Front propagation on a general metric graph

Hiroshi Matano; Shuichi Jimbo

REVIEW 3 major objections 6 minor 19 references

Front propagation on a general metric graph

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper proves that a bistable front arriving from infinity along one arm of a metric graph either propagates into another arm or is fully blocked, with the outcome decided by whether the value of the limit profile at the exit point…

desk verdict Solid general framework for front propagation on metric graphs, but the load-bearing existence theorem is deferred to a star-graph paper and all results are conditional on it until a general proof appears. read the letter →

arxiv 2505.24418 v1 pith:3VCUJFPT submitted 2025-05-30 math.AP

Hiroshi Matano , Shuichi Jimbo This is my paper

classification math.AP MSC 35R0235K5735K5835C0735B08

keywords metricgraphreaction-diffusionequationfrontpropagationtravelingwavelong-timebehaviorbistablenonlinearitylimitprofileKirchhoffcondition

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 establishes a pass-or-block dichotomy for bistable reaction-diffusion fronts on metric graphs made of a finite central graph with several infinite arms. For a front entering from infinity along one arm, the long-time outcome toward any other arm is decided by a single number: the value of the limit profile at that arm's exit point, compared with a threshold beta fixed by the nonlinearity alone. A value above beta means the front propagates along that arm; a value at or below beta means it is blocked, with no intermediate behavior. The paper further proves that propagation is transitive across arms and robust under small perturbations of the central graph, and it delivers sharp criteria for star graphs while exhibiting graphs where propagation is partial or one-way.

What carries the argument

The limit profile v_i(x) = lim_{t->infinity} u_i(t,x) is defined from the unique entire solution u_i that converges as t->-infinity to the traveling wave $\varphi$(-x_i - ct) on the incoming arm and to 0 elsewhere. Each restriction of v_i to another outer path is a one-dimensional stationary trajectory, so the phase portrait of u'' + f(u)=0 decides everything: the value of v_i at the exit point relative to $\beta$ selects either the stable manifold of (1,0) (propagation) or the homoclinic pulse orbit (blocking). The minimality theorem - v_i is the smallest supersolution tending to 1 along Omega_i - is the engine that converts this local phase-portrait fact into the transitivity result and into the perturbation and robustness theorems.

What would settle it

Take a symmetric 3-star graph with equal edge thickness and any bistable nonlinearity satisfying (1.8), and compute the limit profile v_1 at the center. The dichotomy theorem predicts that for each other arm, v_1 at the exit point exceeds beta if and only if the restriction of v_1 to that arm tends to 1 at infinity, while a value at or below beta forces the restriction to decay to 0. A direct numerical construction of the front-like solution, or an explicit stationary solution whose exit-point value equals beta but whose restriction does not tend to 0, would falsify the central claim.

Watch

Extended reading notes

Core claim

The paper's central claim is the propagation-blocking dichotomy (Theorem 2.1): for any two outer paths Omega_i and Omega_j, the restriction of the limit profile v_i to Omega_j converges at infinity to either 1 or 0, and it converges to 1 exactly when v_i(P_j) > $\beta$, while it converges to 0 when v_i(P_j) <= $\beta$, with the same threshold $\beta$ defined by F($\beta$)=0. This makes the propagation index P(i,j) a genuine binary invariant of the graph and the nonlinearity, well-defined by the uniqueness of the underlying front-like solution. The authors prove universal consequences that hold for every center graph: propagation is transitive in the sense that P(i,j)=1 and P(j,k)=1 imply P(i,k)=1 (Theorem 2.5); the limit profile v_i is the minimal nonnegative supersolution that tends to 1 along Omega_i (Theorem 2.2); and v_i is linearly stable on compact subgraphs (Theorem 2.3). They also prove a closedness-robustness pair: limits of blocking graphs or blocking nonlinearities are blocking, so propagation survives sufficiently small perturbations (Theorems 2.7-2.9). For star graphs, they recover and sharpen the earlier criterion to unequal edge thickness: F(1) + ($R_i^{2}$ - 1)F(a) > 0 gives propagation from Omega_i to all other arms, while <= 0 gives blocking everywhere from Omega_i. Finally, they construct explicit graphs showing partial propagation and one-way propagation, and they show that a graph with a reservoir-type subgraph can exhibit incomplete invasion, where the front reaches every outer path yet the limit profile stays small inside the reservoir.

