Pith. sign in

REVIEW 3 major objections 4 minor 1 cited by

Nonlocal Hamilton-Jacobi Equations on a network with Kirchhoff type conditions

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper proves that nonlocal Hamilton-Jacobi equations on networks with Kirchhoff junction conditions have a unique continuous viscosity solution when the nonlocal operators have order below 1 and the Hamiltonian grows at least linearly.

desk verdict Solid single-junction theory for nonlocal Hamilton-Jacobi on networks; the advertised general-network extension is sketched, not fully proved. read the letter →

arxiv 2411.13126 v1 pith:AUA6T5DH submitted 2024-11-20 math.AP

Guy Barles (IDP) , Olivier Ley (IRMAR) , Erwin Topp (IM / UFRJ) This is my paper · ORCID
classification math.AP MSC 35K5535R0947G2035B4033B20
keywords NonlocalHamilton-JacobiequationsNetworksKirchhoffconditionsViscositysolutionsComparisonprincipleFlux-limitedIntegro-differentialoperatorsJunction
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

The paper proves that a stationary nonlocal Hamilton-Jacobi equation on a network, with a Kirchhoff flux condition at interior vertices and Dirichlet data at boundary vertices, has a unique viscosity solution. The key advance is that the nonlocal operator may receive contributions from several edges at once, not just the edge where the equation is written, as long as the kernels have order strictly below 1. The authors construct the solution in two ways, by vanishing viscosity and by Perron's method, prove a strong comparison principle, and show that the Kirchhoff solution is exactly a flux-limited solution for a suitable flux limiter. If correct, this gives a well-posedness theory for a large class of nonlocal junction problems and connects it to control-theoretic solution notions.

What carries the argument

The central object is the junction viscosity solution defined through test functions $\varphi \in C^1(\Gamma)$ and the truncated operator $G^\delta_i$; at the junction $O$ the subsolution inequality is a minimum of the PDE inequalities on all incident edges and the Kirchhoff flux condition. The nonlocal operator $I_i$ with kernels $\nu_{ij}$ satisfying (2.7) is the device that carries information across edges, and its Levy integrability condition $\sigma<1$ makes $I_i u(O)$ finite for Holder functions $u$. The coercivity inequality $C_H^{-1}|p|-C_H \le H_i(x,p)$ is the workhorse: it builds barriers, gives Lipschitz estimates, and rules out large slopes in the comparison proof. The vanishing viscosity family $u^\varepsilon$ solves a Dirichlet problem at $O$ with value $\theta$, and the Kirchhoff data $B$ is recovered by choosing $\theta$ through continuity, or by the Poincare-Miranda theorem on general networks, after which $\varepsilon \to 0$ yields the solution.

What would settle it

On a two-edge junction with $H_i(x,p)=p^2/2$, kernels $\nu_{ij}(r)=\Lambda r^{-1-\sigma}$ with $\sigma=1/2$, $B=0$, and $h_1=h_2=0$, compute the vanishing-viscosity limit for several choices of the junction value $\theta$ and check that the limit is the same function and satisfies the Kirchhoff condition; any dependence on the approximation path, or any discrepancy between this limit and the Perron solution, would refute Theorem 4.4.

Watch

Extended reading notes

Core claim

The central claim is that, under assumptions (2.15), the Kirchhoff-Dirichlet problem (2.12)-(2.14)-(2.13) has a viscosity solution $u \in C(\Gamma)$, Holder continuous on $\Gamma$ and locally Lipschitz away from the boundary vertices, and that any viscosity subsolution lies below any viscosity supersolution, so the solution is unique. The proof's core is that order-$\sigma<1$ nonlocal terms evaluated at the junction remain finite and continuous for Holder functions, so the junction condition can be stated in the Lions-Souganidis junction-viscosity sense; coercivity of the Hamiltonian then substitutes for ellipticity and forces Lipschitz behaviour near the junction. Section 8 closes the circle by proving that Kirchhoff solutions are flux-limited solutions for a flux limiter built from the nonincreasing part of the Hamiltonian, and vice versa.

Load-bearing premise

The proofs need the Hamiltonian to grow at least linearly in the slope on every edge, namely $C_H^{-1}|p|-C_H \le H_i(x,p)$, because that growth is what keeps subsolutions Lipschitz near the junction and rules out slopes that would violate the flux; if coercivity is dropped, the existence and comparison arguments no longer go through.

