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

Cyril Imbert; Régis Monneau

doi:10.3934/dcds.2017278

Article Contents

2017, Volume 37, Issue 12: 6405-6435. Doi: 10.3934/dcds.2017278

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

Cyril Imbert^1, and
Régis Monneau^2,

1.
CNRS & Déepartment de Mathématiques et Applications, École Normale Supérieure (Paris), 45 rue d'Ulm, 75005 Paris, France
2.
70, rue du Javelot, 75013 Paris, France

Received: February 28, 2017

Published: October 2017

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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 penalization 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.

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Mathematics Subject Classification: 35F21, 49L25, 35B51.

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Figure 1. A Hamilton-Jacobi equation posed on a multi-dimensional junction. Here there are 3 branches (or sheets — $N=3$ ) and the tangential dimension is $1$ ( $d=1$ ). We did not illustrate the junction condition on the junction hyperplane $\Gamma$ (which is a line in this example)

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Figure 2. Monotone parts $H_i^\pm$ of a Hamiltonian $H_i$ ($H_i^-$ on the left, $H_i^+$ on the right). The Hamiltonian is in black, monotone parts in red. The tangent variable $p'$ is not shown. In this example, the minimum $A_i$ of $H_i$ is lower than $A_0$. The "inverse" functions $\pi_i^\pm$ of $H_i$ are also shown

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