Load-bearing premise

The entire framework rests on Theorem 1.3, which asserts that for each incoming arm there exists a unique entire solution with the prescribed traveling-wave behavior in the far past; the paper says existence follows from the same super-subsolution argument as the star-graph case but does not reproduce that construction for a general center graph, so if such a solution failed to exist or failed to be unique, the limit profile and everything built on it would be undefined.

Editorial extensions

If this is right

Propagation and blocking become well-defined binary properties for every ordered pair of outer paths, independent of the detailed shape of the initial front.
Propagation is transitive: if the front from Omega_i reaches Omega_j and the front from Omega_j reaches Omega_k, then the front from Omega_i necessarily reaches Omega_k.
Propagation is robust: whenever a front propagates from one outer path to another for a given center graph or nonlinearity, the same propagation persists for every sufficiently close graph or nonlinearity; equivalently, limits of blocking graphs or blocking nonlinearities are blocking.
On star graphs with unequal edge thickness, the sharp criterion F(1)+(R_i^2-1)F(a)>0 settles propagation to all other arms and <=0 settles blocking, so partial propagation cannot occur on a star graph.
General center graphs can display partial propagation and one-way propagation, and a reservoir-shaped subgraph can force incomplete invasion even when the front reaches every outer path.

Reading between the lines

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

The dichotomy suggests a computational strategy: compute only the stationary limit profile on the finite center graph with the appropriate boundary behavior at the exit points, then read off every propagation index from whether the exit values exceed beta, avoiding full time-dependent simulations. This is an editorial extension, not a claim made in the paper.
The transitivity of the propagation index means the outer paths can be partially ordered by the relation 'reaches', which could yield a hierarchy of sources and sinks within a large network and might connect to notions of directionality in graph-structured dynamical systems. This is our inference.
The reservoir result points toward a design principle: attaching a sufficiently large, high-first-eigenvalue subgraph can absorb an invading front so that the network is neither completely invaded nor completely blocked, and the quantitative bound in Theorem 2.25 could serve as a tuning rule for such absorbers. This is our inference.

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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

Plainly: this is a serious paper that generalizes Jimbo-Morita star graph theory to arbitrary compact center graphs, and most of the architecture is sound. The new material is real: the limit-profile definition of propagation/blocking, dichotomy theorem (2.1), minimality (2.2), transitivity (2.5), and perturbation robustness (2.7–2.9) are all stated for general D; the reservoir result (2.25) and partial/one-way propagation examples are substantive. The proofs I checked are careful and use standard tools; the perturbation argument in 2.7 is detailed, and the reservoir estimate is a nice energy argument. The literature handling is honest too: the star graph criterion is explicitly labeled a restatement of Jimbo–Morita and Pauwelussen, not sold as new.

The soft spot is structural. Theorem 1.3 — existence and uniqueness of the front-like entire solution \hat u_i on general D — is the foundation of everything. Uniqueness and time-monotonicity are proved in 4.1, but existence is not. The text says the star graph proof in [10] carries over "without much modifying" and then omits it. That is not a small omission. The uniqueness proof assumes an entire solution satisfying (1.13) already exists, so it cannot supply existence. A star graph has one junction; a general D has multiple vertices, possibly cycles, and the Kirchhoff condition must hold at every vertex. The super/subsolution construction from [10] is not obviously preserved under that added structure. If Theorem 1.3 fails for some D, then \hat v_i, the propagation index, dichotomy, transitivity, and perturbation results are all undefined. I am not claiming the theorem is false; I am saying the paper's central claim is conditional on an unproved, nontrivial existence result. That is a load-bearing gap, not a cosmetic one.

There are smaller omissions in the same direction: the comparison principle (Prop. 3.3) is stated without proof, and Theorem 2.8 (limit of blocking nonlinearities) is left to the reader. These are minor by comparison, but they add to the sense that some foundational bricks are presupposed.

