{"id":"f92419a5-3f73-4c1e-b1e8-b8d8df9533f0","arxiv_id":"1908.09113","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For annuli in the plane, the BV least gradient problem with BV boundary data is shown to be equivalent to a boundary-to-boundary optimal transport problem, and under admissibility conditions a unique solution with W^{1,p} regularity exists.","lead":"The paper proves that the least gradient problem on a two-dimensional annulus is equivalent to an optimal transport problem on the boundary, and gives conditions under which a unique solution exists with W^{1,p} regularity. It is the first existence and regularity theory for this variational problem on a domain with a hole, where the boundary has two components and the domain is not convex.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 3.3, the measure-valued converse used to convert Beckmann solutions into least gradient solutions, is stated without proof; Theorem 3.4 and the existence results of Section 4 depend on it.","rationale":"Proposition 3.3 is the linchpin of the paper's central equivalence, and the reader's weakest_assumption identifies exactly this unproved measure-valued representation. I agree with that assessment. The rest of the paper is largely well-structured: assumptions (H1)-(H4) are explicit, Lemma 4.2's cyclical monotonicity argument is sound in outline, and Theorem 4.4 constructs a Beckmann solution with zero boundary mass. The W^{1,p} section (Theorem 5.1) has sketched approximation and endpoint arguments, but those are refinements of the regularity claim; the equivalence theorem is the foundation. Since Proposition 3.3 is likely provable by standard distributional potential theory (the boundary condition makes the rotated 1-form exact on the annulus), the appropriate verdict is conditional rather than reject: the paper should either supply the proof or cite a complete reference for the measure-valued representation. Other flagged issues, such as the convexity slip in Lemma 2.4 and the atomic approximation in Theorem 5.1, are secondary and do not change the central verdict.","tokens_in":22279,"tokens_out":21505,"duration_ms":228452,"concrete_test":"Provide a complete proof of Proposition 3.3: for v∈M(Ω;R^2) satisfying |v|(∂Ω)=0 and ∇·v=f with f(∂Ω±)=0, define the distributional 1-form ω=-v_2 dx+v_1 dy; prove dω=0 on Ω and compute the period of ω around ∂Ω- using the boundary condition to show it vanishes; then define u(x)=∫_{x0}^{x}ω along paths, prove path-independence, verify u∈BV(Ω) and R_{π/2}Du=v. If the period computation fails or the BV estimate cannot be established, Theorem 3.4's reverse implication and the existence results in Section 4 are not justified.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing gap is Proposition 3.3 (Section 3), which is stated without proof: any v∈M(Ω;R^2) with |v|(∂Ω)=0 and ∇·v=f, where f∈M(∂Ω) and f(∂Ω±)=0, is claimed to be representable as v=R_{π/2}Du for some u∈BV(Ω). This proposition is the mechanism by which Theorem 3.4 converts a solution of the Beckmann problem (3.1) into a solution of the variational problem (3.2). Without it, the 'provided |v|(∂Ω)=0' direction of Theorem 3.4 has no proof, so Theorem 4.4 cannot pass from a Beckmann solution to a least gradient solution, and Theorem 4.5 does not follow. The L1 predecessor Proposition 3.2 is proved by reducing to the convex domains Ω± with [13, Prop 2.1], but the measure case is not a routine limiting argument: one must justify that the rotated 1-form -v_2 dx+v_1 dy is exact on the annulus, show that its period around ∂Ω- vanishes because v is tangential on the boundary, construct the BV potential by path integration, and control the boundary mass. The statement is probably true, but as written it is an unproved lemma at the center of the paper's main equivalence.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":22679,"tokens_out":5929,"duration_ms":63218,"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":[{"comment":"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":"Section 3, Proposition 3.3"},{"comment":"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":"Section 4, Theorem 4.4"},{"comment":"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.","section":"Section 5, Theorem 5.1"}],"minor_comments":[{"comment":"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}.","section":"Lemma 2.4"},{"comment":"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":"Problem (3.1) and Theorem 4.4"},{"comment":"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).","section":"Section 5, notation"}],"recommendation":"major_revision","confidential_remarks":"The paper relies heavily on the authors' own earlier works [6,10,13], but those are published and independently checkable, so I do not see a circularity problem. The decisive issue is the missing proof of Proposition 3.3; if the authors supply that proof and complete the estimates in Section 5, the paper would be a solid contribution to the least gradient and optimal transport literature."