Least gradient problem on annuli

Samer Dweik; Wojciech G\'orny

REVIEW 3 major objections 3 minor 24 references

Least gradient problem on annuli

T0 review · 3 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper proves that on an annulus the least-gradient problem is equivalent to a boundary optimal transport problem, and gives admissibility conditions for existence, uniqueness, and W1,p regularity.

desk verdict A genuinely new least-gradient/optimal-transport framework for annuli, held back by an unproved measure-valued lemma at the center of the equivalence. read the letter →

arxiv 1908.09113 v1 pith:V3S6RI4S submitted 2019-08-24 math.AP math.OC

Samer Dweik , Wojciech Górny This is my paper · ORCID

classification math.APmath.OC MSC 35J2035J2535J7535J92

keywords leastgradientproblemBVfunctionsoptimaltransportBeckmannannulusW1pregularitydensitynon-convexdomain

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 studies the planar least-gradient problem (minimize $\int_\Omega|Du|$ among functions of bounded variation with a prescribed boundary trace) on an annulus, a domain between two nested strictly convex curves. The main claim is that on an annulus the problem is equivalent to a Beckmann optimal-transport problem whose source and target measures live on the boundary, even though the domain is non-convex and its boundary has two components. Under explicit admissibility conditions (H1)-(H4) on the boundary datum, the paper proves existence of a solution to the relaxed problem with prescribed tangential derivative, then passes to existence for a least-gradient datum obtained by adding constants on each boundary component; the optimal flow, and hence the solution, is unique when the positive part of the tangential derivative is atomless. It also obtains $W^{1,p}$ regularity of the solution from $L^p$ bounds on the transport density, for every $p\in[1,\infty]$ when the relevant boundary pieces are flat, and for $p\le 2$ otherwise.

What carries the argument

The load-bearing identity is the rotation formula $v=R_{\pi/2}Du$ in $\mathbb{R}^2$: rotation by $\pi/2$ interchanges gradients with divergence-free fields and normal traces with tangential derivatives, so the least-gradient functional becomes the Beckmann cost. The flow is then studied through an optimal transport problem between the positive and negative parts of the boundary measure $f=\partial_\tau g$, whose optimal plans move mass along transport rays, i.e. maximal segments on which a Kantorovich potential $\varphi$ satisfies $\varphi(x)-\varphi(y)=|x-y|$. The admissibility conditions (H1)-(H4) guarantee that every such ray is a segment contained in the annulus and connects the prescribed monotonicity arcs of the boundary datum; from the optimal plan $\gamma$ the paper builds the flow $v_\gamma=-\sigma\nabla\varphi$, shows that $\sigma(\partial\Omega)=0$, and uses that boundary-free property to pass back to a least-gradient solution.

What would settle it

Look at the vortex field $v=(-x_2/|x|^2, x_1/|x|^2)$ on the annulus $B(0,2)\setminus B(0,1)$: it is divergence-free, its normal component vanishes on both boundary circles, and $|v|(\partial\Omega)=0$, but its circulation around the inner circle is $2\pi$. If this field cannot be written as $R_{\pi/2}Du$ for any $u\in BV(\Omega)$, then Proposition 3.3 is false and the equivalence theorem lacks its key step.

Watch

Extended reading notes

Core claim

The central discovery is the equivalence, on an annulus $\Omega=\Omega_+\setminus\overline{\Omega_-}$, between the Beckmann problem $\inf\{\int_{\overline\Omega}|v|:\nabla\cdot v=f\}$ and the relaxed least-gradient problem $\inf\{\int_\Omega |Du|:\partial_\tau(Tu)=f\}$; the two infima coincide, and optimal objects transfer in both directions. In dimension two the transfer is carried by the rotation operator $v=R_{\pi/2}Du$, which turns a gradient into a divergence-free vector measure and the normal boundary component into the tangential derivative of the trace. Because the annulus is not simply connected, the proof needs an extension argument through the inner hole and a representation lemma for measure-valued flows. With the admissibility conditions (H1)-(H4) in force, every transport ray between $f^+$ and $f^-$ lies inside the annulus, the optimal transport plan is unique when $f^+$ is atomless, the induced flow has no boundary mass, and the associated $u$ solves the least-gradient problem for a boundary datum obtained by vertical shifts on the two boundary components. In addition, $L^p$ summability of the transport density $\sigma$ translates directly into $W^{1,p}$ regularity of the solution.

Load-bearing premise

The argument rests on an unproved representation lemma: every measure-valued flow on the annulus that has zero flux across each boundary component and puts no mass on the boundary must be the 90-degree rotation of the gradient of a function of bounded variation; if that fails, the construction of least-gradient solutions from optimal transport flows collapses.