Editorial extensions

If this is right

  • For each set of data satisfying (2.15), the Kirchhoff-Dirichlet problem has a unique Holder-continuous viscosity solution on any finite connected network.
  • The vanishing viscosity limit and Perron's construction give the same solution, with explicit Lipschitz control near interior vertices.
  • The Kirchhoff solution coincides with the flux-limited solution for the flux limiter defined in Definition 8.1, extending the local equivalence to nonlocal equations.
  • Strictly elliptic viscous versions of the problem have unique classical $C^{2,1-\sigma}(\Gamma)$ solutions satisfying the vertex conditions pointwise.
  • Other boundary conditions, including unbounded edges, exterior data, state constraints, and Neumann conditions, can be handled by the same junction-local arguments.

Reading between the lines

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

  • The comparison proof only needs Lipschitz regularity of subsolutions, so the reliance on coercivity suggests that a non-coercive Hamiltonian with, say, superlinear growth away from the junction might still admit comparison if slope control comes from another source; this is an extension, not a paper claim.
  • The flux-limited equivalence is proved for Hamiltonians that are convex with a unique minimum; a natural test is whether the same equivalence holds for merely quasiconvex Hamiltonians, as in the local Imbert-Monneau setting.
  • Because the junction arguments are local, the same framework should apply to time-dependent problems obtained by adding a time derivative, although the paper treats only the stationary equation.
  • A numerical implementation on a two-edge junction with $\sigma=1/2$ and $H_i(x,p)=p^2/2$ would provide a concrete check of the predicted uniqueness and of the flux-limited characterization.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper studies the stationary nonlocal Hamilton-Jacobi equation λu − I_i u + H_i(x,u_{x_i}) = 0 on each edge of a network, complemented by a Kirchhoff-type condition at interior vertices and Dirichlet conditions at boundary vertices. The nonlocal operators are of Lévy type with order σ < 1 and may integrate over the whole network, including contributions from several different edges. For a single junction (Theorem 4.4 and Theorem 5.1), the authors prove existence of a continuous viscosity solution by vanishing viscosity and Perron's method, and a strong comparison principle, under coercivity and Lipschitz assumptions on the Hamiltonians. They also introduce a nonlocal analogue of flux-limited solutions and prove its equivalence with Kirchhoff-type solutions (Section 8). The final section claims an extension of the well-posedness results to general networks (Proposition 7.3 and Theorem 7.4).

Significance. The single-junction results are a genuinely new contribution: they are, to my knowledge, the first to treat nonlocal Hamilton-Jacobi equations on networks where the nonlocal operator couples different edges, and the comparison proof avoids the usual doubling-of-variables procedure by exploiting the Lipschitz regularity of subsolutions. The flux-limited equivalence in Section 8 is also new in the nonlocal setting. The paper is clearly organized, provides detailed estimates for the core theorems, and includes a useful self-contained comparison principle for censored problems in Appendix B. However, the advertised general-network extension is only sketched via 'minor changes' and 'readily follows', and the specific multi-vertex issues are not addressed in the written proof; as a result the paper's central claim in the abstract and Section 7 is not fully supported.

