be periodic. In the extremal case μ equal to the periodic Dirac measure on Z^n, solutions (modulo linear functions) are placed in bijection with periodic Dirichlet-Voronoi tilings.

Significance. If the central claims hold, the result removes all density or regularity assumptions on μ that were present in the theorems of Caffarelli-Li and Li-Lu, thereby covering fully general periodic measures that may be degenerate or singular. The dichotomous Harnack inequality supplies a new tool that bypasses the failure of doubling and engulfing properties; this is likely to be reusable in other degenerate fully nonlinear settings. The explicit bijection with Voronoi tilings supplies a concrete geometric realization of the extremal case.

minor comments (2)
  1. [Abstract] Abstract: the phrase 'dichotomous Harnack-type inequality' is used without a one-sentence indication of what the two alternatives in the dichotomy are; adding this would improve immediate readability for readers outside the immediate subfield.
  2. The statement that the quadratic term is 'unique up to an additive constant' for the full decomposition should be cross-referenced to the precise normalization (e.g., vanishing at the origin or zero average on the period cell) once it is introduced in the body.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the recognition of its significance in extending prior Liouville-type results to fully general periodic measures, and the recommendation of minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via new inequality

full rationale

The paper's central Liouville decomposition is obtained by first establishing a new dichotomous Harnack inequality for the linearized Monge-Ampère operator under periodic measures (independent of the target theorem), then applying it to control oscillations and extract the quadratic term whose Hessian matches the average measure mass, with the remainder shown periodic. This chain relies on the newly proven inequality rather than any self-definition, fitted input renamed as prediction, or load-bearing self-citation. Prior works (Caffarelli-Li, Li-Lu) are cited only for context on the regular-density case; the extension to singular periodic measures is carried by the fresh Harnack result and the Voronoi-tiling bijection for the Dirac case, both developed internally without reducing to inputs by construction. The argument is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard assumptions from convex analysis and the theory of Monge-Ampère equations. No free parameters are introduced or fitted. No new entities are postulated.

axioms (2)
  • domain assumption Solutions u are convex functions on R^n
    Required for the classical Monge-Ampère equation to make sense in this context.
  • domain assumption μ is a nonnegative locally finite periodic Borel measure
    This is the given data in the problem statement.

pith-pipeline@v0.9.0 · 5750 in / 1229 out tokens · 30063 ms · 2026-05-25T07:34:05.309240+00:00 · methodology

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Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

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