Editorial extensions

If this is right

On an annulus, the least-gradient problem and the Beckmann boundary-transport problem have the same infimum, and any optimal flow with no boundary mass produces a least-gradient solution.
Under the admissibility conditions (H1)-(H4), a solution exists for some boundary datum obtained from the original datum by adding a constant on each boundary component; under an equal-total-variation condition the datum is identified explicitly and the solution is unique.
When the positive part of the tangential derivative is atomless, the underlying optimal transport plan is unique, so the least-gradient solution is unique.
Regularity transfers from boundary to interior: $g\in W^{1,p}(\partial\Omega)$ implies $u\in W^{1,p}(\Omega)$ for every $p\in[1,\infty]$ when the flat pieces are flat, and for $p\le 2$ when some flat piece is not flat and the outer domain is uniformly convex.

Reading between the lines

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

If the unproved representation lemma is valid, the same Beckmann-to-least-gradient transfer should extend to domains with several inner holes, because the proof of the trace variation bound only needs positive distances between the inner components and the outer boundary.
The rotation representation on a non-simply-connected domain implicitly requires the optimal flow to have zero circulation around each hole; checking whether the flows constructed from boundary-to-boundary transport plans automatically satisfy this would settle the status of the unproved lemma.
The admissibility inequalities (H4) are purely geometric, so for a fixed annulus and fixed monotonicity arcs one can verify them by computing distances between arcs, as the paper does in its worked example; this makes the existence criterion directly testable.

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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

Colleague,

You should know two things. First, this paper gives the first existence, uniqueness, and W^{1,p} regularity theory for the BV least gradient problem on an annulus, worked through the Beckmann problem as a bridge. That is a real step beyond the strictly convex domain literature. Second, the bridge has a missing plank: Proposition 3.3, which converts a measure-valued Beckmann solution into a BV least gradient solution, is stated without proof. The stress-test is right to call this load-bearing.

What is genuinely good: the trace results in Section 2—Lemma 2.4, TV(g−) ≤ TV(g+), finite monotonicity changes—are new structural facts about least gradient functions on annuli. The equivalence in Theorem 3.4 between the Beckmann problem and the tangential-derivative formulation is the right way to handle the non-simply-connected domain. The admissibility conditions (H1)–(H4) are explicit, and the examples in Section 6 show awareness of their scope. Section 5's method—Lp bounds on the transport density giving W^{1,p} regularity—is a natural extension of the strictly convex case.

Now the soft spots. Proposition 3.3 is the real problem. For L^1 fields, Proposition 3.2 patches the convex-domain result by subtracting the inner field. For measures, you cannot just repeat that trick; you need to prove that the rotated 1-form is exact on the annulus and that the period around ∂Ω− vanishes. It may well be true, but it is not proved, and Theorem 3.4 and Theorem 4.5 lean directly on it. This needs to be written out. Lemma 2.4 also has a serious slip: the proof writes P(Ω,R2) ≥ M dist(∂Ω−,∂Ω+), but the quantity that should be bounded below is the perimeter of the level set E_t, not the perimeter of Ω. As written that inequality is not justified. Theorem 5.1 has a standard-looking approximation argument (atomic target measures) whose endpoint details are compressed; that is minor by comparison.

The citation pattern is fine. The authors lean on their own prior work [6,10,13], but those are published and checkable theorems, and the new results are not defined in terms of the target theorem. No circularity.

My take: the architecture is right and the results are probably correct, but the manuscript as submitted is not fully rigorous. It deserves a serious referee, not a desk reject. If the referee insists on a proof of Proposition 3.3 and a corrected Lemma 2.4, the paper would be a solid contribution.

Referee Report

3 major / 3 minor

Summary. The paper studies the BV least gradient problem on a planar annulus Ω = Ω+ \ Ω−, where Ω− is compactly contained in the strictly convex domain Ω+. The central claims are: (i) the least gradient problem with boundary datum g ∈ BV(∂Ω) is equivalent, in an appropriate sense, to a Beckmann optimal transport problem and to a Monge-Kantorovich problem with boundary-supported measures; (ii) under structural admissibility conditions (H1)–(H4), the Beckmann problem has a solution with zero boundary mass, unique when f+ is atomless, and this yields existence of a solution to the least gradient problem for some boundary datum g̃ with ∂τ g̃ = f; and (iii) under an additional condition (H5), transport densities are Lp and the corresponding least gradient solutions are W^{1,p} for all p ∈ [1, ∞] when the relevant boundary arcs are flat. The main results are Theorem 3.4, Theorem 4.4, Theorem 4.5, Theorem 5.1, and Corollary 5.2.