major comments (3)
  1. [Section 7, Proposition 7.3] Proposition 7.3 is asserted to follow from Lemma 4.2 and Proposition 4.3 'with minor changes', but the multi-vertex setting introduces genuinely new structure. In Lemma 4.2 the iterative construction and the contraction estimate (4.12) use a single scalar parameter θ at the unique interior vertex and split indices into censored and non-censored edges; for a general network the Dirichlet data are a vector Θ=(θ_\bar v)_{\bar v\in V_i}, an edge can have interior vertices at both endpoints, and the zero-order coefficient Λ^η_i in the counterpart of (4.10) must account for contributions from all edges incident at both endpoints. In addition, the uniqueness of the solution to (7.7)-(7.8) requires the network version of the comparison principle from Lemma B.1, which is only covered by a remark. As written, the well-posedness of the viscous Dirichlet problem on a general network is not established.
  2. [Section 7, Theorem 7.4 (existence part)] The existence proof for a general network relies on the Poincaré-Miranda theorem applied to the map Θ ↦ F_j(Θ) = ∑_{E∈Inc(v_j)} −∂_E u^{ε,Θ}(v_j) − B_{v_j}. For this application one needs the continuity of Θ ↦ u^{ε,Θ} with respect to the whole vector Θ, and one needs uniform (in ε) Lipschitz estimates of u^{ε,Θ*_ε} in a neighborhood of every interior vertex in order to pass to the limit ε→0. The first property is only sketched for the scalar case in Proposition 4.3(v), and the second is proved in the star case using the special test function (4.27) whose kink satisfies the single junction condition K−(N−1)L≤B. The paper does not prove the network analogues of these two ingredients, so the existence half of Theorem 7.4 is not secured by the written argument.
  3. [Section 7, Theorem 7.4 (uniqueness part)] The uniqueness assertion says it 'follows readily' from Theorem 5.1 'arguing locally as in the junction case.' This is a nontrivial reduction. Theorem 5.1 relies on Lemma 5.2, whose proof uses the star-shaped geometry: the maximum of u(x)−u(x_0)−φ(x) is considered with φ defined as in (4.27), and the possibility of the maximum at O is excluded using the single Kirchhoff inequality. On a network, after localizing at an interior vertex v, the auxiliary function must be extended over the whole graph, and the maximum can be attained at a different interior vertex w; the slope constants must then simultaneously satisfy flux inequalities at v and w. The required modification is not provided, and the network version of the local Lipschitz estimate for subsolutions is not proved. Consequently the comparison principle for general networks is not established.
minor comments (4)
  1. [Section 3, proof of Lemma 3.3] In the displayed equality in the converse part of the proof, the subscript of φ in the first term is written as φ_{x_{i0}}(O); this should presumably be φ_{x_i}(O). The intended identity G_i^δ = G_i + I_i u(O) − I_i[B_c^δ(O)]u(O) − I_i[B_δ(O)]φ(O) is otherwise clear.
  2. [Section 4, Theorem 4.4 and Remark 4.5(ii)] The theorem states that u solves the Dirichlet problem (2.12)-(2.14)-(2.13), with (2.14) written as u_i(a_i)=h_i pointwise. The proof only establishes the relaxed Dirichlet condition for the supersolution side, as acknowledged in Remark 4.5(ii). The statement should clarify that the Dirichlet condition is understood in the viscosity sense of Definition 3.1, not necessarily pointwise.
  3. [Section 5, Theorem 5.1] The final sentence says 'there exists a unique continuous viscosity solution u ∈ C^{0,1}(Γ_δ)', but Γ_δ is defined in (4.26) and is a set that depends on δ; the sentence appears to refer to the solution from Theorem 4.4. For readability, the notation should distinguish the Lipschitz-regularity statement from the comparison conclusion.
  4. [Section 7.1, equation (7.1)] The general network is introduced in the introduction as made of curves, but (7.1) defines every edge as a straight segment. If curved edges are intended, the parametrization and the definition of the inward derivative (7.2) need to be adapted; otherwise the introductory wording should be corrected to 'straight segments'.

Circularity Check

0 steps flagged · score 1.0 of 10

No circularity: the central existence and comparison theorems are proved directly from the stated assumptions; the only blemishes are minor self-citations for supporting lemmas and a delegated general-network proof, neither of which makes the derivation reduce to its inputs.