Who should read this: anyone working on bistable fronts on networks will want the framework, and the examples are illuminating. The paper deserves a serious referee, but the referee should demand a complete proof of Theorem 1.3 (or a precise reference that covers arbitrary center graphs) before it is accepted. I would engage with it, but I would not cite the main theorems as established until that gap is closed.

Referee Report

3 major / 6 minor

Summary. The paper studies front propagation for a bistable reaction-diffusion equation on a metric graph consisting of a bounded compact center graph D and N semi-infinite outer paths. It introduces a unique front-like entire solution arriving from each outer path, defines its limit profile, and uses the profile to define propagation and blocking indices. The main results are a propagation/blocking dichotomy, a minimality theorem, linear stability, transitivity of propagation, robustness of propagation under small perturbations of the center graph, a sharp criterion for star graphs, examples of partial and one-way propagation, a reservoir estimate establishing incomplete invasion, and convergence of a class of Cauchy-problem solutions to the limit profile. The paper is a substantial extension of earlier star-graph work and proposes a general framework for propagation on arbitrary compact core graphs.

Significance. If the stated results hold, this is a significant contribution to the theory of reaction-diffusion equations on metric graphs. The definition of the limit profile gives an unambiguous notion of propagation and blocking, and the dichotomy, transitivity, and perturbation theorems are natural and potentially widely applicable. The paper also contains explicit parameter-free criteria for star graphs, examples that exhibit partial and one-way propagation, and a novel reservoir mechanism for incomplete invasion. Strong points include the detailed treatment of comparison principles, the Gauss-Green estimates on graphs, the harmonic-function flux estimate used in the blocking perturbation theorem, and the explicit construction of barriers in the reservoir estimate. The main caveat is that the central existence theorem for general center graphs is deferred to the star-graph paper [10], and the proof of the dichotomy theorem has a phase-portrait gap; these issues are load-bearing because the limit profile is the object on which all propagation/blocking definitions rest.

major comments (3)

[§4.1, Theorem 1.3] The existence part of Theorem 1.3 is not proved in this paper. The text states that existence was proved in [10] for star graphs and that "basically the same proof applies to our case without much modifying the arguments," and then only uniqueness and time-monotonicity are proved (Proposition 4.1). This is a load-bearing omission: the limit profile bv_i, the propagation index P(i,j), and every subsequent theorem (dichotomy, transitivity, perturbation, reservoir, Cauchy convergence) are defined only after this entire solution is known to exist. A star graph has a single junction vertex, whereas a general center graph D is a compact graph with several vertices and possibly cycles, and the super- and subsolution pair used in [10] must satisfy the appropriate sub-/super-Kirchhoff condition at every vertex of D. The manuscript should either include the full existence construction for general D or give a detailed reduction showing exactly which parts of [10] adapt and how the multiple-vertex and unequal-thickness cases are handled.
[§4.2, proof of Theorem 2.1] The proof of the dichotomy theorem is incomplete. It asserts that if bv_i(P_j)>β, then because bv_i|Ω_j stays between 0 and 1 for all x_j≥0, it "must lie on the stable manifold of (1,0)" and therefore tends to 1. This does not follow from the phase portrait alone. For the equation v''+f(v)=0, an orbit with energy in (0,F(1)) is periodic and remains bounded in (0,1) with values on both sides of β; moreover, the heteroclinic orbit connecting 1 to 0, restricted to a half-line with starting value above β, is strictly decreasing, stays in (0,1), and tends to 0. Both types of profiles satisfy the same pointwise bounds as bv_i|Ω_j yet do not converge to 1. The proof needs an additional argument specific to the limit profile, for example using the stability result (Theorem 2.3) or the dynamic selection of the front-like solution, to rule out these alternatives and to justify the asserted monotonicity of bv_i on Ω_j. As written, the dichotomy and the equivalence with bv_i(P_j)>β are not established.
[§4.4, proof of Theorem 2.7] The passage to the limit in the proof of Theorem 2.7 has a closely related gap. After extracting a subsequence of limit profiles v_m converging to a stationary solution v_∞, the proof asserts that v_∞|Ω_i tends to 1 at infinity because each v_m lies on the stable manifold of (1,0) on Ω_{m,i}. Compact convergence on Ω_i does not preserve behavior at infinity; the limit could in principle be a periodic orbit or a reversed front with the same compact limit. This issue is downstream of the gap in Theorem 2.1, but it should be addressed explicitly, for instance by proving a uniform-in-m tail estimate for v_m on Ω_{m,i} or by first establishing a stronger convergence property of limit profiles.

