Convex and quasiconvex functions in metric graphs

Leandro M. Del Pezzo; Nicolás Frevenza; Julio D. Rossi

doi:10.3934/nhm.2021019

Article Contents

2021, Volume 16, Issue 4: 591-607. Doi: 10.3934/nhm.2021019 open access

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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: January 31, 2021

Revised: April 30, 2021

Early access: June 2021

Published: December 2021

L.D.P. and J.D.R. partially supported by CONICET grant PIP GI No 11220150100036CO (Argentina), PICT-2018-03183 (Argentina) and UBACyT grant 20020160100155BA (Argentina). J.D.R. is also supported by MINECO MTM2015-70227-P (Spain)

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Abstract

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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Mathematics Subject Classification: Primary: 05C12, 52A41.

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