full rationale

The paper's main claims are the vanishing-viscosity existence result (Theorem 4.4) and the strong comparison/uniqueness result (Theorem 5.1) for the nonlocal Kirchhoff-Dirichlet problem, plus the general-network extension (Theorem 7.4). These are not obtained by fitting, renaming, or definitional identification. The existence proof constructs solutions u^{epsilon,theta} of the viscous Dirichlet problem, derives uniform estimates independent of epsilon, and then uses the Intermediate Value Theorem on F(theta)=sum_i -u^{epsilon,theta}_{x_i}(O)-B to enforce the Kirchhoff condition; the limit epsilon->0 is then shown to satisfy the generalized Kirchhoff and Dirichlet conditions. No step reuses the target conclusion as an input. The comparison proof proves Lipschitz regularity of subsolutions (Lemma 5.2) from coercivity and the Kirchhoff condition, then adapts the Lions-Souganidis argument; it does not import a uniqueness theorem from the authors' earlier work. Section 8's flux-limited equivalence is a genuine two-way implication: the flux limiter is defined independently following Barles-Chasseigne, and Theorems 8.4 and 8.5 prove that Kirchhoff subsolutions/supersolutions are flux-limited and conversely. The section is not a renaming of the Kirchhoff condition, because the flux-limited notion uses the H^+ / H^- decomposition and an auxiliary minimization that is not simply the Kirchhoff inequality. The paper does rely on several lemmas from prior work with author overlap, e.g., [11, Lemma 15.1], [11, Lemma 5.3.1], [14, Corollary 1], and [15, Theorem 2.1]. These are supporting, external results with stated assumptions that do not include the paper's target theorems; they therefore count as legitimate evidence, not load-bearing self-citation. The only notable gap is the general-network extension: Proposition 7.3 is asserted to follow from the junction case 'with minor changes', and Theorem 7.4 says uniqueness 'follows readily' by localizing at a vertex. A two-junction network can place the same trial maximum at a second interior vertex, so the written proof does not fully exhibit the required modifications. That is a rigor/completeness concern, not circularity: no equation is being reduced to a prior result by construction, and no fitted parameter is being relabelled as a prediction. Accordingly, the circularity score is 1 rather than 0, reflecting minor self-citation and the under-proofed extension, but no actual circular step is present.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper postulates no new physical entities or fitted constants. All assumptions are standard coercivity, Lipschitz, and Lévy-kernel conditions on the data. The new 'nonlocal flux limiter' is a definition internal to the paper without independent empirical content.

assumptions (7)
  • domain assumption Kernels νij satisfy (2.7): 0≤νij≤Λ r^{-1-σ}, |νij(x,r)-νij(y,r)|≤Λ|x-y| r^{-1-σ}, with σ∈(0,1).
    Standing assumption (2.15); ensures Lévy integrability and Lemma 2.2 Hölder regularity of Iu; central for evaluating Ii at the junction O.
  • domain assumption Hamiltonians Hi satisfy (2.6): Lipschitz in x and p, and coercive C_H^{-1}|p|-C_H ≤ Hi(x,p) ≤ C_H(1+|p|).
    Used for barriers, Lipschitz estimates, and comparison; absence breaks the proofs in Sections 4 and 5.
  • domain assumption λ>0.
    Zeroth-order term gives contraction in Lemma 4.2 and strict inequalities in Theorem 5.1; λ=0 is not covered.
  • domain assumption Network is a finite star-shaped or connected graph with arc-length parametrization and geodesic distance (Sections 2.1, 7.1).
    The PDE and Kirchhoff condition depend on this topology; general curved edges are not treated.
  • standard math Standard viscosity solution machinery: Perron's method, stability, doubling variables, Sobolev embeddings, Han-Lin regularity theorem.
    Invoked in Proposition 4.3 and Appendix B; not proved in the paper.
  • standard math Cited junction sub/superdifferential characterizations and flux-limiter lemma from Barles-Chasseigne [11] and Lions-Souganidis [32].
    Used in Lemma 5.3 and Section 8; these are background results from the same research community, not derived here.
  • standard math Poincaré-Miranda theorem for existence of interior vertex values Θ in Theorem 7.4.
    Higher-dimensional intermediate value theorem used to select Dirichlet values at all junction vertices.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Nonlocal Hamilton-Jacobi Equations on a network with Kirchhoff type conditions." pith.science (2026). https://pith.science/paper/AUA6T5DH

@misc{pith2026241113126,
  author       = {Pith},
  title        = {Pith review of: Nonlocal Hamilton-Jacobi Equations on a network with Kirchhoff type conditions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AUA6T5DH}},
  note         = {Machine review of arXiv:2411.13126}
}
read the original abstract