Significance. If the proof gaps identified below are filled, this would be a valuable contribution to the least gradient literature. The paper extends existence and regularity theory beyond strictly convex domains, makes a nontrivial connection between least gradient problems and optimal transport on domains with holes, and is careful to state structural hypotheses and discuss their optimality with concrete examples. The authors also provide explicit examples of existence and non-existence, and they are transparent about the limitations of their approach. The main unresolved issue is the missing proof of the measure-valued converse in Proposition 3.3, which is load-bearing for the equivalence theorem and hence for the existence results in Section 4; Section 5 also contains several unfinished estimates.

major comments (3)

[Section 3, Proposition 3.3] Proposition 3.3 is stated without proof, but it is the measure-valued converse used in the proof of Theorem 3.4. The statement that every v ∈ M(Ω; R^2) with |v|(∂Ω)=0 and ∇·v=f can be written as v=R_{π/2}Du for some u∈BV(Ω) is not a routine limiting case of Proposition 3.2. On the annulus, the rotated one-form -v_2 dx + v_1 dy is not automatically exact, and one must prove that its period around ∂Ω− vanishes, that the BV potential can be constructed by path integration, and that the condition |v|(∂Ω)=0 controls the boundary contribution to Du. Without this proof, the direction of Theorem 3.4 constructing a solution of (3.2) from a solution of (3.1) has no justification, and Theorem 4.5 inherits the gap. Please supply the full proof.
[Section 4, Theorem 4.4] The uniqueness claim in Theorem 4.4 depends on the assertion, made after Proposition 3.7, that 'every solution w for the Beckmann problem (3.1) is of the form w=w_γ for some optimal transport plan γ.' This representation is attributed to [20, Chapter 4] but is not proved in the present setting, where the measures f+ and f− are supported on the boundary and the domain is not convex. The representation is necessary to conclude that uniqueness of the optimal transport plan implies uniqueness of the Beckmann solution. Please provide a proof or a precise sufficient condition from the literature that covers this boundary-supported, non-simply-connected case.
[Section 5, Theorem 5.1] The proof of the Lp estimate for the transport density is incomplete. The displayed computation of ||σ++_{i,j}||_{L^p} uses an atomic approximation of f+ and a change of variables, but the Jacobian computation, the integration bounds, and the passage to the limit n→∞ are only sketched. The estimate for σ− is dismissed with the sentence that it follows from 'an approximation of f+ by an atomic sequence,' but σ− is supported on different segments and requires a separate argument. In addition, the theorem states p∈[1,∞] while the proof only addresses p<∞. Since Corollary 5.2 is the paper's W^{1,p} regularity claim, these gaps must be repaired.

minor comments (3)

[Lemma 2.4] In the proof of Lemma 2.4, the sentence 'As Ω is a convex subset of the plane, ∂Ω is homeomorphic to a circle' should refer to Ω±, since Ω is an annulus. Also, the inequality P(Ω,R^2) ≥ M dist(∂Ω−,∂Ω+) requires an explanation of how the number of transition points of {g≥t} on ∂Ω− forces at least that many disjoint segments of the level set ∂{u≥t}.
[Problem (3.1) and Theorem 4.4] The domain of the measures in the Beckmann problem is inconsistent: problem (3.1) is written with v∈M(Ω;R^2) and integral over ̅Ω, while Theorem 4.4 writes v∈M(̅Ω;R^2). The authors should fix one convention and state whether |v|(∂Ω) is meaningful by extension or by allowing measures on the closure.
[Section 5, notation] The notation F_i^+ in Theorem 5.1 conflicts with the decomposition F_i^{++} ∪ F_i^{+-} used in condition (H2); the first paragraph of Theorem 5.1 says 'assume that F_i^+ is a flat part for each i,' but later discusses the case where F_i^+ is not flat. Please clarify which arcs are assumed flat and how this relates to (H2).

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Beckmann/least-gradient equivalence is proved by explicit rotation identities, and the prior self-citations used are independent published theorems.

full rationale

The paper's central equivalence (Theorem 3.4) is not circular. Proposition 3.1 proves the forward direction directly by integration by parts: v=R_{pi/2}Du is divergence-free in Omega and has v·nu = d_tau(Tu). Proposition 3.2 proves the L^1 converse by subtracting the inner hole and applying the published convex-domain result [13, Prop. 2.1]; the annulus case is non-trivial because the domain is not simply connected. Although [13] and [6] are self-citations, they are published, parameter-free theorems whose assumptions do not include the annulus conclusion, so they are independent support. The admissibility conditions (H1)-(H4) are hypotheses, not fitted parameters. The L^p estimates in Theorem 5.1 are derived from the transport map structure with explicit constants, not assumed. One flagged issue is that Proposition 3.3, the measure-valued converse used in Theorems 3.4 and 4.5, is stated without proof; this is an omitted-proof and correctness risk, but it is not a circular reduction because the statement is not defined in terms of the target result and no fitted quantity is renamed as a prediction. Overall, the derivation is self-contained; circularity score is 0.