minor comments (6)

[Throughout] The term "transient properties" is used repeatedly (e.g., Section 2.2, Theorem 2.5, Eq. (1.18)) where the intended meaning is "transitive properties"; the statement P(i,j)=1 and P(j,k)=1 implies P(i,k)=1 is transitivity.
[Theorem 2.5] The theorem states "Let i,j,k∈{1,...,N}" without specifying distinctness; since P(i,i) is not defined in Definition 1.5, the statement should either assume the indices are mutually distinct or state a convention for the diagonal case.
[Theorem 2.13] In the paragraph after (2.10), "evi ≡ 1 on Ω" should read "bv_i ≡ 1 on Ω".
[Theorem 2.25, Eq. (2.22)] The notation "1 2L" in (2.22a) and in the proof is ambiguous; it should be written as 1/(2L) to distinguish it from (1/2)L.
[§4.4, Eq. (4.16)] Equation (4.16) contains a typo: "∂xk eVi eQj0)" is missing a parenthesis and should read "∂x_k eV_i(eQ_{j0})".
[Throughout] There are numerous typographical errors that should be corrected in a revision: "complate invasion", "incomplate invasion", "pertubartions", "ege", "fucntion", "blockign", "transiend", "engenvalue", among others.

Circularity Check

1 steps flagged · score 4.0 of 10

Core framework is conditional on a deferred self-cited existence proof for Theorem 1.3; the remaining derivation is self-contained.

self citation load bearing [Section 4.1, proof of Theorem 1.3 (front-like solution existence)]
"The existence is already proved in [10] for star graphs, and basically the same proof applies to our case without much modifying the arguments. ... So we omit the existence proof and focus on the uniqueness and time monotonicity."

Every subsequent object — the limit profile (1.14), the propagation index (1.17), the dichotomy theorem, transitivity, and the perturbation theorems — is defined in terms of the front-like entire solution u_i whose existence is Theorem 1.3. The paper does not construct u_i for a general center graph D; it defers to [10], which covers only star graphs, and asserts without proof that the same super-/subsolution argument carries over. No explicit verification is given that the Kirchhoff conditions at the extra vertices or cycles of D are satisfied by the would-be super-/subsolution pair. Thus the foundational input is imported from a self-cited special case rather than derived from the stated assumptions, making the propagation/blocking framework conditional on that unproved generalization.

full rationale

Apart from the deferred existence proof of Theorem 1.3, the derivation chain is self-contained. Uniqueness and time-monotonicity of the front-like solution are proved in Proposition 4.1 using the comparison lemmas developed in Section 4.1. The dichotomy and minimality theorems are proved from the phase portrait and the comparison arguments. Transitivity follows directly from minimality, and the perturbation and reservoir results are derived within the paper. The star-graph criterion is restated from [11] but is reproven in Section 4.5 using the paper's own framework, so it is not merely imported. The only load-bearing self-citation is the existence part of Theorem 1.3: the text cites [10] for star graphs and asserts the same proof works for arbitrary D without presenting the construction. This makes the central framework conditional, but it does not make the later theorems circular. Hence the score reflects a load-bearing self-citation gap rather than an equivalence-by-construction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claims rest on standard parabolic PDE theory on metric graphs plus the specific bistable structure (1.8). No new physical entities or fitted parameters are introduced. The main unpaid input is the existence of the front-like solution, which is imported from prior work by the authors.

assumptions (4)

domain assumption Existence of a unique front-like entire solution for arbitrary center graph D (Theorem 1.3), with existence proof deferred to [10].
The entire propagation/blocking framework is defined through the limit profile of this solution. Existence is stated to follow from the star graph case [10] with the same proof, but the details are not given in this paper.
standard math Comparison principle for super- and subsolutions on metric graphs (Proposition 3.3).
Used throughout for uniqueness, monotonicity, and minimality. The proof is stated to be similar to the Euclidean case and is omitted.
domain assumption Bistable nonlinearity condition (1.8), including the integral condition integral_0^1 f > 0.
The traveling wave and pulse solutions used in the phase portrait analysis exist only under these hypotheses. This is the standard bistable assumption in the literature.
standard math Poincare and Poincare-Wirtinger inequalities on bounded metric graphs (Section 3.5).
Used in the reservoir estimate (Theorem 2.25) and in the gradient estimates (Proposition 3.11). These are standard results on metric graphs.