In this article, we consider nonlocal Hamilton-Jacobi Equations on networks with Kirchhoff type conditions for the interior vertices and Dirichlet boundary conditions for the boundary ones: our aim is to provide general existence and comparison results in the case when the integro-differential operators are of order strictly less than 1. The main originality of these results is to allow these nonlocal terms to have contributions on several different edges of the network. The existence of Lipschitz continuous solutions is proved in two ways: either by using the vanishing viscosity method or by the usual Perron's method. The comparison proof relies on arguments introduced by Lions and Souganidis. We also introduce a notion of flux-limited solution, nonlocal analog to the one introduced by Imbert and Monneau, and prove that the solutions of the Kirchhoff problem are flux-limited solutions for a suitable flux-limiter. After treating in details the case when we only have one interior vertex, we extend our approach to treat general networks.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Stochastic scattering control of spider diffusion governed by an optimal diffraction probability measure selected from its own local-time

    math.AP 2025-01 conditional novelty 5.0 of 10

    For a spider diffusion with controlled, local-time-dependent branch selection, the value function is characterized as the unique viscosity solution of a HJB system with a nonlinear local-time Kirchhoff boundary condition.

Reference graph

Works this paper leans on

40 extracted references · 39 canonical work pages · cited by 1 Pith paper

  1. [1]

    Hamilton-Jacobi equations constrained on networks

    Yves Achdou, Fabio Camilli, Alessandra Cutr ` ı, and Nicoletta Tchou. Hamilton-Jacobi equations constrained on networks. NoDEA Nonlinear Differential Equations Appl. , 20(3):413–445, 2013

    work page 2013

  2. [2]

    A class of infinite horizon mean field games on networks

    Yves Achdou, Manh-Khang Dao, Olivier Ley, and Nicoletta Tchou. A class of infinite horizon mean field games on networks. Netw. Heterog. Media , 14(3):537–566, 2019

    work page 2019

  3. [3]

    Finite horizon mean field games on networks

    Yves Achdou, Manh-Khang Dao, Olivier Ley, and Nicoletta Tchou. Finite horizon mean field games on networks. Calc. Var. Partial Differential Equations , 59(5):Paper No. 157, 34, 2020

    work page 2020

  4. [4]

    Hamilt on-Jacobi equations for optimal control on junctions and networks

    Yves Achdou, Salom´ e Oudet, and Nicoletta Tchou. Hamilt on-Jacobi equations for optimal control on junctions and networks. ESAIM Control Optim. Calc. Var. , 21(3):876–899, 2015

    work page 2015

  5. [5]

    Barles, A

    G. Barles, A. Briani, and E. Chasseigne. A Bellman approa ch for two-domains optimal control problems in RN . ESAIM Control Optim. Calc. Var. , 19(3):710–739, 2013

    work page 2013

  6. [6]

    Barles, A

    G. Barles, A. Briani, and E. Chasseigne. A Bellman approa ch for regional optimal control problems in RN . SIAM J. Control Optim. , 52(3):1712–1744, 2014

    work page 2014

  7. [7]

    Barles, A

    G. Barles, A. Briani, E. Chasseigne, and C. Imbert. Flux- limited and classical vis- cosity solutions for regional control problems. ESAIM Control Optim. Calc. Var. , 24(4):1881–1906, 2018

    work page 1906

  8. [8]

    An introduction to the theory of viscosity so lutions for first-order Hamilton-Jacobi equations and applications

    Guy Barles. An introduction to the theory of viscosity so lutions for first-order Hamilton-Jacobi equations and applications. In Hamilton-Jacobi equations: approxi- mations, numerical analysis and applications , volume 2074 of Lecture Notes in Math. , pages 49–109. Springer, Heidelberg, 2013

    work page 2013

Show all 40 references
  1. [9]

    Backw ard stochastic differen- tial equations and integral-partial differential equation s

    Guy Barles, Rainer Buckdahn, and Etienne Pardoux. Backw ard stochastic differen- tial equations and integral-partial differential equation s. Stochastics Stochastics Rep., 60(1-2):57–83, 1997

    1997

  2. [10]

    (Almost) everythi ng you always wanted to know about deterministic control problems in stratified dom ains

    Guy Barles and Emmanuel Chasseigne. (Almost) everythi ng you always wanted to know about deterministic control problems in stratified dom ains. Netw. Heterog. Me- dia, 10(4):809–836, 2015

    2015

  3. [11]

    On Modern Approaches of Hamilton-Jacobi Equations and Control Problems with Discontinuities , volume 104 of Progress in Nonlinear Differential Equations and their Applications

