How a Tübingen Dissertation Helped Found an Entire Research Field

In 2006, Dirk Schieborn defended his doctoral thesis at the Eberhard Karls University of Tübingen, titled “Viscosity Solutions of Hamilton-Jacobi Equations of Eikonal Type on Ramified Spaces.” From that work emerged, in collaboration with Italian mathematician Fabio Camilli (Sapienza Università di Roma), the paper “Viscosity solutions of Eikonal equations on topological networks,” published in 2013 in Calculus of Variations and Partial Differential Equations (Springer). It is also freely available on arXiv.

What at first glance looks like a specialist paper became one of the foundational texts of an entire research area. This blog post traces soberly the reception that work received — who picked it up, where, and in which fields.

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What is it about?

Broadly speaking: the work asks how a certain class of differential equations — so-called eikonal equations — can be solved on networks. A network here consists of curves meeting at junction points (think of a road network or a pipe system).

In the classical case, such equations are solved on smooth surfaces or in Euclidean space. Schieborn and Camilli were the first to define an appropriate notion of solution (so-called viscosity solutions) for the case where the underlying space is a topological network. They proved the existence and uniqueness of the solution. The central contribution was the formulation of the correct transition conditions at the junction points — that is, the mathematical description of what happens at the branching locations.

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Direct follow-up work by the authors

Schieborn and Camilli extended their results in several subsequent papers of their own:

• “An approximation scheme for a Hamilton–Jacobi equation defined on a network” (Camilli, Festa, Schieborn; Applied Numerical Mathematics, 2013): Development of a semi-Lagrangian numerical scheme for the concrete computation of solutions on networks. Adriano Festa (then at Sapienza Rome, now at Politecnico di Torino, Italy) was a co-author.
• “The vanishing viscosity limit for Hamilton–Jacobi equations on networks” (Camilli, Marchi, Schieborn; Journal of Differential Equations, 2013): An investigation of what happens when the equations on the network are regularized by a small viscosity term that is then sent to zero. Claudio Marchi (Università di Padova, Italy) was a co-author.
• “Eikonal equations on ramified spaces” (Camilli, Schieborn, Marchi; Interfaces and Free Boundaries, 2013): Generalization of the results to higher-dimensional ramified spaces (so-called “ramified manifolds”).

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Reception in the international research community

The work of Schieborn and Camilli was widely taken up within the research community. An overview of the most important threads:

France: Traffic flow modeling and optimal control

One of the most influential strands building directly on this work comes from France. Cyril Imbert (CNRS/École Normale Supérieure, Paris), Régis Monneau (École des Ponts ParisTech), and Hasnaa Zidani developed a Hamilton–Jacobi approach to junction problems with applications to traffic flows:

“A Hamilton-Jacobi approach to junction problems and application to traffic flows” (Imbert, Monneau, Zidani; ESAIM: COCV, 2013): This widely cited paper references both Schieborn’s dissertation and the Schieborn–Camilli paper as foundational works. It applies the approach to the Lighthill–Whitham–Richards traffic model.

Yves Achdou (Université Paris Cité), Alessandra Cutrì (Università Roma Tor Vergata), and Nicoletta Tchou (Université de Rennes 1, France) also built on the work:

• “Hamilton–Jacobi equations constrained on networks” (Achdou, Camilli, Cutrì, Tchou; NoDEA, 2013): Extension to state-constrained control problems on networks.
• “Hamilton–Jacobi equations for optimal control on junctions and networks” (Achdou, Oudet, Tchou; ESAIM: COCV, 2015): Further development of the theory for optimal control problems at junctions. Cites Schieborn–Camilli as a central reference.
• “Hamilton-Jacobi Equations on Networks as Limits of Singularly Perturbed Problems in Optimal Control: Dimension Reduction” (Achdou, Tchou; Comm. PDE, 2015): Dimension reduction of control problems on thin domains toward network equations.

Jessica Guerand used the work in her dissertation “Équations de Hamilton-Jacobi discontinues et régularité parabolique à la De Giorgi” (2018, Paris), identifying the Schieborn–Camilli paper as one of the early key results for numerical methods on networks.

Germany: Traffic simulation

Simone Göttlich (University of Mannheim), Michael Herty (RWTH Aachen), and Ute Ziegler developed a numerical discretization scheme for Hamilton–Jacobi equations on networks, motivated by traffic flow modeling:

“Numerical discretization of Hamilton–Jacobi equations on networks” (Göttlich, Ziegler, Herty; Networks and Heterogeneous Media, 2013): Application of an adapted Lax–Friedrichs scheme to the Lighthill–Whitham–Richards traffic model, for instance to compute travel times in a roundabout.

Italy: From networks to fractals and CAT(0) spaces

The work inspired the transfer of the theory to fractals in Italy:

“Eikonal equations on the Sierpinski gasket” (Camilli, Capitanelli, Marchi; Mathematische Annalen, 2016): Eikonal equations on the Sierpinski triangle, a classical fractal. Raffaela Capitanelli works at Sapienza Roma. The authors cite the Schieborn–Camilli work as the direct foundation of their network approach.