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

No numerically fitted parameters appear. The paper postulates structural admissibility conditions (H1)-(H5) which are domain assumptions on the boundary datum, not discovered constants. No new physical or mathematical entities are introduced; transport densities and Kantorovich potentials are standard objects.

assumptions (5)

domain assumption The domain is an annulus Ω = Ω+ \ Ω− with Ω± open bounded strictly convex and Ω− compactly contained in Ω+.
Definition 2.1 restricts the entire analysis; strict convexity is used for uniqueness of transport rays and trace structure.
standard math Optimal transport duality, cyclical monotonicity, and the L^p estimates for convex domains from Dweik-Santambrogio [6] and Górny-Rybka-Sabra [13] are taken as known.
Used throughout; [13, Proposition 2.1] is the core tool in Proposition 3.2, and [6] is used in Section 5.
ad hoc to paper Admissibility conditions (H1)-(H4): the boundary datum g ∈ BV(∂Ω), the boundary can be decomposed into matching increasing/decreasing arcs with equal total variation, visibility of corresponding arcs, and strict inequalities (H4) bounding distances between arcs.
These hypotheses are introduced in Section 4 specifically to make the proof work; the paper does not prove they are necessary, only gives examples showing failure when some are absent.
ad hoc to paper Condition (H5): there exists c>0 such that (y-x)·ν(x) ≥ c for x on target arcs and y on corresponding source arcs, where ν is the outward normal to ∂Ω−.
Introduced in Section 5 to make the Jacobian estimate in the L^p regularity proof work.
standard math The trace operator and BV theory on one-dimensional boundaries (coarea formula, precise representatives, superlevel sets minimality, Theorem 2.3).
Background from [2,8,9,11] used throughout.

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Pith. "Pith review of Least gradient problem on annuli." pith.science (2026). https://pith.science/paper/V3S6RI4S

@misc{pith2026190809113,
  author       = {Pith},
  title        = {Pith review of: Least gradient problem on annuli},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/V3S6RI4S}},
  note         = {Machine review of arXiv:1908.09113}
}

abstract

We consider the two dimensional BV least gradient problem on an annulus with given boundary data $g \in BV(\partial\Omega)$. Firstly, we prove that this problem is equivalent to the optimal transport problem with source and target measures located on the boundary of the domain. Then, under some admissibility conditions on the trace, we show that there exists a unique solution for the BV least gradient problem. Moreover, we prove some $L^p$ estimates on the corresponding minimal flow of the Beckmann problem, which implies directly $W^{1,p}$ regularity for the solution of the BV least gradient problem.

Figures

Figures reproduced from arXiv: 1908.09113 by the authors.

**Figure 1.** Visibility conditions in practice 5. W1,p regularity of the solution to the least gradient problem The aim of this section is to study the W1,p regularity of the solution u of the least gradient problem (3.2) in the case where the domain Ω is an annulus. First, we note that this question has already considered in [6], but in what concerns the case where the domain Ω is uniformly convex, where the authors proved the … view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Beckmann, A continuous model of transportation,Econometrica 20, 643–660, 1952

M. Beckmann, A continuous model of transportation,Econometrica 20, 643–660, 1952

work page 1952

[2] [2]

Bombieri, E

E. Bombieri, E. de Giorgi and E. Giusti , Minimal cones and the Bernstein problem,Invent. Math. 7, 243–268, 1969.v

work page 1969

[3] [3]

De Pascale, L

L. De Pascale, L. C. Ev ans and A. Pratelli, integral estimates for transport densities,Bull. of the London Math. Soc.. 36, n. 3,pp. 383-395, 2004

work page 2004

[4] [4]

De Pascale and A

L. De Pascale and A. Pratelli, Regularity properties for Monge Transport Density and for Solutions of some Shape Optimization Problem,Calc. Var. Par. Diﬀ. Eq. 14, n. 3, pp.249-274, 2002

work page 2002

[5] [5]

De Pascale and A

L. De Pascale and A. Pratelli, Sharp summability for Monge Transport density via Interpolation,ESAIM Control Optim. Calc. Var.10, n. 4, pp. 549-552, 2004

work page 2004

[6] [6]

Dweik and F

S. Dweik and F. Santambrogio , Lp bounds for boundary-to-boundary transport densities, andW 1,p bounds for the BV least gradient problem in 2D,Calc. Var. Partial Diﬀerential Equations58, no. 1, 31, 2019

work page 2019

[7] [7]

Summability estimates on transport densities with dirichlet regions on the boundary via symmetrization techniques

S. Dweik and F. Santambrogio, Summability estimates on transport densities with Dirichlet regions on the boundary via symmetrization techniques,preprint arXiv:1606.00705, 2016

work page Pith review arXiv 2016

[8] [8]

Ev ans and R.F

L.C. Ev ans and R.F. Gariepy , Measure theory and ﬁne properties of functions, CRC Press, Boca Raton, 1992

work page 1992