    Guy Barles and Emmanuel Chasseigne. On Modern Approaches of Hamilton-Jacobi Equations and Control Problems with Discontinuities , volume 104 of Progress in Nonlinear Differential Equations and their Applications . Birkh¨ auser/Springer, Cham,

  4. [12]

    H¨ o lder continuity of solutions of second-order non-linear elliptic integro-differential equations

    Guy Barles, Emmanuel Chasseigne, and Cyril Imbert. H¨ o lder continuity of solutions of second-order non-linear elliptic integro-differential equations. J. Eur. Math. Soc. (JEMS), 13(1):1–26, 2011. 55

    2011

  5. [13]

    On the generalized Diri chlet problem for viscous Hamilton-Jacobi equations

    Guy Barles and Francesca Da Lio. On the generalized Diri chlet problem for viscous Hamilton-Jacobi equations. J. Math. Pures Appl. (9) , 83(1):53–75, 2004

    2004

  6. [14]

    Second-order elliptic int egro-differential equations: viscosity solutions’ theory revisited

    Guy Barles and Cyril Imbert. Second-order elliptic int egro-differential equations: viscosity solutions’ theory revisited. Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire , 25(3):567–585, 2008

    2008

  7. [15]

    Regularity results and large time behavior for integro-differential equations with coer cive Hamiltonians

    Guy Barles, Shigeaki Koike, Olivier Ley, and Erwin Topp . Regularity results and large time behavior for integro-differential equations with coer cive Hamiltonians. Calc. Var. Partial Differential Equations , 54(1):539–572, 2015

    2015

  8. [16]

    Lipschitz regularity for cen sored subdiffusive integro- differential equations with superfractional gradient term s

    Guy Barles and Erwin Topp. Lipschitz regularity for cen sored subdiffusive integro- differential equations with superfractional gradient term s. Nonlinear Anal., 131:3–31, 2016

    2016

  9. [17]

    Optimal control proble ms on stratified domains

    Alberto Bressan and Yunho Hong. Optimal control proble ms on stratified domains. Netw. Heterog. Media , 2(2):313–331, 2007

    2007

  10. [18]

    Regularity theory f or fully nonlinear integro- differential equations

    Luis Caffarelli and Luis Silvestre. Regularity theory f or fully nonlinear integro- differential equations. Comm. Pure Appl. Math. , 62(5):597–638, 2009

    2009

  11. [19]

    Stationary mean field games systems defined on networks

    Fabio Camilli and Claudio Marchi. Stationary mean field games systems defined on networks. SIAM J. Control Optim. , 54(2):1085–1103, 2016

    2016

  12. [20]

    The vanishing viscosity limit for Hamilton-Jacobi equations on networks

    Fabio Camilli, Claudio Marchi, and Dirk Schieborn. The vanishing viscosity limit for Hamilton-Jacobi equations on networks. J. Differential Equations , 254(10):4122– 4143, 2013

    2013

  13. [21]

    M. G. Crandall, H. Ishii, and P.-L. Lions. User’s guide t o viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) , 27(1):1–67, 1992

    1992

  14. [22]

    L. C. Evans. Partial differential equations , volume 19 of Graduate Studies in Mathe- matics. American Mathematical Society, Providence, RI, 1998

    1998

  15. [23]

    Homogenizatio n of a discrete model for a bifurcation and application to traffic flow

    Nicolas Forcadel and Wilfredo Salazar. Homogenizatio n of a discrete model for a bifurcation and application to traffic flow. J. Math. Pures Appl. (9) , 136:356–414, 2020

    2020

  16. [24]

    On Neumann problems for nonlocal Hamilto n-Jacobi equations with dominating gradient terms

    Daria Ghilli. On Neumann problems for nonlocal Hamilto n-Jacobi equations with dominating gradient terms. Calc. Var. Partial Differential Equations , 56(5):Paper No. 139, 41, 2017

    2017

  17. [25]

    Reflected symmetric α -stable processes and re- gional fractional Laplacian

    Qing-Yang Guan and Zhi-Ming Ma. Reflected symmetric α -stable processes and re- gional fractional Laplacian. Probab. Theory Related Fields , 134(4):649–694, 2006

    2006

  18. [26]