Antonio Siconolfi (Sapienza Roma) and Alfonso Sorrentino (Università di Roma Tor Vergata) substantially extended the theory:

• “Global Results for Eikonal Hamilton-Jacobi Equations on Networks” (Siconolfi, Sorrentino; Analysis & PDE, 2018): Global solution theory for eikonal equations on networks. The authors describe Schieborn’s and Camilli’s model as a direct predecessor that they generalize.
• “Discounted Hamilton-Jacobi Equations on Networks and Asymptotic Analysis” (Pozza, Siconolfi; Indiana Univ. Math. J., 2019): Discounted equations on networks without geometric restrictions.

Marco Pozza (also at Sapienza Roma) extended this further most recently:

• “Aubry Set of Eikonal Hamilton-Jacobi Equations on Networks” (Pozza; Preprint, 2024): A new generalization of the network theory, citing Schieborn–Camilli as one of the founding works of the field.
• “Homogenization of Hamilton-Jacobi equations on networks” (Pozza, Siconolfi; Preprint, 2024): Homogenization on periodic networks, with connections to Mather theory.

A recent paper shows the influence reaching into geometric analysis:

“Viscosity Solutions of Hamilton-Jacobi Equations in Proper CAT(0) Spaces” (Journal of Geometric Analysis, 2024): Cites both Schieborn’s dissertation and the Schieborn–Camilli paper in developing a viscosity solution theory in CAT(0) spaces (a class of non-positively curved metric spaces).

Japan: Generalization to general metric spaces

Yoshikazu Giga (University of Tokyo), Nao Hamamuki (Hokkaido University), and Atsushi Nakayasu developed the theory for general metric spaces:

“Eikonal equations in metric spaces” (Giga, Hamamuki, Nakayasu; Transactions of the AMS, 2015): Introduction of metric viscosity solutions, which include topological networks as a special case. The work of Schieborn and Camilli is named as an important application and motivation.

Further Japanese follow-up work concerns stability and long-time behavior of solutions in metric spaces (Nakayasu and Namba, Nonlinearity, 2018).

Further countries

The work was also taken up by researchers in the United States (Pierre-Louis Lions, Columbia/Collège de France; Panagiotis Souganidis, University of Chicago — both associated with Fields Medal laureates), who worked on the well-posedness of Hamilton–Jacobi equations with Kirchhoff conditions at junctions. Qing Liu, Nageswari Shanmugalingam, and Xiaodan Zhou (University of Cincinnati / OIST, Japan/USA) proved in 2021 the equivalence of various notions of solutions for eikonal equations in metric spaces, drawing on the chain of works that began with Schieborn–Camilli.

A recent preprint (2024) from Chile/Germany by Trí Minh Lê and Sebastián Tapia-García on “On (discounted) global Eikonal equations in metric spaces” continues to work within the framework co-founded by Schieborn–Camilli.

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Relevance beyond nonlinear analysis

The work is relevant not only within the theory of nonlinear partial differential equations. It touches and influences several neighboring fields:

• Traffic flow modeling and transportation science: Traffic on road networks is described by conservation laws that, via a change of variables, become Hamilton–Jacobi equations on networks. The work of Imbert, Monneau, Göttlich, Herty, and others makes direct use of this connection. The computation of travel times, congestion propagation, and optimal traffic control at intersections benefits directly from it.
• Optimal control: The Hamilton–Jacobi equation is the central equation of dynamic programming. Extending it to networks allows the treatment of control problems where an agent moves on a network — for instance in logistics or robotics.
• Graph theory and shortest paths: Schieborn and Camilli showed that their theory contains the classical shortest-path problem on graphs as a special case, while generalizing it: instead of restricting start and destination to nodes, they can lie at any point along the edges, and costs can vary continuously along edges.
• Analysis on fractals: The transfer to the Sierpinski triangle (Camilli, Capitanelli, Marchi) shows that the network approach serves as a bridge to differential equations on fractals — an active area of modern analysis.
• Metric geometry: The Japanese generalizations to general metric spaces and the Italian work on CAT(0) spaces show that the ideas from Schieborn–Camilli reach far beyond the classical setting and have proven fruitful in geometric analysis.
• Numerical mathematics: Several works developed concrete computational schemes (semi-Lagrange methods, Lax–Friedrichs schemes, finite difference schemes) built on the theory. These are directly implementable and relevant to engineering applications.

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Assessment

Schieborn’s dissertation and the paper with Camilli that arose from it were among the first works to systematically transfer the theory of viscosity solutions to topological networks. In parallel and independently, related works emerged (in particular from Imbert and Monneau for junctions, and from Achdou, Camilli, Cutrì, and Tchou for state-constrained problems). Fabio Camilli and Claudio Marchi compared the various solution concepts in a 2013 paper of their own, “A comparison among various notions of viscosity solution for Hamilton–Jacobi equations on networks” (J. Math. Anal. Appl.), establishing their equivalence.

The work was taken up in France, Italy, Germany, Japan, and the United States. It stimulated research in traffic modeling, optimal control, graph theory, analysis on fractals, metric geometry, and numerical mathematics. As recently as 2024, papers appear that build directly on it.

The EU supported related research under the FP7 project SADCO (Sensitivity Analysis for Deterministic Controller Design), in whose publication list the Schieborn–Camilli paper is included.

Dirk Schieborn is today a Professor of Mathematics, Data Analytics, and Machine Learning at Reutlingen University, as well as Managing Partner of a Steinbeis Transfer Center. Fabio Camilli is a Professor at Sapienza Università di Roma.