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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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Submitted on : Friday, August 25, 2017 - 3:24:57 AM
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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, American Institute of Mathematical Sciences, 2017, 37, pp.6405 - 6435. ⟨10.3934/dcds.2017278⟩. ⟨hal-01073954v3⟩

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