    Elliptic partial differential equations , volume 1 of Courant Lecture Notes in Mathematics

    Qing Han and Fanghua Lin. Elliptic partial differential equations , volume 1 of Courant Lecture Notes in Mathematics . New York University, Courant Institute of Mathemat- ical Sciences, New York; American Mathematical Society, Pr ovidence, RI, 1997

    1997

  19. [27]

    Flux-limited solutio ns for quasi-convex Hamilton- Jacobi equations on networks

    Cyril Imbert and R´ egis Monneau. Flux-limited solutio ns for quasi-convex Hamilton- Jacobi equations on networks. Ann. Sci. ´Ec. Norm. Sup´ er. (4), 50(2):357–448, 2017

    2017

  20. [28]

    A Hami lton-Jacobi approach to junction problems and application to traffic flows

    Cyril Imbert, R´ egis Monneau, and Hasnaa Zidani. A Hami lton-Jacobi approach to junction problems and application to traffic flows. ESAIM Control Optim. Calc. Var. , 19(1):129–166, 2013

    2013

  21. [29]

    Effective junction con ditions for degenerate parabolic equations

    Cyril Imbert and Vinh Duc Nguyen. Effective junction con ditions for degenerate parabolic equations. Calc. Var. Partial Differential Equations , 56(6):Paper No. 157, 27, 2017

    2017

  22. [30]

    The Poincar´ e-Miranda theorem

    Wladyslaw Kulpa. The Poincar´ e-Miranda theorem. Amer. Math. Monthly , 104(6):545–550, 1997

    1997

  23. [31]

    Viscos ity solutions for junctions: well posedness and stability

    Pierre-Louis Lions and Panagiotis Souganidis. Viscos ity solutions for junctions: well posedness and stability. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. , 27(4):535– 545, 2016

    2016

  24. [32]

    Well-p osedness for multi-dimensional junction problems with Kirchoff-type conditions

    Pierre-Louis Lions and Panagiotis Souganidis. Well-p osedness for multi-dimensional junction problems with Kirchoff-type conditions. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. , 28(4):807–816, 2017. 56

    2017

  25. [33]

    Peter S. Morfe. Convergence & rates for Hamilton-Jacob i equations with Kirchoff junction conditions. NoDEA Nonlinear Differential Equations Appl. , 27(1):Paper No. 10, 69, 2020

    2020

  26. [34]

    Quasi linear parabolic pde posed on a netwo rk with non linear Neumann boundary condition at vertices

    Isaac Ohavi. Quasi linear parabolic pde posed on a netwo rk with non linear Neumann boundary condition at vertices. J. Math. Anal. Appl. , 500(1):Paper No. 125154, 29, 2021

    2021

  27. [35]

    Viscosity solutions of Eikonal equations on topo- logical networks

    Dirk Schieborn and Fabio Camilli. Viscosity solutions of Eikonal equations on topo- logical networks. Calc. Var. Partial Differential Equations , 46(3-4):671–686, 2013

    2013

  28. [36]

    Time-dependent Hamilton-Jacobi eq uations on networks

    Antonio Siconolfi. Time-dependent Hamilton-Jacobi eq uations on networks. J. Math. Pures Appl. (9) , 163:702–738, 2022

    2022

  29. [37]

    Optimal control with state-space con straint

    Halil Mete Soner. Optimal control with state-space con straint. I. SIAM J. Control Optim., 24(3):552–561, 1986

    1986

  30. [38]

    Existence and uniqueness for integro-diffe rential equations with domi- nating drift terms

    Erwin Topp. Existence and uniqueness for integro-diffe rential equations with domi- nating drift terms. Comm. Partial Differential Equations , 39(8):1523–1554, 2014

    2014

  31. [39]

    Classical solvability of linear par abolic equations on networks

    Joachim von Below. Classical solvability of linear par abolic equations on networks. J. Differential Equations , 72(2):316–337, 1988. Guy Barles: Institut Denis Poisson (UMR CNRS 7013) Universi t´e de Tours, Universit´e d’Orl´eans, CNRS. P arc de Grandmont. 37200 Tours, France....

    1988

  32. [2024]

    A Guide to Theory, Applications, and Some Open Problem s, PNLDE Subseries in Control, 104

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.