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nYou 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.\n\nWhat 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.\n\nNow 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.\n\nThe 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.\n\nMy 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.","headline":"A genuinely new least-gradient/optimal-transport framework for annuli, held back by an unproved measure-valued lemma at the center of the equivalence.","tokens_in":23076,"tokens_out":2321,"would_cite":true,"duration_ms":24180,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35J20","35J25","35J75","35J92"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["least gradient problem","BV functions","optimal transport","Beckmann problem","annulus","W1,p regularity","transport density","non-convex domain"],"falsifier":"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.","tokens_in":22114,"feed_emoji":"🔄","tokens_out":12198,"duration_ms":109311,"temperature":0.7,"pith_summary":"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.","feed_headline":"Least-gradient problem on annuli is a transport problem","feed_subtitle":"Admissibility conditions give existence, uniqueness, and W1,p regularity for non-convex rings.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the rotated-gradient representation $v=R_{\\pi/2}Du$ for convex domains and the equivalence between least gradient and Beckmann problems that the annulus proof adapts.","marker":"[13]"},{"why":"Supplies the boundary-to-boundary transport-density estimates and $W^{1,p}$ regularity results for uniformly convex domains that Section 5 extends to annuli.","marker":"[6]"},{"why":"States the Beckmann-Kantorovich duality and the form $v=-\\sigma\\nabla\\varphi$ used to construct optimal flows from optimal transport plans.","marker":"[20]"},{"why":"Gives existence and uniqueness for strictly convex domains through level-set construction, the baseline that motivates the annulus treatment.","marker":"[22]"},{"why":"Establishes that superlevel sets of least-gradient functions are minimal sets, used for the trace restrictions in Section 2.","marker":"[2]"},{"why":"Introduces admissibility conditions for convex but not strictly convex domains, which motivate hypotheses (H1)-(H4).","marker":"[18]"}],"fun_headline_variants":["Annulus least-gradient equals optimal transport","Transport equivalence solves least-gradient on rings","Least-gradient on annuli: transport yields uniqueness and regularity","Rings: least-gradient problem recast as transport"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Annulus least-gradient equals optimal transport","Transport equivalence solves least-gradient on rings","Least-gradient on annuli: transport yields uniqueness and regularity","Rings: least-gradient problem recast as transport"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000279,"raw_usage":{"total_tokens":1630,"prompt_tokens":892,"completion_tokens":738,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":508,"completion_tokens_details":{"reasoning_tokens":680}},"tokens_in":508,"tokens_out":738,"duration_ms":7047,"temperature":1.0,"reasoning_tokens":680,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:23:05.287766+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Górny, P","cited_arxiv_id":null,"evidence_quote":"Supplies the rotated-gradient representation $v=R_{\\pi/2}Du$ for convex domains and the equivalence between least gradient and Beckmann problems that the annulus proof adapts."},{"cited_title":"Dweik and F","cited_arxiv_id":null,"evidence_quote":"Supplies the boundary-to-boundary transport-density estimates and $W^{1,p}$ regularity results for uniformly convex domains that Section 5 extends to annuli."},{"cited_title":"Santambrogio,Optimal Transport for Applied Mathematicians, inProgress in Nonlinear Differential Equa- tions and Their Applications87, Birkhäuser, Basel, 2015","cited_arxiv_id":null,"evidence_quote":"States the Beckmann-Kantorovich duality and the form $v=-\\sigma\\nabla\\varphi$ used to construct optimal flows from optimal transport plans."},{"cited_title":"Sternberg, G","cited_arxiv_id":null,"evidence_quote":"Gives existence and uniqueness for strictly convex domains through level-set construction, the baseline that motivates the annulus treatment."},{"cited_title":"Bombieri, E","cited_arxiv_id":null,"evidence_quote":"Establishes that superlevel sets of least-gradient functions are minimal sets, used for the trace restrictions in Section 2."},{"cited_title":"Rybka and A","cited_arxiv_id":null,"evidence_quote":"Introduces admissibility conditions for convex but not strictly convex domains, which motivate hypotheses (H1)-(H4)."}],"review_version":1}