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Pith. "Pith review of Front propagation on a general metric graph." pith.science (2026). https://pith.science/paper/3VCUJFPT

@misc{pith2026250524418,
  author       = {Pith},
  title        = {Pith review of: Front propagation on a general metric graph},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3VCUJFPT}},
  note         = {Machine review of arXiv:2505.24418}
}

abstract

We consider a bistable reaction-diffusion equation on a metric graph that is a generalization of the so-called star graphs. More precisely, our graph $\Omega$ consists of a bounded finite metric graph $D$ of arbitrary configuration and a finite number of branches $\Omega_1,\ldots,\Omega_N\,(N\geq 2)$ of infinite length emanating from some of the vertices of $D$. Each $\Omega_i\,(i=1,\ldots,N)$ is called an ``outer path''. Our goal is to investigate the behavior of the front coming from infinity along a given outer path $\Omega_i$ and to discuss whether or not the front propagates into other outer paths $\Omega_j\,(j\ne i)$. Unlike the case of star graphs, where $D$ is a single vertex, the dynamics of solutions can be far more complex and may depend sensitively on the configuration of the center graph $D$. We first focus on general principles that hold regardless of the structure of the center graph $D$. Among other things, we introduce the notion ``limit profile'', which allows us to define ``propagation'' and ``blocking'' without ambiguity, then we prove transient properties, that is, propagation $\Omega_i\to \Omega_j$ and $\Omega_j\to \Omega_k$ imply propagation $\Omega_i\to \Omega_k$. Next we consider perturbations of the graph $D$ while fixing the outer paths $\Omega_1,\ldots,\Omega_N$ and prove that if, for a given choice of $i,j$, propagation $\Omega_i\to \Omega_j$ occurs for a graph $D$, then the same holds for any graph $D'$ that is sufficiently close to $D$ (robustness under perturbation). We also consider several specific classes of graphs, such as those with a ``reservoir'' type subgraph, and study their intriguing properties.

Figures

Figures reproduced from arXiv: 2505.24418 by the authors.

**Figure 1.** Kirchhoff condition. By Fick’s law, the mass flux along each edge is proportional to −∂xu, therefore (1.2) implies that the total mass flux flowing into V is zero; in other words, mass is preserved at the junction point V . This condition is crucial when one considers diffusion equations on a metric graph. Note that, as we see from the conditions (a), (b), (c) above, the direction of the edges of G does not play any… view at source ↗

**Figure 2.** Graph with a degree 1 vertex. where ρ denotes the thickness of E. Accordingly, the mass flux on each edge is proportional to −ρ ∂xu. For simplicity we assume that the thickness ρ is constant on each edge. In this case, the Kirchhoff condition is given in the following form: Xm i=1 ρi ∂u ∂νi (V ) = 0 (Kirchhoff condition for unequal thickness), (1.4) where ρi denotes the thickness of the ege Ei (i = 1, . . . , m). In… view at source ↗

**Figure 3.** Examples of Ω for N = 5. (left) a star graph; (right) a more general graph. We consider the following reaction-diffusion equation on the graph Ω: ∂tu = ∆Ωu + f(u). (1.7) Here f ∈ C 1 is a bistable nonlinearity satifsying the following for some 0 < a < 1: f(0) = f(a) = f(1) = 0, f(s) < 0 (0 < s < a), f(s) > 0 (a < s < 1), f ′ (0) < 0, f′ (a) > 0, f′ (1) < 0, Z 1 0 f(s)ds > 0. (1.8) 5 [PITH_FULL_IMAGE:figures/full_fi… view at source ↗

Figures from the paper (15 more)

**Figure 4.** Figure 4: Two star graphs connected by an edge [9] ( [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

**Figure 5.** Figure 5: Replacing a point on an edge or a vertex by an arbitrary small graph Σ [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]

