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Networks and Heterogeneous Media

2021, Volume 16, Issue 4: 591-607. doi: 10.3934/nhm.2021019
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Convex and quasiconvex functions in metric graphs

  • 1.
    CONICET and Departamento de Matemática, FCEyN, Universidad de Buenos Aires, Pabellón I, Ciudad Universitaria (1428), Buenos Aires, Argentina
  • 2.
    Departamento de Métodos Matemáticos y Cuantitativos, FCEA, Universidad de la República, Gonzalo Ramírez 1926 (11200), Montevideo, Uruguay
  • Received: 01 February 2021 Revised: 01 May 2021 Published: 28 June 2021

  • Primary: 05C12, 52A41

  • We study convex and quasiconvex functions on a metric graph. Given a set of points in the metric graph, we consider the largest convex function below the prescribed datum. We characterize this largest convex function as the unique largest viscosity subsolution to a simple differential equation, u″=0 on the edges, plus nonlinear transmission conditions at the vertices. We also study the analogous problem for quasiconvex functions and obtain a characterization of the largest quasiconvex function that is below a given datum.

    Citation: Leandro M. Del Pezzo, Nicolás Frevenza, Julio D. Rossi. Convex and quasiconvex functions in metric graphs[J]. Networks and Heterogeneous Media, 2021, 16(4): 591-607. doi: 10.3934/nhm.2021019

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  • We study convex and quasiconvex functions on a metric graph. Given a set of points in the metric graph, we consider the largest convex function below the prescribed datum. We characterize this largest convex function as the unique largest viscosity subsolution to a simple differential equation, u″=0 on the edges, plus nonlinear transmission conditions at the vertices. We also study the analogous problem for quasiconvex functions and obtain a characterization of the largest quasiconvex function that is below a given datum.





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