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Journal Articles Discrete and Continuous Dynamical Systems - Series A Year : 2017

Quasi-convex Hamilton-Jacobi equations posed on junctions: The multi-dimensional case

Abstract

A multi-dimensional junction is obtained by identifying the boundaries of a finite number of copies of an Euclidian half-space. The main contribution of this article is the construction of a multidimensional vertex test function G(x, y). First, such a function has to be sufficiently regular to be used as a test function in the viscosity solution theory for quasi-convex Hamilton-Jacobi equations posed on a multi-dimensional junction. Second, its gradients have to satisfy appropriate compatibility conditions in order to replace the usual quadratic pe-nalization function |x − y| 2 in the proof of strong uniqueness (comparison principle) by the celebrated doubling variable technique. This result extends a construction the authors previously achieved in the network setting. In the multi-dimensional setting, the construction is less explicit and more delicate. Mathematical Subject Classification: 35F21, 49L25, 35B51.
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Dates and versions

hal-01073954 , version 1 (10-10-2014)
hal-01073954 , version 2 (06-07-2016)
hal-01073954 , version 3 (25-08-2017)

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Cyril Imbert, R Monneau. Quasi-convex Hamilton-Jacobi equations posed on junctions: The multi-dimensional case. Discrete and Continuous Dynamical Systems - Series A, 2017, 37, pp.6405 - 6435. ⟨10.3934/dcds.2017278⟩. ⟨hal-01073954v3⟩
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