**Figure 6.** Figure 6: Unification of outer paths Ωj1 , . . . , Ωjm beyond the points Qj1 , . . . , Qjm. Theorem 2.11 (Unification of outer paths). Let i, j1, . . . , jm ∈ {1, . . . , N} be mutually distinct indices and let Qj1 , . . . , Qjm be points on the outer paths Ωj1 , . . . , Ωjm. Le…

**Figure 7.** Figure 7: 2-star graph. Corollary 2.17. Let N = 2 and assume ρ1 ≤ ρ2. Then P(2, 1) = 1. Furthermore, P(1, 2) = 0 ⇐⇒ F(1) + (ρ2/ρ1) 2 − 1 F(a) ≤ 0, (2.12) The above corollary gives a necessary and sufficient condition for a one-way propagation on a 2-star graph. With a suitable…

**Figure 8.** Figure 8: An example of a star graph (left) and its small perturbation (right). Theorem 2.19 (Perturbation of a non-blocking star graph). Let Ω be a star graph with outer paths Ω1, . . . , ΩN , and assume that (2.11a) holds. Let Ω ′ be a graph which is obtained by replacing the …

**Figure 9.** Figure 9: Joining an arbitrary graph D0 to a star graph; in front (left) and behind (right). In both cases, the star graph has the center point P and N′ outer paths. For simplicity, we only consider the case where all the edges have the same thickness, but more general cases can…

**Figure 10.** Figure 10: An example of partial propagation (above) and one-way propagation (below). Next we discuss the example given in [PITH_FULL_IMAGE:figures/full_fig_p017_10.png]

**Figure 11.** Figure 11: A graph with a “reservoir”. We first give a detailed description of the reservoir, which we call R0. As shown in [PITH_FULL_IMAGE:figures/full_fig_p018_11.png]

**Figure 12.** Figure 12: Detailed view of the reservoir. Next we introduce some constants associated with the nonlinearity f. Let F(s) = R s 0 f(r)dr and W(s) := −F(s). Then, by (1.8), (1.9) and (1.10), we have W(0) = 0, W(s) > 0 (0 < s < β), W(β) = 0, W(s) < 0 (β < s ≤ 1). Let δ, be a consta…

**Figure 13.** Figure 13: Phase portrait for ∂ 2 x v + f(v) = 0 on R. on this line, all the orbits in the phase portrait cross this line horizontally, and, along each orbit, vx attains either the maximum or the minimum on this line. The orbit that is marked in a thick line in [PITH_FULL_IMAGE…

**Figure 14.** Figure 14: Symmetrically decreasing stationary solution (pulse solution). [PITH_FULL_IMAGE:figures/full_fig_p021_14.png]

**Figure 15.** Figure 15: Graph with exit points Q1, . . . , Qm. Let G be a metric graph and D0 be its bounded finite subgraph that are connected to the rest of G by edges E1, . . . , Em that do not belong to D0 but stretch from some vertices Q1, . . . , Qm of D0, as shown in [PITH_FULL_IMAGE…

**Figure 16.** Figure 16: Image of the subgraph D0 created by the edge E0.. Let Q0 be an endpoint of the edge E0. By (3.12) (a), we have −ρˆ0 ∂xu(Q0) +Xm j=1 ρˆj∂νu(Qj ) +X k r=1 ρir ∂νu(Pir ) = 0. Here ∂ν denotes the derivative along the edges Ej (j = 1, . . . , m) or along the outer paths Ωi…

**Figure 17.** Figure 17: Melon-shaped graph. 3.6 Local energy Let D0 be a bounded finite metric graph with boundary points Q1, . . . , Qm. As we explained before (3.13), these are the points where the Kirchhoff condition is not required to hold. We consider the equation ∂tu = ∆D0 u + f(u) on …

**Figure 13.** Figure 13: Consequently vbi(xj ) → 1 as xj → ∞. This proves the dichotomy (2.1). Furthermore, if vbi(Pj ) ≤ β, the solution vbi|Ωj must lie on the homoclinic orbit since otherwise it cannot stay positive. Therefore vbi|Ωj (xj ) = V (xj +b) for some b ≥ 0. Finally, the monotonici…

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